Nonlinear dispersive equations: local and global analysis

Last edited: Dec 5, 2017

Nonlinear dispersive equations: local and global analysis

Terence Tao

CBMS regional conference series in mathematics, July 2006

Softcover, 373 pages. ISBN-10: 0-8218-4143-2, ISBN-13: 978-0-8218-4143-3

These lecture notes try (perhaps ambitiously) to introduce the reader to techniques in analyzing solutions to nonlinear wave, Schrödinger, and KdV equations, in as self-contained a manner as possible. It is a six-chapter book; the first three chapters and an appendix can be found here. It is based on these lectures.

— Errata —

Page xi, bottom: “certain many” should be “certainly many”.
Page xii: Shaunglin should be Shuanglin.
Page xiv: $End(V \to W)$ should be $Hom(V \to W)$ throughout the text (e.g. on pages 33, 34). Frechet should be Fréchet.
Page xv: Frechet should be Fréchet.
Page 1: “is still its infancy” should be “is still in its infancy”.
Page 2: “A Study in Scarlet” should be “A Scandal in Bohemia”.
Page 3: In the first equation, $G(u_0,\ldots,u_k)$ should be $G(u_0,\ldots,u_k,s_0)$ . After (1.4), “the domain ${\mathcal Y}$ ” should be “the range ${\mathcal Y}$ “.
Page 4: In the first paragraph, (6.4) should be 6.4. In the last paragraph, “G is real analytic” should be “F is real analytic”. On line 5, $Y$ should be ${\mathcal Y}$ .
Page 5: “a open interval” should be “an open interval”. $u_t = u^2$ should be $\partial_t u = u^2$ .
Page 6: In the definition of weak solution, $L^\infty$ should be $L^\infty_{loc}$ .
Page 8: In the proof of Theorem 1.4, $C^0(I \to \Omega_\varepsilon)$ should be $C^0(I \to {\mathcal D})$ . Similarly on line 17.
Page 10: In the last line of Exercise 1.1, G should be F.
Page 11: In Exercise 1.4, S(t) should be $S_{t_0}$ . “Kowaleski” should be “Kowalevski”.
Page 12: In Theorem 1.12, $B$ should take values in ${\mathbb R}$ (and the hypothesis that $B$ is non-negative should be dropped.)
Page 13: $|F(u(s))-F(v(s))| \leq M |u(s)-v(s)|$ should be $\|F(u(s))-F(v(s))\|_{\mathcal D} \leq M \|u(s) - v(s) \|_{\mathcal D}$ .
Page 17: In Corollary 1.1, “for all $t \in {\mathbb R}$ should be $n \in {\mathbb Z}$ .
Page 18: In Exercise 1.14, in order for the supplied hint to work, $F, G$ would need to be $C^2_{loc}$ rather than $C^1_{loc}$ . However, the exercise is still true as stated; one needs to apply Gronwall’s inequality in $t$ to expressions such as $\frac{u(s+ds,t) - u(s,t)}{ds} - F(u(s,t))$ for small $dt$ .
Page 19: In Exercise 1.15, $M_n({\mathbb C})$ should be $\hbox{End}(H)$ .
Page 20: In the second part of Exercise 1.19 (“Show that $u$ in fact extends…”), the additional hypothesis “If F is continuously differentiable at 0” is needed, and $\partial_t u(0)$ should be $G(0)/(1-F'(0))$ . “built your castles in the air” should be “built castles in the air”.
Page 22: “be such that such that” should be “be such that”.
Page 25: In Exercise 1.24, the inequality $\langle F(v), dH_j(v)\rangle \geq 0$ should be $\langle F(v), dH_j(v)\rangle > 0$ . At the end of the exercise, add “Give a counterexample to show that the result fails if the strict inequality $\langle F(v), dH_j(v)\rangle > 0$ is weakened to $\langle F(v), dH_j(v)\rangle \geq 0$ “.
Page 27: In the formula for the Poisson bracket {H,E} in Example 1.27, the $p_j$ and $q_j$ should be swapped (or equivalently, the equation is off by a sign).
Page 28: In the definitions of $\frac{\partial H}{\partial z_j}$ and $\frac{\partial H}{\partial \overline{z}_j}$ in Example 1.28, there are factors of 1/2 missing. In the definition of the symplectic form (both (1.31) and the following equation), there is a negative sign missing.
Page 29: In (1.33), there should be a minus sign on the RHS. Just before (1.34), $H(u(t_0))$ should be $H(u(t))$ .
Page 30: In (1.35), the $m_i$ should be on the denominator.
Page 31: In Exercise 1.27, add the hypothesis that J is skew-adjoint. Also, $\nabla_\omega H = - J \nabla H$ should be $\nabla_\omega H = J \nabla H$ .
Page 32: In the 10th line from the bottom, Louville should be Liouville.
Page 33: In Exercise 1.37, $G : C^0_{loc} \ldots$ should be $G \in C^0_{loc}\ldots$ .
Page 34: In Exercise 1.41, “exists real numbers” should be “exist real numbers”, and $z_d$ should be $z_n$ .
Page 36: in Example 1.31, $C \geq 2$ should be $0 < C \leq 2$ .
Page 40: In the ODE in Exercise 1.48, there is a unit vector $\frac{x_j(t)-x_i(t)}{|x_j(t)-x_i(t)|}$ missing in the right-hand side.
Page 41: In (1.42), $N(\|u\|_D)$ should be $\|N(u)\|_D$ .
Page 46: In the definition of ${\mathcal N}$ , the word “then” after $F \in C^0([0,+\infty) \to {\mathcal D})$ should be “whose norm”, and $e^{\sigma t}$ should be $e^{2\sigma t}$ .
Page 47: After the fourth display, “ ${\mathcal N}$ is bounded” should be “ $N_k$ is bounded”.
Page 48: In Exercise 1.51, $\varepsilon$ should equal $1 / (2k C_0 C_1)^{1/(k-1)}$ rather than $1/(2kC_0C_1)$ .
Page 53: In Exercise 1.56, “commute with a given Hamiltonian” should be “commute with each other”. “Torii” should be “tori” (two occurrences). In Exercise 1.58, “uppose that” should be “Suppose that”. In Exercise 1.57, $P(u(t))$ should be $-P(u(t))$ .
Page 54: In Exercise 1.59, “Exercise 1.27” should be “Example 1.27”. In the last line (above the footnotes), ${\Bbb Z}^n$ should be ${\Bbb Z}^d$ .
Page 57: In the first line, $E, B: \mathbf{R}^{1+3} \times \mathbf{R}^3$ should be $E, B: \mathbf{R}^{1+3} \to \mathbf{R}^3$ . After equation (2.6), in the formula for $dg^2$ the space index should run from 1 to d rather than from 1 to 3.
Page 58: In the “Conversely” portion of Exercise 2.2, one must assume the Lorenz gauge condition $\partial^\alpha A_\alpha = 0$ .
Page 59: In the first display of Exercise 2.3, $u$ should be $\overline{u}$ . Exercise 2.4 the second line should be $L:= ih(D)$ . For the Schrödinger equation in Exercise 2.4, the phase velocity is half the group velocity rather than twice the group velocity (i.e. $v/2$ instead of $2v$ ). In Exercise 2.5, in the second line the range of $\tilde u$ is V rather than $\mathbf{C}$ . Same for Exercise 2.6, and 2.10. In the display of Exercise 2.5, the term $e^{i m t |v|^2/ 2\hbar}$ should be $e^{-i m t |v|^2/ 2\hbar}$ .
Page 60: In the last display, $x_n$ should be $x_d$ .
Page 61: In exercise 2.12, the hypothesis that $u$ is radial should be added. In the second display of Exercise 2.14, the exponent $- \frac{d-1}{2}-1$ should be $- \frac{d-1}{2}-2$ .
Page 62: In the second paragraph of Section 2.1, $P:{\mathbb R}^d \to {\mathbb R}$ should be $h: {\mathbb R}^d \to {\mathbb R}$ .
Page 63: In the 8th line from the bottom, “propagator” should be “propagators”, and there is a semicolon missing in the preceding display.
Page 64: In the definition of the spacetime Fourier transform, $dt dx$ should be $dx dt$ . Similarly, in the inversion formula, $d\tau d\xi$ should be $d\xi d\tau$ .
Page 65: After Principle 2.1, $\hbar \xi/v$ should be $\hbar \xi/m$ . In the last paragraph, “thi principle” should be “this principle”. 5th line from top, “to the solution” should be “on the solution”.
Page 66: In Exercise 2.18, $\delta(\tau-|\xi|)$ should be $\delta(|\tau|-|\xi|)$ . In the second to last display, the closing right parenthesis should be deleted.
Page 67: In Exercise 2.19, the normalisation $\hbar = 1$ is missing. In the two-sheeted hyperboloid, $|\xi|2$ should be $|\xi|^2$ .
Page 67, bottom: “forall” should be “for all”.
Page 70: In the second display, $x/t^{-1/3}$ should be $x/t^{1/3}$ .
Page 71: Two lines before (2.19), $\lambda^n$ should be $\lambda^d$ . In the first display, $|t|^{n-1}$ should be $|t|^{d-1}$ .

Page 72: In Exercise 2.28, the Laplacian $\Delta$ in the third display should be $\frac{\Delta}{2}$ , and $v(0,x)$ should equal $\frac{1}{(2\pi)^d/2} \bar{\hat{u_0}}(x)$ rather than $\frac{1}{(2\pi)^d/2} \hat{u_0}(x)$ ; also, “psedoconformal” should be “pseudoconformal”. For the extra challenge, one needs to use separation of variables and consider solutions to Schrödinger of the form $u(t,x) = u(x) e^{iEt}$ for some $E$ (and some rescaling of the wave-Schrödinger correspondence may also be necessary). In Exercise 2.30, “Airy function” should be “Airy equation”.
Page 73: In Exercise 2.33, $x_n$ should be $x_d$ .
Page 74: In (2.26), $ds$ should be $dt'$ . In the discussion after Theorem 2.3, it should be noted that the estimates of Strichartz are based on the earlier restriction theorems obtained by Stein (unpublished, 1968, though mentioned in the thesis of Charles Fefferman) and Tomas (in the cited reference [Tomas]), and in particular on a subsequent unpublished interpolation argument of Stein that leads to what is now known as the Tomas-Stein restriction theorem (and which is discussed for instance in Stein’s book Harmonic analysis, or in Stein’s Beijing lecture notes). Marcinkeiwicz should be Marcinkiewicz. In the second paragraph after (2.23), “than on the left” should be “than on the right”.
Page 75: In the proof of Theorem 2.3, $q,q' \neq 2$ should be $q,r,\tilde q, \tilde r \neq 2$ .
Page 76: In Figure 1, the role of $1/r$ and $1/q$ should be interchanged. “Applying Holder’s inequality” should be “Applying Holder’s inequality twice”.
Page 77: On the fifth line, add “(after replacing $q,r$ with $\tilde q, \tilde r$ )” after “which is (2.25)”. In the second display, $e^{-is\Delta}$ should be $e^{i(t-s)\Delta}$ . After invoking Christ-Kiselev, add the parenthetical “(strictly speaking, this lemma does not apply directly because $e^{i(t-s)\Delta}$ need not be bounded from $L^r$ to $L^{\tilde r'}$ , but this technicality can be dealt with by a standard regularisation argument, e.g. replacing $e^{i(t-s)\Delta}$ with $P_{\leq N} e^{i(t-s)\Delta}$ , applying Christ-Kiselev, and then taking the limit $N \to \infty$ .)”.
Page 78: In Figure 2, the role of $1/r$ and $1/q$ should be interchanged.
Page 80: In Exercise 2.35, “(2.34)” should be “Exercise 2.34”. “for all $u_0$ ” should be “holds for all $u_0$ “. In Exercise 2.3.7, “ $=0$ ” needs to be appended to $\lim_{t \to \pm \infty} \|u(t) \|_{L^p_x}$ .
Page 81: In Exercise 2.43, the space-time domain “ $|t|\ll 1/\varepsilon, |x_1-t|\ll \varepsilon$ and $|x_2|, \cdots,|x_n|\ll 1$ ” should be “ $|t|\ll 1/\varepsilon^2, |x_1+t|\ll 1$ and $|x_2|, \cdots, |x_n|\ll 1/\varepsilon$ “.
Page 81-82: In Exercise 2.46, the hypothesis $r \leq \infty$ should be replaced with $r < \infty$ (and so the claim is not quite true for all Schrödinger-admissible exponents). Also, to use complex interpolation to prove this estimate requires the theory of BMO (and the Fefferman-Stein interpolation theorem); it is easier to use the Littlewood-Paley inequality (A.7) instead.
Page 83: two lines above (2.33), “transation” should be “translation”.
Page 84: In the display after (2.35), the minus sign should be deleted. Three lines above (2.36), “multiplying first equation” should be “multiplying the first equation”. On the 8th line from bottom, delete the second “the useful identity”.
Page 85: Before (2.40), $4 \pi$ should be $8 \pi$ . In (2.40), $4\pi$ should be $2\pi$ .
Page 87: In Exercise 2.52, add “to” after $H^{k,k}_x({\mathbb R}^d)$ . At the end of Exercise 2.54, “in homogeneous” should be “inhomogeneous”.
Page 92: In the equation just below (2.54), $T^{00}(t,x)$ should be $T^{00}(0,x)$ . In (2.54), $u(t.x)$ should be $u(t,x)$ .
Page 94: In the first display, $\int^{S_{d-1}}$ should be $\int_{S^{d-1}}$ . In the second and third display, $t^{-1-d}$ should be $t^{1-d}$ .
Page 99: in the definition of $X^{s,b}$ norm with the torus as spatial domain around the middle of the page the $\xi$ should be replaced by k. In the formula following it $\xi$ should be replaced by x. In the last line of Lemma 2.8, $\eta \in {\mathcal S}_x({\mathbf R})$ should be $\eta \in {\mathcal S}_t({\mathbf R})$ .
Page 100: In the first line, “ $s'\le s$ and $b'\le b$ ” should be “ $s'\ge s$ and $b'\ge b$ “. In the penultimate display, $f_\tau$ should be $f_{\tau_0}$ .
Page 101: In the last line of Lemma 2.11, the condition $\sigma > 0$ may be deleted. In the penultimate display, $\tau - \tau_0 - h(\xi)$ should be $\tau + \tau_0 - h(\xi)$ .
Page 102: The case $b'=b$ in the proof of Lemma 2.11 is not as trivial as claimed. However, once the $b'=0$ case is proven, the $b'=b$ case can then be deduced as follows. Observe that the $b'=0$ bound suffices to control the portion of $\| \eta(t/T) u\|_{X^{s,b}}$ for which $\langle \tau-h(\xi) \rangle \leq 1/T$ , so it suffices to control $\| P( \eta(t/T) u) \|_{X^{s,b}}$ , where P is the Fourier projection to the region $\langle \tau-h(\xi) \rangle \geq 1/T$ . We split this into $\| P(\eta(t/T) Pu)\|_{X^{s,b}}$ and $\| P(\eta(t/T) (1-P)u)\|_{X^{s,b}}$ . For the former term, we can observe that $\| P (e^{it\tau_0} P u) \|_{X^{s,b}} \lesssim_b \langle T \tau_0 \rangle^b \|u\|_{X^{s,b}}$ for any frequency $\tau_0$ (improving the bound in the proof of the first estimate), and then by repeating the proof of the first estimate one obtains an acceptable estimate for this term. As for the final term $\| P(\eta(t/T) (1-P)u)\|_{X^{s,b}}$ , we bound this by $T^{1-b} \| (\partial_t -L) (\eta(t/T) (1-P) u) \|_{X^{s,0}}$ . By the Leibniz rule, the expression inside the norm splits into $\eta(t/T) (\partial_t -L) (1-P) u$ and $T^{-1} \eta'(t/T) (1-P) u$ . The first term contributes at most $\lesssim T^{1-b}\|(\partial_t -L) (1-P) u\|_{X^{s,0}} \lesssim \|u\|_{X^{s,b}}$ , while from the b’=0 theory the second term contributes at most $T^{1-b} T^{-1} T^b \| (1-P) u \|_{X^{s,b}}$ , and both terms are acceptable. Finally, the composition argument to treat the $b' \leq 0 \leq b$ case may be elaborated as follows. Firstly, by a smooth partition of unity it suffices to establish the claim for smooth compactly supported $\eta$ (as long as the bounds depend only on the width of the support and on a $C^k$ norm for finite $k$ ). It is then easy to factorise $\eta = \eta_1 \eta_2$ where $\eta_1,\eta_2 \in C^\infty_c$ obey similar bounds to $\eta$ . Now one can compose easily.
Page 102: In the last line of fourth display, the $X^{s,b}$ norm should be $X^{0,b}$ . In the fourth to last display, $m(\xi) f(\xi)$ should be $m(\xi) \hat f(\xi)$ .
Page 103: In the 9th last line, $\tau-\xi$ should be $\tau-h(\xi)$ . In the third-to-last display, the $X^{0,b}$ norm of F should be $X^{0,b-1}$ . In the last display, the plus sign should be a minus sign.
Page 104: In the fourth display, the right-hand side should be $( \int_{\mathbb R} \langle \tau-h(\xi) \rangle^{2(b-1)} |\tilde F(\tau,\xi)|^2\ d\tau)^{1/2}$ . In the third line of the proof of Lemma 2.13, $2^M$ and $2^{M+1}$ should be $M$ and $2M$ respectively (and $M$ should range over powers of two, rather than integer powers of two), and the display after this is missing a final period.
Page 105: In the fourth display, $(3/4 -2 \varepsilon m)$ should be $(3/4 -2 \varepsilon ) m$ . In the first line after the fifth display, $2\tau-k$ should be $2\tau-k^2$ . Moreover, in the display of Exercise 2.70, one should interchange the role of u and v.
Page 106: In Exercise 2.75, the hypothesis $b > 1/2$ is missing. In Exercise 2.74, $C^0_t L^4_x$ should be $C^0_t L^2_x$ , and all occurrences of $latex{\mathbb T}^2$ should be ${\mathbb T}$ .
Page 107: In the second display of Exercise 2.77, the $L^2_t L^2_x$ norm should be an $L^6_t L^6_x$ norm. In Exercise 2.78, “Periodic Airy $L^6$ estimate, II” should be “Periodic Schrödinger $L^6_{t,x}$ estimate”.
Page 109: “defocusing, absent, or focusing” should be “focusing, absent, or defocusing”.
Page 110: In the second paragraph, $F(zu)$ should equal $|z|^{p-1} z F(u)$ rather than $|z|^p F(u)$ .
Page 112: In the second paragraph, “the Laplacian $\xi$ ” should be “the Laplacian $\Delta$ “, and “in order to solve the NLS” should be “in order for $u$ to solve the NLS”. After (3.5), ${\Bbb Z}^d$ should be ${\Bbb R}^d$ . In (3.5), the expression of u should be $u=\alpha e^{i \xi x}e^{-i |\xi|^2 t/2} e^{-i \mu |\alpha|^{p-1} t}$ . In the text after equation (3.5), anticlockwise should be clockwise, and “compared the frequency” should be “compared to the frequency”.
Page 113: Before (3.6), $\mu=+1$ should be $\mu=-1$ . After (3.6), $\mu=-1,0$ should be $\mu=+1,0$ . After (3.7), $\omega \in {\Bbb R}$ should be $\tau \in {\Bbb R}$ . In (3.8), $|Q|^p$ should be $|Q|^{p-1}$ . After (3.8), “defocusing” should be “focusing”. The discussion for NLW is inaccurate (the sign of $\beta$ is unfavorable) and all references for NLW ground states should be deleted. (There is a ground state for critical NLW, or for NLKG, but it would be rather complicated to discuss those cases here.) Before Exercise 3.1, “In Section 3.5” should be “in Section 3.5”.
Page 114: In (3.10), $e^{it|v|^2/2}$ should be $e^{-it|v|^2/2}$ .
Page 116: In (3.15), $e^{- i t/\tau}$ should be $e^{- i \tau/t}$ . In(3.16), $\Delta$ should be $\frac{\Delta}{2}$ . In the formula before (3.18), “ $- i |\alpha|^{p-1}$ ” should be “ $- i \mu |\alpha|^{p-1}$ “. In (3.19), “ $+i \mu |\alpha|^{p-1}$ ” should be “ $-i \mu |\alpha|^{p-1}$ “.
Page 117: In (3.20) and the following equation, $e^{i|\xi|^2 t/2}$ and $e^{i\mu|\alpha|^{p-1} t}$ should be $e^{-i|\xi|^2 t/2}$ and $e^{-i\mu|\alpha|^{p-1} t}$ .
Page 119: In the end of the first main paragraph, “if Principle 3.1” should be “of Principle 3.1”.
Page 120: In Exercise 3.4, the exponents for the predicted time $T$ should have a minus sign. In Exercise 3.5, $\mu=+1$ should be $\mu=-1$ , and “focusing regularity” should be “focusing nonlinearity”.
Page 122: In the first paragraph, “show existence of solution” should be “show existence of a solution”
Page 123: In the proof of Proposition 3.2, Theorem 1.10 is not strictly applicable because $\|v(t)\|_{L^2_x({\mathbb R}^d)}$ need not be continuous. However, using the Lebesgue differentiation theorem one may extend the proof of Theorem 1.10 to the case when the function is bounded measurable rather than continuous.
Page 124: the second line after the proof of Proposition 3.3, “one and nonlinearities” should be “and nonlinearities one”. In (3.22), the final semicolon should be deleted. In the penultimate line, the intersection symbol $\cap$ should be a subset symbol $\subset$ . After (3.23), add “with some polynomial growth bound on the $L^p_t L^p_x$ norm on balls $B(0,R)$ .”
Page 125: In the second line of Definition 3.4, “ $u_0^*\in {\mathbb R}^d$ “should be “ $u_0^*\in H^s_x ({\mathbb R}^d)$ “. Also, “with the $C^0_t H^s_x([-T,T] \times {\mathbb R}^d)$ ” should be “with the $C^0_t H^s_x([-T,T] \times {\mathbb R}^d)$ topology”.
Page 129: In the second-to-last line of the main text, “in one usually needs” should just be “one usually needs”.
Page 130: In the second-to-last sentence of footnote 18, “controlled in” should just be “controlled”. In the third paragraph, “are locally bounded” should be “is locally bounded”. In the first paragraph, the final left parenthesis should be replaced with a semicolon.
Page 131: “Banach space algebra” should be “Banach algebra”. On the last line of the main text, the right-parenthesis after $u_0$ should be omitted.
Page 132: In the fourth and fifth lines, $u$ should be $u^*$ . In the second paragraph after Remark 3.10, add “norm” before “stays bounded”. In (3.25), the exponent $p$ should instead be $p-1$ .
Page 133: In Remark 3.12, the phrase “by Sobolev embedding” should be placed in parentheses and moved to before “and hence in”.
Page 134: In Remark 3.14, “a critical controlling norms” should be “a critical controlling norm”.
Page 135: In Proposition 3.15, $T$ does not depend on $k$ . In (3.26), $\|u(t_0)\|_{L^2_x (I\times R^d)}$ should be $\|u(t_0)\|_{L^2_x (R^d)}$ . Two lines above (3.26), Proposition 2.3 should be Theorem 2.3.
Page 136: “ $1 < p < 1+4/n$ ” should be “ $1 < p < 1 + 4/d$ ”. “ $q<r$ ” should be “ $r<q$ ” (two occurrences), and “ $p/q > 1/q'$ ” should be “ $p/q < 1/q'$ ”.
Page 137: In the formula of Proposition 3.17, $2(n+2)/n$ should be $2(d+2)/d$ . The final parenthetical comment in Proposition 3.17 should be deleted.
Page 138: In (3.28), the $H^1$ norm should be on ${\Bbb R}^d$ , not on $I \times {\Bbb R}^d$ .
Page 139: In the second to last display in the proof of Proposition 3.19, the exponent $5/2p$ should be $5(p-1)/2$ .
Page 140: In Figure 5, $H^1$ should be $\dot H^1$ in both appearances in the caption.
Page 141: In the formula of Exercise 3.16, the $t_0$ in the LHS should be $t$ .
Page 142: In Exercise 3.18, “n” should be “d” throughout (for consistency with the rest of the text).
Page 144: In the line before the first formula, “by by” should be “by”.
Page 145: In Proposition 3.23, “some time interval” should be “the time interval”.
Page 146: In the proof of Proposition 3.23, Proposition 3.23 should be Proposition 3.22. In the first line of the proof, “we” should be capitalised.
Page 147: A period is missing after Footnote 28.
Page 148: second paragraph after Principle 2.34, last line “n>6” should be “d>6”. “Proposition 3.19” should be “(the two-dimensional analogue of) Proposition 3.19”.
Page 150: “subcritical” should be “sub-critical”
Page 151: $H^1$ should be $H^1_x$ . In the formula of Exercise 3.31, the term $\partial_j (\frac{1}{2} Im( \overline{\partial_{jk} u(t,x)} \partial_k u(t,x))$ should be $\partial_j (\frac{1}{2} Im( \overline{\partial_{kk} u(t,x)} \partial_j u(t,x))$ .
Page 152: In exercise 3.35, the first appearance of “defocusing” should be omitted.
Page 153: In the formula of Exercise 3.39, the $H^{k-1}$ norm shouldbe taken for $\partial_t u(t)$ but not $u(t)$ .
Page 154, fourth to last line: $\mu=-1$ should be $\mu=+1$ .
Page 155: In the paragraph before (3.36), “Morawetz inequalities for the NLS and NLW” should be “Morawetz inequalities for the Schrödinger and wave equations”.
Page 156: After (3.37), $\Delta^2 a$ should be $-\Delta^2 a$ . In (3.38), an integration in $dt$ is missing. In (3.37), there should be a (d-1) in front of the $\frac{2(p-1)\mu}{p+1}$ , and similarly for (3.40) and (3.41).
Page 157: In (3.40) and (3.41), $\pi$ should be $2\pi$ . In the penultimate display $1 - \frac{(x_j-y_j)(x_k-y_k)}{|x-y|^2}$ should be $\delta_{jk} - \frac{(x_j-y_j)(x_k-y_k)}{|x-y|^2}$ .
Page 158: In the first display, $\pi$ should be $2\pi$ .
Page 159: In the first display, the first bracket should not be subscripted. In (3.45), an integration in $dx$ is missing. In the second formula of this page, $\partial_t E[v(t),t]=\frac{d}{2} (p-p_{L^2_x})$ should be $\partial_t E[v(t),t]=d (p-p_{L^2_x})$ . In the last formula of this page, the $L^{2(d+2)/2}$ norm should be a $L^{2(d+2)/d}$ norm.
Page 160: After the first formula of this page, $H^1_x$ -criticalshould be $H^{1/2}_x$ -critical. In the third formula of this page, the minus sign should not occur.
Page 161: In Exercise 3.46, the coefficient $+ \frac{p \delta_{jk}}{2(p+1)}$ in the first display should be $- \frac{1}{p+1}$ , and the coefficient $\frac{p}{p+1}$ in the second display should be $\frac{2}{p+1}$ .
Page 162: In line 4 and 7, $p_d$ should be $p_{L^2_x}$ .
Page 166: $L^q_x$ should be $L^q_{t,x}$ ; similarly on (3.51) in page 167.
Page 167: In the third display, $W^{1,10/3}$ should be $W^{1,10/3}_x$ . Near the end of the proof, “yields” should be “yield”. After the display following the proof, “energy give” should be “energy gives”. In the sixth display, the final term should be $\varepsilon^{2\alpha} \|u\|_{S^1(I \times {\mathbb R}^3)}^{3-2\alpha}$ .
Page 168: In the second formula of this page, the denominator shouldbe 2d rather than 4d. In the statement and proof of Proposition 3.32, $R^3$ should be $R^2$ (three occurrences). “pseudoconformal decay laws” should be “pseudoconformal decay law”. In Proposition 3.32, “norm of $u_0$ ” should be “norm of $u$ “.
Page 169: In the second line after the last formula of this page,Exercise 3.35 should be Proposition 3.25. From the last 6 lines onwards,all occurrences of 1/T should be T.
Page 170: In Remark 3.3, “(still open)” should be “(still unproven)” (although this result has in fact been proven by Dodson after the publication of this book).
Page 171: After (3.52), “small some suitable norms” should be “small in some suitable norms”.
Page 173: In (3.55), (3.56) and the second line before (3.55), four occurrences of the exponent 2 should be p-1. Before (3.56), “This equation just” should be “This equation is just”.
Page 174: In the first paragraph, (3.55) should be (3.56). In the second and third displays, the last term $\partial_{xx} \tilde{v}$ should be $\frac{\partial_{xx} \tilde{v}}{2}$ . In the third display, a $e^{i(t-t')\partial_{xx}/2}$ is missing after the integral sign, and a -i should be present before the integral. In (3.57) and the previous formula, $\varepsilon^2|\psi|^2$ should be $\varepsilon^{p-1}|\psi|^{p-1}$ . Moreover, in (3.57), $t^{(p-3)/2}$ should be $t^{-(p-3)/2}$ . In line -7, “long-range case p>3” should be “long-range case p<3”. In the last paragraph, “that the short-range case” should be “that in the short-range case”.
Page 175: In the proof of Proposition 3.35, $\varepsilon^2$ should be $\varepsilon^2/2$ (two occurrences). In the fifth display, “ $\omega(t)=\int_0^t$ ” should be “ $\omega(t)=- i\int_0^t$ “. A period is missing after Footnote 42. Also, at the beginning of the proof of Proposition 3.35, observe that one can assume without loss of generality that $\varepsilon$ is sufficiently small depending on $\psi$ , because the case when $\varepsilon$ is smaller than (say) 1/2 can then be deduced from this case by a scaling argument.
Page 176, first line, “sufficiently small depending on t” should be “sufficiently small depending on $\psi$ “.
Page 178: In the 9th line of the third paragraph, $e^{it \tau+\theta(t)}$ should be $e^{i (t \tau+\theta(t))}$ .
Page 179: In the second display, $u^{(\varepsilon)}$ should be $u$ . In Exercise 3.56, the “ $\max (|u^{(\lambda)}|^{p-1}, \lambda |u^{(\lambda)}|^{4}) u^{(\lambda)}$ ” in the first display and “ $\max (\omega^{p}, \lambda \omega^5)$ ” after the second display should be” $\min (|u^{(\lambda)}|^{p-1}, \lambda^4) u^(\lambda)$ ” in the firstdisplay and “ $\max (\omega^{p}, \lambda^4 \omega^2)$ “, respectively.
Page 180: In the third line, $+ \frac{1}{2} |k|^2 t$ should be $-\frac{1}{2} |k|^2 t$ . The definition of ${\mathcal N}_t$ needs a prefactor of $-i$ , and in the exponent $\frac{i}{2}$ should be $-\frac{i}{2}$ . In the final display, a right-parenthesis is missing in the norm for $b$ , and the first integral sign in that display should be removed.
Page 182: In (3.72), $\mu$ should be $2\mu$ . After (3.72), “ $p_{L^2_x}:=1+\frac{4}{2}$ ” should be “ $p_{L^2_x}:=1+\frac{4}{d}$ “. In the second paragraph, the critical index $\sqrt{2}$ for focusing NLW should be $1+\sqrt{2}$ .
Page 183: After (3.73), Exercise 3.38 should be Exercise 3.35 and Exercise 3.39.
Page 184: Before the first display, $|v|^2 \omega_\varepsilon$ should be $|\omega_\varepsilon|^2 \omega_\varepsilon$ . In the last display, one should replace “p” by “3”.
Page 186: In the quote, “Law” should not be capitalised.
Page 189: After (3.74), “wellposednes” should be “wellposedness”.
Page 190: In the penultimate display, the slash should be a period.
Page 191: In the fourth display, $m(\xi-\eta N)$ should be $m(\frac{\xi-\eta}{ N})$ . In the second display, a right parenthesis is missing inside the norm.
Page 192: In Proposition 3.39, $t\ll_s N^{\frac{1}{2}-2(1-s)}$ should be $t\ll_s N^{\frac{1}{2}-\frac{2(1-s)}{s}}$ . s>3/4 should be replaced by s>4/5, and the first display should be replaced by $\|u(t)\|_{H^s_x}\lesssim_s \langle t \rangle^{(1-s) / (\frac{1}{2}-\frac{2(1-s)}{s})}$ .
Page 198, top: the reflection symmetry claimed for the KdV equation is incorrect and should be deleted.
Page 199: In (4.7), $-5u^4$ should be $+5u^4$ . In the bottom middle box, a right-parenthesis is missing.
Page 200: In Exercise 4.2, $P(t)$ should be $4D^3 - 3(Du(t) + u(t) D)$ , and $P(t) f$ should be $4 \partial_{xxx} f - 3 (\partial_x (u(t) f) + u(t) \partial_x f)$ .
Page 206: In (4.13), $L^2_t L^\infty_x$ should be $L^2_x L^\infty_t$ . In (4.14), $L^4_t L^\infty_x$ should be $L^4_x L^\infty_t$ .
Page 208: Superfluous ) parenthesis on (4.18) and on the preceding equation, as well as the display two equations down.
Page 235: In the definition of the local energy $E_\Omega[u[t_0]]$ , all occurrences of $t$ should be $t_0$ .
Page 236: In (5.5), the limit superior should be to $T_*$ rather than $T_*^+$ .
Page 238: In the last line of Proposition 5.6, insert “is the linear solution” before “with initial data”.
Page 240: The application of Proposition 5.1 in the third display is not correct, as it neglects the linear term. The fix is a little complicated: adding the linear term adds a 1 to the RHS, which prevents a direct continuity argument from working. But one can use a wider range of Strichartz estimates than provided by Proposition 5.1 to place the LHS in, say, $L^3_t L^{18}_x$ norm rather than $L^4_t L^{12}_x$ norm. Interpolating back with the $L^6$ hypothesis one recovers an estimate which is amenable to a continuity argument (with $\epsilon_2$ replaced by a slightly smaller power of $\epsilon_2$ ).
Page 247: In the third line of Theorem 5.1, $H^1 \times L^2$ should be $H^1$ .
Page 249: In the fourth display, $\eta^2$ should be $\eta^4$ .
Page 254: In the sixth to last line, “unexceptional” should be “exceptional”.
Page 261: In the last paragraph above the exercises, $J - O_{E,\eta}(1)$ should be $\gtrsim_{E,\eta} J$ .
Page 275: In the first line after the display in Exercise 5.21, “ $N(t)=1$ ” should be “ $N(0)=1$ “.
Page 280: In (6.3), u should be $\phi$ (two occurrences). In equation (6.5), the $\frac{1}{2}$ should be outside the integral.
Page 281: In the display after (6.7), a factor $\frac{1}{2}$ is missing from the right-hand side.
Page 283: In Exercise 6.2(iii), one of the superscripts $\alpha$ should instead be a subscript.
Page 285: In Exercise 6.6, the $Y \nabla_Y X$ term in the zero torsion property should just be $\nabla_Y X$ .
Page 287: In the last display of Exercise 6.13, $\psi$ should be $\Psi$ .
Page 302: In (6.35), $\tilde \phi_\phi$ should be $\tilde \phi+\phi$ . In (6.36), $\partial_\beta A$ should be $\partial_\beta A_\alpha + [A_\alpha,A_\beta]$ .
Page 334: In (A.7), the condition “for $1 < p < \infty$ ” should be added.
Page 339, second display: $++$ should be $+$ . In the right-hand side of the fifth display, $N^s M^{-s}$ should be $N^{2s} M^{-2s}$ , and $L^2$ should be $H^s$ . (The latter correction should also apply to the second line of the fourth display.)
Page 340, equation (A.20): $N^{-2k}$ should be $N^{-k}$ . In the last display, the $L^\infty_x$ norm should be $L^2_x$ .
Page 341, last display in proof of Lemma A.9: The $L^2$ norm on the LHS should be squared, and the $(N')^k N^{-k}$ term should be $(N')^{2k-\varepsilon} N^{-2k+\varepsilon}$ , where $\varepsilon > 0$ is arbitrary (and the implied constant now depends of course on $\varepsilon$ . When we sum in N, we have to assume $\varepsilon$ sufficiently small depending on k and s.
Page 343, Exercise A.8: In the endpoint Sobolev inequality, both instances of the exponent $d$ should be replaced by $d/(d-1)$ . (Also, $d$ needs to be strictly greater than 1.) In Exercise A.12, there is a term missing on the right-hand side, and the correct bound is $\|F(f)-F(g)\|_{H^s_x({\mathbb R}^d)} \leq O_{\|f\|_{L^\infty_x({\mathbb R}^d)},\|g\|_{L^\infty_x({\mathbb R}^d)},F,V,s,d}(\|f-g\|_{H^s_x({\mathbb R}^d)})$ $+ O_{\|f\|_{H^s_x({\mathbb R}^d)},\|g\|_{H^s_x({\mathbb R}^d)}, \|f\|_{L^\infty_x({\mathbb R}^d)}, \|g\|_{L^\infty_x({\mathbb R}^d)},F,V,s,d}(\|f-g\|_{L^\infty_x({\mathbb R}^d)})$ .
Page 344, Exercise A.18: The hypothesis that $u$ is spherically symmetric is missing.
Page 347: The quote by Antoine de Saint-Exupery is slightly inaccurate; the correct quote is “la perfection soit atteinte non quand il n’y a plus rien à ajouter, mais quand il n’y a plus rien à retrancher.“. In the third paragraph, “model example of positive solution” should be “model example of a positive solution”. In the last line, $\alpha$ should equal $\frac{2}{p-1}(\frac{-2p}{p-1}+d)$ rather than $\frac{2}{p-1}(\frac{2}{p-1}+d)$ .
Page 348: Before (B.3): “a positive and finite” should be “positive and finite”. In second paragraph: closing parenthesis before “we conclude that”. In Lemma B.1, one can remark that the hypothesis $Q \not \equiv 0$ is redundant since $W_{max}$ is known to be positive. The formula for $\alpha$ should be $\frac{2(p+1)}{d(p-1)} \|Q\|_{L^{p+1}}^{-p-1} \| \nabla Q \|_{L^2}^2$ .
Page 349: In Lemma B.2: $-|\nabla Q| \leq \nabla |Q| \leq |\nabla Q|$ should be $|\nabla |Q|| \leq |\nabla Q|$ , with a similar modification within the proof of that lemma. In the proof of Lemma B.1, there is a factor of $\int |Q|^{p+1}$ missing in the second and third terms of the right-hand side of the first display. “Q is maximiser of W” should read “Q is a maximiser of W”. In the proof of Lemma B.3, add the following clarification in the second sentence: “(since $P_N u_n(x)$ is the inner product of $u_n$ against a Schwartz function for any fixed $x,N$ )”.
Page 351: In the second line from the top, “On the other hand” should be “On the one hand”. In the last line of the proof of Lemma B.4, W(u) should be W(Q). In Theorem B.5, the hypothesis that u is non-zero may be omitted (since $W_{max}$ is strictly positive).
Page 352: In the second display, $1/p+1$ should be $1/(p+1)$ for clarity.
Page 353, Proposition B.7: “Let Q be non-negative solution” should be “Let Q be a non-negative solution”.
Page 354, Proposition B.8: “Let Q be non-negative solution” should be “Let Q be a non-negative solution”. All occurrences of $t\xi - x$ should be replaced with $t\xi + \overline{x}$ , where $\overline{x}$ denotes the reflection of $x$ across the plane $\{ x \cdot \xi = 0 \}$ . Similarly for $t \xi - x_t$ .
Page 359: In Exercise B.2, $O_k$ should be $O_{k,d}$ , and the condition $|x| \geq 1$ should be added.
Page 360: In the hint for Exercise B.3, $Q$ and $x \cdot \nabla_x Q$ should be $\overline{Q}$ and $x \cdot \nabla_x \overline{Q}$ .
Page 362: A right parenthesis is missing at the end of Exercise B.13. In the end of Exercise B.14, the parentheses around B.13 should be removed.
Page 365: In reference [CS], “disperives” should be “dispersives”.

Many thanks to Adam Azzam, Jordan Bell, Sebastien Breteaux, Bjorn Bringmann, Cattle, James Fennell, Eric Foxall, Danny Goodman, Zaher Hani, Khang Huynh, itaibn, Rowan Killip, Soonsik Kwon, Liu Xiao Chuan, Georg Meyer, Jason Murphy, Timothy Nguyen, Guilio Pasqualetti, Guillermo Reyley, Tristan Roy, Shuanglin Shao, Paul Smith, Elias Stein, Monica Visan, Haokun Xu, Chengbo Wang, Fan Zheng, Shijun Zheng, and Zuchong Zhi for corrections!

159 comments

Comments feed for this article

25 August, 2008 at 5:10 pm

Tricks Wiki article: The tensor product trick « What’s new

[…] Remark. A similar trick allows us to deduce “interaction” or “many-particle” Morawetz estimates for the Schrödinger equation from their more traditional “single-particle” counterparts; see for instance Chapter 3.5 of my book. […]

13 November, 2008 at 5:34 am

liuxiaochuan

Dear Professor:
Here are two corrections.

1, In page 28, I think on the formula for $\nabla_\omega H$ in Example 1.28, the original 2i is correct, instead of -2i.

2, In page 29, just before (1.34), I think it should be $H(u(t))=H(u_0)$ there.

13 November, 2008 at 11:31 am

Terence Tao

Thanks for the corrections!

10 December, 2008 at 10:54 pm

Anonymous

For what it is worth, this text might greatly benefit from an index.

28 February, 2009 at 12:13 am

Tricks Wiki: Give yourself an epsilon of room « What’s new

[…] We will sketch (omitting several technical details, which can be found for instance in my PDE book) a very typical instance. Consider a nonlinear PDE, e.g. the nonlinear wave […]

26 January, 2010 at 12:13 pm

Anonymous

Is a second edition planned?

2 April, 2010 at 1:42 pm

Amplitude-frequency dynamics for semilinear dispersive equations « What’s new

[…] It turns out that one can similarly analyse the behaviour of nonlinear dispersive equations on a similar heuristic level, as that of understanding the dynamics as the amplitude and wavelength (or frequency ) of a wave. Below the fold I give some examples of this heuristic; for sake of concreteness I restrict attention to the nonlinear wave equation (1), though one can of course extend this heuristic to many other models also. Rigorous analogues of the arguments here can be found in several places, such as the book of Shatah and Struwe, or my own book on the subject. […]

20 August, 2010 at 10:30 am

Spielman, Meyer, Nirenberg « What’s new

[…] (Now it turns out that there are some technical issues in making the above sketch precise, mainly because of the non-compact nature of the half-space , but these can be fixed with a little bit of fiddling; see for instance Appendix B of my PDE textbook.) […]

16 September, 2010 at 4:05 pm

Anonymous

Dear Professor Tao,
I think that Exercise 1.24 is false if the inequality $\langle F(v), dH_j(v) \rangle \geq 0$ is not strict (but true with a strict inequality there). Here is a counterexample: We take $D=\mathbb{R}$ , $F(x) = x$ , $u(t) = e^t$ , $H(x) = -(x-1)^2$ . Then the conditions are satisfied and $H(u(0)) = 0$ but $H(u(t)) < 0$ everywhere else.

[Good point! I’ve added an erratum for this. -T.]

13 March, 2011 at 2:46 am

Sebastien

Dear Professor Tao,

I think there is an “error” p.12 in theorem 1.12,
“B:[t0,t1]->R+ is continous and nonnegative.”
the nonnegative hypothesis can be dropped (as you say just before the theorem).

[Added, thanks – T.]

28 March, 2011 at 4:34 am

Sebastien

Dear Professor Tao,
A correction p.41 in equation (1.42): ||N(u)||_D and not N(||u||_D).

[Added to the errata, thanks -T.]

28 March, 2011 at 6:58 am

Sebastien

Dear professor Tao,
Pp 46-47 in the proof of proposition 1.41, I think there is something inconsistent. -The norm N is defined on C^0([0,+∞)->D) and you assert that then
sup_t exp(sigma t) ||u(t)||_D
is finite which is generally false.

And with this norm I don’t see how one can hope to get the estimate (p.47)
||DF||_S ≤ ||F||_N / sigma.

So I guess there is a mistake somewhere.

[Sorry, there were a number of typos in the text; I’ve added them to the errata. -T.]

28 March, 2011 at 7:07 am

Anonymous

I think sigma is supposed to be 2sigma

1 April, 2011 at 4:49 am

Sebastien

“the first display of Page 47, the first factor of exp(-sigma t) should be exp(sigma t).”
No, I think that with the definition of the norm N with 2 sigma everything is fine as it is. [Ah, I overcorrected for this problem :-). Thanks! – T.]

15 May, 2011 at 9:39 am

Filipe

Dear Prof. Tao,
just a small typo in view of a possible future re-edition of your book:
on the top of page 101, inside the integral, it is ^2b and not <\tau_o)^2b.

15 May, 2011 at 9:43 am

Filipe

I mean, > instead of ), sorry.

27 July, 2011 at 6:37 am

Anonymous

I think there is an error in exercise 2.3 (page 58). The Schrödinger equation should be $\partial_t u + i \partial_x^2 u = 0$

[Corrected, thanks – T.]

29 December, 2011 at 1:18 pm

Anonymous

On page 331, you give a definition of the fractional Sobolev spaces using Bessel potentials. It seems to me this is not the standard definition: usually $W^{s,p}$ is defined as the real interpolation space between the $W^{k,p}$ (i.e. the Besov space $B^s_{p,p}$ ). This probably doesn’t make any difference, but it might be worth mentioning.

30 December, 2011 at 7:19 pm

Gandhi Viswanathan

I noticed the same thing on page 331. The Fourier multiplier symbol for the inhomogeneous Sobolev norm is not the one I had expected.

31 December, 2011 at 1:17 pm

Gandhi Viswanathan

Please disregard my comment, I had not noticed the “Japanese bracket”…

2 January, 2012 at 3:47 am

Anonymous

I think in page 74, the right hand side of (2.26) should be $$ \|F\|_{L^{q’}_t L^{r’}_x(\mathbb{R}\times \mathbb{R}^d)}$$

2 January, 2012 at 6:12 am

Anonymous

never mind this comment

13 January, 2012 at 12:56 pm

Anonymous

In page 102 (in the proof of Lemma 2.11), you say that $b’ = b$ case is trivial. Could you just point me in the right direction? This does not at all seem easy to prove when $b\neq 0$. I see that there’s a rather involved argument through using $A_p$ weights, but no simple ones come to mind. Thanks.

13 January, 2012 at 2:26 pm

Terence Tao

Hmm, this is indeed less trivial than I had thought. Note though that from the $0=b< b'$ case that one can already treat the portion of $\| \eta(t/T) u\|_{X^{s,b}}$ for which $\langle \tau-h(\xi) \rangle \leq 1/T$ , so it suffices to control $\| P( \eta(t/T) u \|_{X^{s,b}}$ , where P is the Fourier projection to the region $\langle \tau-h(\xi) \rangle > 1/T$ . We can observe that $\| P e^{it\tau_0} u \|_{X^{s,b}} \lesssim_b \langle T \tau_0 \rangle^b \|u\|_{X^{s,b}}$ for any frequency $\tau_0$ (improving the bound in the proof of the first estimate), and then by repeating the proof of the first estimate one obtains the claim. I’ll add an erratum with this argument.

14 January, 2012 at 11:30 am

Anonymous

Thanks for the reply. But I still don’t understand your claim that $\|P e^{it\tau_0} u \|_{X^{s,b}} \lesssim_{b} \langle T\tau_0\rangle^b \|u\|_{X^{s,b}}$ . For example, if $\widetilde{u}$ is supported on $\tau,\xi \sim 1$ , then $\mathcal{F}_{t,x} [P e^{i\cdot\tau_0} u](\tau,\xi) = \chi_{[1/T,\infty)}(\tau-\xi^2) \widetilde{u} (\tau-\tau_0,\xi)$ . If $\tau_0 \sim 1/T$ , then the left-side of the inequality is $O(1/T^b) \|u\|_{L^2_{t,x}}$ , while the right side is $O(1)\|u\|_{L^2_{t,x}}$ . Am I mistaken somewhere? Thanks again for your time.

14 January, 2012 at 12:04 pm

Terence Tao

Gah, you’re right; the argument I sketched only deals with the $P \eta(t/T) P u$ component of $P \eta(t/T) u$ . (This is essentially enough if $\eta$ has a compactly supported Fourier transform, but we are not quite able to assume this.) The final term $P \eta(t/T) (1-P) u$ requires some additional argument which I have sketched in a revised erratum in the main post.

3 August, 2012 at 7:21 pm

7starsea

In Exercise 1.27, Do we have $\nabla_\omega H = J \nabla H$ instead of $\nabla_\omega H = -J \nabla H$

3 August, 2012 at 10:37 pm

Terence Tao

No, I believe there should be a minus sign here. (Note that there is an additional hypothesis that J should be skew-adjoint, as noted in the errata, which is the source of the minus sign.)

4 August, 2012 at 5:46 am

7starsea

Here is my calculation:
By definition, we have $\langle\nabla_\omega H, Jv\rangle=\omega(\nabla_\omega H, v)=\langle v, \nabla H\rangle$ , and since $J^2=-id$ and $J$ is skew-adjoint, we also have $\langle v, \nabla H\rangle=-\langle J^2 v,\nabla H\rangle = -\langle Jv, J^t\nabla H\rangle = \langle Jv, J\nabla H\rangle$ . Hence, we can conclude that $\nabla_\omega H = J\nabla H$ .

On the other hand, $\langle v, \nabla H\rangle = \langle\nabla_\omega H, Jv\rangle=\langle J^t\nabla_\omega H, v\rangle$ , which shows that $\nabla H = J^t\nabla_\omega H$ . This gives us $\nabla_\omega H = -J\nabla H$ .

This gives us a contradiction !!!

Thanks.

4 August, 2012 at 8:33 am

Terence Tao

Ah, I see the issue now: you are indeed correct that $\nabla_\omega H = J \nabla H$ . (The other calculation gives $\nabla H = - J \nabla_\omega H$ , which is what confused me.)

4 August, 2012 at 9:08 am

7starsea

I think the definition of the symplectic form should be $\omega(u,v):=\langle Ju, v\rangle$ . This coincides with that $\nabla_\omega H = -J\nabla H$ .

See http://www.math.psu.edu/tabachni/courses/symplectic.pdf

Thanks.

11 August, 2012 at 8:06 pm

7starsea

Dear Professor Tao,

There is a typo (‘neighborhood’ instead of ‘neighbourhood’) in the assertion (c) in pp. 21. [I was unable to locate this issue – T.]

In Exercise 2.19, I think you missed the assumption that $\hbar =1$ . [Correction added, thanks – T.]

In Exercise 1.59, I do not know how to do the calculation since you did not specify the specific simplectic form (i.e. $J$ in Exercise 1.27) for the Hamiltonian, is that right? [Sorry, “Exercise 1.27” should read “Example 1.27”. – T.]

By the way, in Exercise 1.27, I think $\omega(u,v):=\langle Ju, v \rangle$ instead of $\omega(u,v):=\langle u, Jv \rangle$ , see my previous comments. [Either sign convention would be acceptable here – there is no universally agreed upon convention. -T]

Thanks so much!

25 August, 2012 at 1:09 am

Bootstrap arguments « Hydrobates

[…] 10.3 of my book. Another description can be found in section 1.3 of Terry Tao’s book ‘Nonlinear dispersive equations: local and global analysis‘. A related and more familiar concept is that of the method of continuity. Consider a […]

13 February, 2013 at 3:41 pm

Hello, I was trying to do Exercise 2.28 (the part where we are asked to link the pseudoconformal transformation for the Schrodinger equation to the conformal transformation for the wave equation) but got really stuck. Are there any references where this is explained?

Thank you very much.

[See the errata for this question in the body of this post – T.]

14 February, 2013 at 7:02 pm

Thank you!

14 February, 2013 at 1:41 pm

The pseudoconformal and conformal transformations « What’s new

[…] an “extra challenge” posed in an exercise in one of my books (Exercise 2.28, to be precise), I asked the reader to use the embeddings (or more generally ) to […]

13 March, 2013 at 2:43 pm

Eric Foxall

Dear Professor Tao,

In Chapter 1, when proving for ODEs that weak solution –> classical solution, I am a bit puzzled by the step “integral of F(u) Lipschitz implies u Lipschitz”; this would be obvious if u appeared alone on the LHS of equation (1.8). Is there a typo here, or am I missing an estimation step? Thanks in advance.

13 March, 2013 at 3:12 pm

Terence Tao

I am afraid I don’t see the problem, since u(t) does indeed appear alone on the LHS of (1.8).

14 March, 2013 at 8:15 am

Anonymous

Oh, pardon me, I think I see my mistake. I had thought, while reading, that (1.8) had been referring to the equation for a weak solution. I suppose I am unclear, then, what it means for u(t) to solve the equation for a strong solution “in the sense of distributions”, since a distribution may not always be a function.

14 March, 2013 at 8:58 am

Terence Tao

Not every distribution is a function, but every (locally integrable) function is a distribution, and two locally integrable functions agree as distributions iff they are equal almost everywhere. The interpretation of (1.8) in the distributional sense is given explicitly in the display after (1.8).

14 March, 2013 at 10:19 am

Anonymous

I think it is clearer now. If I understand correctly, given a weak solution u, then since the integral of F(u) is Lipschitz, we conclude from the equation for a strong solution that there is a “version” of u which is a Lipschitz function, and plugging in once more, conclude that u is C1.

16 March, 2013 at 1:52 pm

Eric Foxall

Hi, a note on exercise 1.57: the result appears to follow easily if one assumes that dL/dt = PL – LP = -[L,P], but I cannot see how to do it if dL/dt = [L,P].

[Erratum added – T.]

16 March, 2013 at 2:33 pm

Jérôme Chauvet

Dear Pr. Tao,

I feel very flattered you quoted a french author for this one. It would be however more appropriate in this case to quote the whole sentence:

“Il semble que la perfection soit atteinte non quand il n’y a plus rien à ajouter, mais quand il n’y a plus rien à retrancher.“

“soit” is a form of the verb “être” (= to be) which cannot be dissociated from “Il semble que” without producing an awkwardness for the french reader (who’s flattered anyway). You get more or less the same feeling in english when conjugating with subjunctive :

“It seems as if perfection be attained not when etc.” turned into “perfection be attained not when etc.”

I just wanted to help.

Quite a c $\infty$ l b $\infty$ k. Thanks.

Best regards,

11 December, 2013 at 11:20 pm

Anonymous

Possible error on page 198. The KdV does not seem to have the stated reflection symmetry.

[Oops, you’re right; I’ve added an erratum accordingly. -T.]

24 March, 2014 at 9:31 am

Anonymous

Dear Professor Tao,

I may be wrong, but I think on page 339 in the fourth display in the last L^2 norm there should be a |\nabla|^s derivative to compensate for the appearing dyadic number M^{-s}.
Concerning the Lipschitz-estimate in Exercise A.12. It is clear, as indicated in the book, that the above given corrected estimate is a consequence of Lemma 8 and 9 with the fundamental theorem. However I wonder if the “uncorrected” version could still be true. In his book “Tools for PDE” Taylor gives a counterexample for the analogous statement if R^d is replaced by the two dimensional Torus. But maybe the situation on R^d or other manifolds is better?

Thanks in advance for any comment on this and many thanks for the well-explained heuristics in the book.

Best regards

[Correction added, thanks. As these estimates are local in nature, I would expect that the counterexample in Taylor can be adapted to other domains than the torus. -T.]

28 March, 2014 at 5:32 am

Anonymous

Thanks for the comment. In the meantime I realized that there is way to define Sobolev spaces on certain (e.g. compact) manifolds via the corresponding Sobolev norm on R^n, via localization. So an estimate like the “uncorrected” Lipschitz estimate could be carried over to these manifolds. Thus indeed the “uncorrected” estimate cannot hold, since it would lead to the analogous estimate on such manifolds and this would be a contradiction to the counterexample of Taylor.
One more thing: On page 340 in the last display, I think the first norm should be $L^2_x$ instead of $L^\infty_x$ .

[Corrected, thanks – T.]

Best regards

23 April, 2014 at 7:53 am

Anonymous

In the corrected version of the Lipschitz estimate on page 343, I think there should be an additional dependency on $\|u\|_\infty$ and $\|v\|_\infty$ of the “constant” for the additional second term. This comes from the application of the prior nonlinear estimate on $F'$ after using the fundamental theorem and the product estimate. Also $F'(0)$ should be $0$ to do this.

Best regards.

[Corrected, thanks. The normalisation $F'(0)=0$ can easily be removed by subtracting a linear function $x \mapsto F'(0) x$ from $F$ . -T.]

5 June, 2014 at 12:23 pm

zuchongzhi

Dear Professor Tao:

May I ask for a hint for Exercise 1.15? I can show that the solution exists for all times $t\in \mathbb{R}$ , but I do not know how to show $g(t)\in G$ . You suggested the reader to use Gronwall’s inequality again, but I just could not see how it helps me to show $g$ stays in $G$ .

Another trivial question I want to ask is in Example 1.28 we should have that

$\omega(\nabla_{\omega}H(\mu),v)=\frac{d}{d\epsilon}H(u+\epsilon v)$

by (1.26). Consider the simplest case $n=1$ we should have

$\omega(2i\frac{\partial H}{\partial \overline{z}}(\mu), e_{1})=\frac{d}{d\epsilon}H(\mu+\epsilon e_1)$

where $e_1$ is some vector in $\mathbb{C}$ . Since you have

$\omega(z_1, w_1)=\frac{1}{2}\text{Im}(z\overline{w})$

this should imply

$Im (i \frac{\partial H}{\partial \overline{z}}(\mu)\overline{e_1})=\frac{d}{d\epsilon}H(\mu+\epsilon e_1)$

But at this step I got lost, since the left hand is

$Im (i \overline{\frac{\partial H}{\partial \overline{z}}}(\mu){e_1})) \neq \frac{d}{d\epsilon}H(\mu+\epsilon e_1)$

I do not know if I made some mistake somewhere.

5 June, 2014 at 4:35 pm

Terence Tao

Thanks for the correction!

For Exercise 1.15, you might try first with the specific case of the special linear group $G = SL_n({\bf R})$ , which is cut out by a single equation $\hbox{det}=1$ , and to use Gronwall to show that $\hbox{det} X(t)=1$ for all times t if this is true at time 0, using the description of the Lie algebra $\mathfrak{g}$ as the trace zero matrices. The general case is similar (except that one may have multiple defining equations to cut out the group, rather than just a single one).

5 June, 2014 at 6:40 pm

zuchongzhi

Dear Professor Tao:

Thank you for the reply. I was taken away with my belief Lie groups may not be matrix groups, and I ignored the explicit condition given. By the way I think you meant $\det G(t)=1,\forall t\in \mathbb{R}$ , not $\det X(t)=1$ . I think the strategy is to use the matrix identity $d(\det(m))=\det(m)Tr(m^{-1}dm)$ , since we know $\det(m)=1$ and $Tr(dm)\approx 0$ , we can use Gronwall’s inequality on $h(t)=\det(g(t))$ as the $Tr(dg(t))$ term would be essentially bounded for small $t$ near $t_0$ . Then we iterates this procedure. I do not know if this is the right way (it feels overly complicated).

May I also ask for hint for Problem 1.16? I was thinking you wanted me to differentiate $|\mu(t)|^{2}=2\langle \mu(t), \frac{d}{dt}\mu(t)\rangle=2\langle \mu(t), L(t)\mu(t)+F(t)\rangle\le 2(|L(t)|_{op}|\mu(t)|^{2}+\langle \mu(t), F(t)\rangle)$ . Since we want to bound $|\mu(t)|^{2}$ ‘s size, I could have assumed that time $t=t_0$ we have $|\mu(t)|\ge 1\rightarrow |\mu(t)|^{2}\ge |\mu(t)|$ . Therefore we have $\frac{d}{dt}|\mu(t)|^{2}\le 2(|L(t)|_{op}|+|F|)|\mu(t)|^{2}$ for large enough $\mu$ at $t\ge t_0$ . And from this we can apply Gronwall’s inequality. But I feel this bound is too coarse, and it does not give us the right result we wanted after I go back to $|\mu|$ . Further I failed to use your hint that I need to make use of the self-adjoint component of $L$ . Sorry I was kind of lost.

6 June, 2014 at 8:16 am

Terence Tao

Use the fact that $\langle u(t), L(t) u(t) \rangle$ is equal to $\langle u(t), L^*(t) u(t) \rangle$ .

6 June, 2014 at 9:51 am

zuchongzhi

Dear Professor Tao:

Thanks for the hint! I did not think about that.

In Page 36, Example 1.31, do you mean $C\le 2$ instead of $C\ge 2$ ? Also in Page 37, Example 1.33, I think there is an extra $\frac{|\partial_{t}x(t)|^{2}}{\langle x(t)\rangle^{3}}$ on the third line, it seems to have been incorporated into the second term already. I do not know if I missed something in the reading.

[The remarks in Example 1.31 are in fact valid for any positive C. For Example 1.33, what is going on is that $\frac{1}{\langle x \rangle}$ is being split as the sum of $\frac{1}{\langle x \rangle^3}$ and $\frac{|x|^2}{\langle x \rangle^3}$ .

6 June, 2014 at 6:07 pm

zuchongzhi

Dear Professor Tao:

Sorry for commenting at here as I cannot comment my own post. I am kind of lost with your comment that Example 1.31 works for any $C$ . I think by virial identity we have
$\partial^{2}_{t}(|x(t)|^{2})\ge 2|\partial_{t}x(t)|^{2}+2CV(x)\ge 2C(\frac{1}{2}|\partial_{t}x(t)|^{2}+V(x))$ . And it would require we have $2|\partial_{t}x(t)|^{2}\ge 2C(\frac{1}{2}|\partial_{t}x(t)|^{2}$ , which after cancellation should imply $C\le 2$ . I do not know if I am missing something really trivial. And thanks for pointing out my mistake in Example 1.33.

I have a trivial question to ask. For Exercise 1.16, consider the most trivial case where $L\in End(\mathcal{D})$ is a self-adjoint linear operator, we should recover Duhamel’s formula in Proposition 1.35. However the two formulas (1.48 and the one appeared in Levinson’ theorem) seems rather different, for in (1.48) the $f(s)e^{(t-s)L}$ term is inside the integral, and I do not know how to modify it to the form of Levinson’s theorem.

For your hint, I think you meant $\langle \mu, L\mu\rangle=\langle L^{*}\mu, \mu\rangle\rightarrow \langle \mu, L\mu\rangle=\overline{\langle \mu, L^{*}\mu\rangle}$ . Therefore if the ground field is $\mathbb{R}$ they are equal. Now I have $\partial_{t}|\mu(t)^{2}|=\langle \mu, (L+L^{*})\mu\rangle+2\langle \mu, F(t)\rangle$ . I do not really know how to control the term $2\langle \mu, F(t)\rangle$ without using some Duhamel type argument or some superficial argument (like assume $\mu^{2}\ge \mu$ for $t>t_0$ ). And I do not know how to get $|\mu_0|+\int^{t}_{0}|F(s)|_{\mathcal{D}}ds$ as I wanted. Sorry for being really slow for following your guidance.

6 June, 2014 at 6:28 pm

Terence Tao

Yes, you’re right, C should actually be between 0 and 2, and there should be a conjugation sign for $\langle u(t), L^* u(t) \rangle$ in the complex case.

For Duhamel’s formula to apply in the form of (1.48), one needs L to be independent of t, in which case the bound in Levinson’s theorem will indeed follow from (1.48).

To control $\langle u, F(t)\rangle$ , use the Cauchy-Schwarz inequality. There will be factors of $\|u\|$ on both sides that one can cancel out (or, if one is concerned about division by zero, one can work instead with $\partial_t (\varepsilon + \|u\|^2)^{1/2}$ for some small $\varepsilon > 0$ ).

8 June, 2014 at 2:01 pm

Adam Azzam

Small Typo: On Page 235, in your definition of the local energy $E_{\Omega}[u[t_0]]$ , I believe that each appearance of $(t,x)$ should be $(t_0,x)$ .

16 June, 2014 at 10:46 pm

Son

Professor Tao,
On page 74, equation (2.25), is it $||F||_{L_t^{\tilde{q}'}L_x^{\tilde{r}'}}$ ? If it is the case then there is a typo on page 77, line 4. If the equation on line 4 is true then this is a typo.

[Corrected, thanks – T.]

24 June, 2014 at 12:32 pm

Adam Azzam

In Exercise 3.46 [Morawetz for NLW] on Page 161, the identity in the first display seems to be incorrect. The coefficient in front of $|u|^{p+1}$ in $T^{jk}$ is $-\frac{p}{p+1}$ , whereas the coefficient in front of $|u|^{p+1}$ on the RHS of the identity is $-\frac{1}{2(p+1)}$ .

24 June, 2014 at 12:34 pm

Adam Azzam

Of course, I meant that the coefficient in front of $|u|^{p+1}$ in $T^{jk}$ is $-\frac{1}{p+1}$ .

[Correction added, – T.]

8 October, 2014 at 10:42 am

Anonymous

I’ve been confused about what “global” and “local” solutions to a differential equation mean. Some textbook define a global solution for an ODE as something like “the solution $x(t)$ is valid for $t\in{\bf R}$ “. But questions like “do we have a solution to some ODE (PDE) on $t\in[0,\infty)$ ” is sometimes also called question about global existence. I tried to search lots of textbooks (this one included), but I don’t see any formal definitions of these two terms. You mentioned before that not every term in mathematics has a definition. So, what do people mean when they say “global” and “local” solutions for a PDE (ODE)?

8 October, 2014 at 12:57 pm

Terence Tao

As you say, these terms do not have a single formal definition, but generally speaking, a local solution to a differential equation is one defined in some (small) neighbourhood of the initial time $t_0$ , or in some cases the initial position $x_0$ or initial spacetime point $(t_0,x_0)$ . If the equation is not time reversible (e.g. a parabolic equation), then one often considers one-sided neighbourhoods such as $[t_0, t_0 + \varepsilon)$ rather than two-sided neighbourhoods $(t_0-\varepsilon,t_0+\varepsilon)$ . Global solutions are defined on the whole domain of interest, which in the case of the time variable may be either ${\bf R}$ or $[0,+\infty)$ , depending on the situation (again, one usually sees the latter for non-time-reversible equations such as parabolic equations, and the former for time-reversible equations such as Hamiltonian equations).

12 November, 2014 at 12:45 pm

jamesfennell

There’s a small typo in Exercise 2.37, page 80, which concerns solutions of Schrodiner’s equation with finite energy. It reads “show that $\lim_{t\rightarrow \pm \infty} \| u(t) \|_{L^p_x}$ for $2<p\leq 6$ …".

I thought the limit should be 0, but I'm not sure: the best I can get out of Strichartz and Sobolev is that $\| u(t) \|_{L^6_x}$ is bounded in time, and this holds for the other values by interpolating between $p=6$ and $p=2$ .

[Corrected, thanks. For the exercise, think about the standard proof of the Riemann-Lebesgue lemma, and see if you can prove the exercise in the case that the initial data is a Schwartz function. -T.]

31 March, 2015 at 10:04 am

Fan

Some additional typos: In (1.26) the inner product on the LHS has an angled bracket on the left and a parenthesis on the right. Besides, I think the expression (1.27) is correct, but there is a sign error when substituting for the symplectic gradient in (1.27) to obtain the expreesion for {H, E} below,.By the way, is this the accepted sign convention? (I hope so.)

[I could not locate the first issue; the second issue has already been listed in the errata. -T.]

2 April, 2015 at 1:37 pm

Fan

Dear Prof. Tao,

For exercise 1.53 (Duhamel v.s. resolvent), I’m having some difficulty in showing the second part of the identity using Duhamel’s identity. The only way I’m able to show it is to verify it satisfies the ODE. Is it how we’re expected to do it? Also, I’m not sure what we are supposed to do for the second identity. I’m only able to obtain it by multiplying the first identity with $e^{-\lambda t}$ and integrating from t=0 to $\infty$ . I’m not sure what is meant by “Fourier duality between t and $\lambda$ .

2 April, 2015 at 1:46 pm

Terence Tao

One can derive the second part of Duhamel’s identity from the first after swapping L and L_0 (and replacing V with -V).

Integrating with $e^{-\lambda t}$ is essentially a Laplace transform, which is closely related to the Fourier transform. (One could rotate the t parameter by i if one wanted to directly use a Fourier transform rather than a Laplace transform.)

2 April, 2015 at 2:14 pm

Fan

Many thanks!

2 April, 2015 at 9:30 pm

Fan

Maybe I’m ignorant to Hamiltonian mechanics, but I can’t understand why there are at most n conserved integrals if the phase space has dimension 2n (Exercise 1.56). For example, for the Harmonic oscillator in 2D (so n=2), with the Hamiltonian $2H=q_1^2+q_2^2+p_1^2+p_2^2$ , we have conserved quantites $E_i=q_i^2+p_i^2$ for i=1, 2, and a mysterious third conserved quantity $E_3=p_1q_2-p_2q_1$ (which seems to measure the phase difference between the oscillations in two directions). Presumeably none of the E_i’s can be expressed in terms of the other two.

2 April, 2015 at 9:33 pm

Fan

I guess the $E_3$ above looks a lot like the angular momentum.

3 April, 2015 at 6:32 am

Terence Tao

Oops, there is an important typo in this exercise, it’s not that the $E_i$ Poisson commute with an external Hamiltonian (which would be a vacuous statement if for instance H=0), but that they Poisson commute with each other.

3 April, 2015 at 7:59 am

Anonymous

Thanks. But I’m still wondering why we require the conserved quantities to commute with each other in the definition of complete integrable system. Is there any physical motivation for that? Or is there any reason why any of the three conserved qualities above do not qualify?

3 April, 2015 at 8:09 am

Anonymous

An unrelated question: In the proof of Proposition 1.46, why is it justified to only differentiate the last factor of L. Or is there a reason why L and its time derivative commute?

3 April, 2015 at 8:43 am

Terence Tao

This comes from the cyclic property of trace: $\hbox{tr}(AB)=\hbox{tr}(BA)$ (so in particular $\hbox{tr}(A^j B A^k) = \hbox{tr}(A^{j+k} B)$ ).

3 April, 2015 at 8:45 am

Terence Tao

Poisson commutation of the conserved quantities is equivalent to the symmetry group generated by these conservation laws being abelian. One can of course have a nonabelian group of symmetries (e.g. rotation symmetry), but this is not considered to be a completely integrable system as one cannot linearise the solution in action-angle variables.

3 April, 2015 at 9:07 am

Anonymous

Sorry, but what do you mean by linearize the solution? Is it discussed in the book? Or where can I find a reference for that?

3 April, 2015 at 9:26 am

Terence Tao

See Example 1.43.

3 April, 2015 at 9:36 am

Anonymous

I guess you are referring to Example 1.43, but I can’t see why the existence of such a linearization is related to the abelianity of the flow generated by the conserved quantities. I suppose for the 2D harmonic oscillator you can also use the 3 conserved quantities for the action variables and the parameter of the integral curve for the angle variable. What’s wrong with such a linearization?

3 April, 2015 at 12:42 pm

Terence Tao

This would not be an action-angle coordinate system, as the angle variables are not the symplectic conjugates of the action variables (in particular, this would not be a system of canonical coordinates).

3 April, 2015 at 6:35 pm

Fan

Thanks very much! Then it makes much more sense!

4 April, 2015 at 5:37 pm

Fan

Another typo: in the equation of the characteristic surface of the Klein Gordon equation, there is an exponent 2 that is not in superscript.

[Correction added, thanks – T.]

4 April, 2015 at 8:38 pm

Fan

This is on p.67.

4 April, 2015 at 8:24 pm

Fan

Also, in the first display in p.71, $|t|^{n-1}$ should be $|t|^{d-1}$ .

[Correction added, thanks – T.]

4 April, 2015 at 11:04 pm

Fan

Prof. Tao, I don’t understand what the expression (2.33) really means: appearantly it adds a scalar $xu$ to a vector $it\nabla u$ .

4 April, 2015 at 11:08 pm

Fan

In particular, are there any cross terms involving the product $xu\cdot t\nabla u$ ? If there are, what do they look like?

4 April, 2015 at 11:12 pm

Fan

Gah, $x$ is also a vector. I should take a nap now…

5 April, 2015 at 6:59 pm

Fan

Back from the nap: in penult display on p.100, should $f_\tau$ be $f_{\tau_0}$ ?

5 April, 2015 at 9:32 pm

Fan

Prof. Tao,

I can’t see why in the proof of Lemma 2.11 the case when $b=b'=0$ implies the case $b=b'$ for $\le 1/T$ (I’m assuming h=0 for simplicity.) When applying the bound in the first part it seems necessary to bound a factor of $latex $ by $latex $, which is only possible when $\le 1$ .

[See the existing errata for p. 102. -T.]

5 April, 2015 at 10:10 pm

Fan

Sorry that the angled brackets have messed up the parser.

5 April, 2015 at 11:03 pm

Fan

From a different perspective: if we take $b=b'>0$ , $s=0$ , $d=0$ and $h=0$ , Then the $X^{s,b}$ norm seems to degenerate to the Sobolev norm $H^b(\mathbb R)$ in time. Then the lemma is claiming that $\|\eta(t/T)u\|_{H^b}\ll_{\eta,b} \|u\|_{H^b}$ , which looks very weird, considering $b>0$ and $T<1$ .

8 April, 2015 at 8:51 pm

Fan

In the last line of the fourth display of p.102, the $s$ had better be 0, because it has already been set to 0 early on.

8 April, 2015 at 8:52 pm

Fan

Ah, you have got this in the errata.

9 April, 2015 at 2:01 pm

Fan

I guess there should be a minus sign before the integral in the last display of p.103, or some clarification of the direction of the integral.

[Correction added – T.]

9 April, 2015 at 3:15 pm

Fan

Also should the $\tilde\eta$ here be actually $1_{[-2R,0]}$ , as the previous $\tilde\eta=1$ on $[-3R,3R]$ .

[No, $\tilde \eta$ is written as intended. Note that $F$ vanishes below $-2R$ . -T.]

9 April, 2015 at 3:18 pm

Fan

Also, the matching of parentheses in the fourth display on p.104 is unfortunately a mess.

[Correction added – T.]

9 April, 2015 at 3:18 pm

Fan

I mean its right hand side.

9 April, 2015 at 3:26 pm

Fan

In general I think the exposition of this part could be improved by setting $h=0$ by conjugating $u$ with $e^{tL}$ (which commutes with localization in time), setting $s=0$ (as already done) and doing the estimates pointwise in the physical space (effectively setting $d=0$ ). Then essentially we’re doing various Sobolev estimates in one-dimension, which could make some of the displays shorter and clearer.

9 April, 2015 at 9:20 pm

Fan

In the last line of Lemma 2.8 on p.99, $\eta\in S_x(\mathbf R)$ should be $\eta\in S_t(\mathbf R)$ .

[Corrected, thanks – T.]

10 April, 2015 at 12:38 pm

Fan

In the last line of Lemma 2.11 on p.101, $\sigma>0$ seems superfluous as $\sigma$ is used nowhere in the lemma or its proof.

[Correction added – T.]

11 April, 2015 at 10:49 am

Fan

In the third line in the proof of Proposition 2.13 on p.104, the frequency support of $u_M$ should be something like $M\le\langle \tau-k^2 \rangle<2M$, as M is already a power of 2.

[Correction added – T.]

11 April, 2015 at 10:57 am

Fan

Just nitpicking, the antepenult display on p.104 is missing a period at the end.

[Correction added – T.]

12 April, 2015 at 9:50 pm

Fan

Prof. Tao,

I can’t somehow solve Exercise 2.73. I’m trying to reduce to the first order case, but the $\tau$ multiplier in the $\|\cdot\|_{s,b}$ norm associated is different than the corresponding multiplier in the $X^{s,b}$ norm. Do you have any hint?

12 April, 2015 at 9:52 pm

Fan

I am trying to use the factorization $\partial_t^2-\Delta=(\partial_t+|\nabla|)(\partial_t-|\nabla|)$ .

14 April, 2015 at 9:47 am

Fan

Prof. Tao,

Do you really mean the $L^2_{t,x}$ norm in the second display on p,107? For one this is more or less trivial, even without the $\langle k \rangle^\epsilon$ on the RHS. For the other, I don’t see how it is related to the $L^6_{t,x}$ estimate in the third display.

[The $L^2_{t,x}$ norm should be $L^6_{t,x}$ . -T.]

14 April, 2015 at 10:48 am

Fan

Also, in exercise 2.74 on the previous page, is the estimate really on $R\times T^2$ instead of $R\times T$ ? Also, on the left hand side is there really a $C^0_t L^4_x$ norm? If we look at the free solution $F=0$ and at time $t=0$ , we would be able to bound $\|u(0)\|_{L^4}$ in terms of $\|u(0)\|_{L^2}$ , for free!

[This should be $C^0_t L^2_x({\bf R} \times {\bf T}^2)$ – T.]

14 April, 2015 at 6:04 pm

Fan

Still, is it on ${\mathbb T}^2$ or $\mathbb T$ ? I find a factor of $\langle k \rangle^\epsilon$ in the cited paper of Bourgain (specifically, proposition 3.6).

[This should be ${\bf T}$ ; a correction has been added. -T.]

14 April, 2015 at 6:05 pm

Fan

I mean the $L^4_{t,x}$ norm.

15 April, 2015 at 10:45 am

Fan

Also why is exercise 2.78 named “period Airy $L^6$ estimate? Isn’t the dispersion relation $\tau=k^2$ that of the Schrodinger equation?

[Correction added – T.]

15 April, 2015 at 10:45 am

Fan

I mean “periodic”.

16 April, 2015 at 2:10 pm

Anonymous

Prof. Tao, I find the discussion after (3.5) of focusing / defocusing a little bit unconvincing. I feel the same argument would apply to the case p=1, but then the equation is linear and the solution is just a modulation of the free solution by $e^{-\mu it}$ . I fail to see the difference between focusing and defocusing in this case. Presumably there is some interaction between the sign and the nonlinearity going on here.

16 April, 2015 at 3:18 pm

Terence Tao

The discussion is an oversimplification for the sake of building intuition. It would be more accurate to look at the relative changes in phase rather than absolute changes in phase, and to consider solutions in which the amplitude and frequency can vary in space, instead of being constant in space in this example. The basic point is that in typical singularity formation scenarios, frequency and amplitude are positively correlated: one has high amplitude, high frequency concentrations of energy, or low amplitude, low frequency dispersals of energy. If $p > 1$ , then the phases |\xi|^2 t/2$ and $\mu \alpha^{p-1} t$ would rise together and fall together in the defocusing case, and work against each other in the focusing case. When $p=1$ , the latter phase is insensitive to amplitude or frequency and thus has no correlation with the linear dispersive phase.

17 April, 2015 at 11:58 am

Fan

Thanks.

17 April, 2015 at 11:58 am

Fan

In the last display on p.60, $x_n$ should be $x_d$ .

[Correction added – T.]

17 April, 2015 at 1:34 pm

Fan

Is it mentioned anywhere in Exercise 2.12 that u is radial? Otherwise the word “thus” on the second line is very confusing to me.

[Correction added – T.]

20 April, 2015 at 8:39 pm

Fan

In the last line of exercise 3.5, “focusing regularity” should probably be “focusing nonlinearity”?

[Correction added – T.]

21 April, 2015 at 10:49 am

Fan

In the first line of the second display of p.352, the exponent is better written as $1/(p+1)$ , so it is not confused with $(1/p)+1$ .

[Clarification added – T.]

21 April, 2015 at 7:57 pm

Fan

In exercise B.2 on p.359, are you suggesting (by using “all $x\in R^d$ “) that the Bessel kernel is bounded even at the origin. AFAIK this is not true: it behaves much the same like the old Newton potential.

[Correction added -0.]

21 April, 2015 at 9:08 pm

Fan

In the middle of p.353 you wrote “Inserting this fact into the above equation and iterating (again using Sobolev embedding) we can
successively enlarge the range of q,” I can’t see how we can use Sobolev embedding (alone). Probably we need some multiplier theorem to relate $\|u\|_{W^{2,p}}$ to $|\Delta u-u\|_{L^p}$ ?

[One can use the scale of Sobolev spaces defined for non-integer exponents: http://en.wikipedia.org/wiki/Sobolev_space#Sobolev_spaces_with_non-integer_k -T.]

22 April, 2015 at 10:19 am

Fan

Thanks, but to show it is the old Sobolev space defined in the physical space we use multipliers anyway.

22 April, 2015 at 2:41 pm

Fan

In the first display on p.354, the two arguments are not symmetric to the plane $\{x\cdot \xi=t\}$ as expected.

[Correction added, thanks -T.]

22 April, 2015 at 5:47 pm

Fan

In equation (3.10) on p.114, there is probably a sign error in the exponent $it|v|^2/2$ , of the same nature as the sign error in equation (3.5).

[Correction added, thanks -T.]

22 April, 2015 at 6:41 pm

Fan

The ground state equation (3.8) for NLW has the wrong sign for $\beta$ as per your correction. Does the ground state soliton still exist then?

[Oops, this is inaccurate and the references to NLW should be deleted. (Ground states exist for NLKG and for critical NLW, but not for NLW in general.) -T.]

22 April, 2015 at 9:36 pm

Fan

In equation (3.20) on p.117, probably there are sign errors in the exponents $i|\xi|^2t/2$ and $i\mu|\alpha|^{p-1}t$ .

[Correction added, thanks -T.]

22 April, 2015 at 10:15 pm

Fan

Just nitpicking: in the proof of Proposition 3.2, at the end you used Gronwall’s inequality (on $\|v\|_{L^2}$ ). However, the theorem as quoted requires $\|v\|_{L^2}$ to be continuous in t, but _a priori_ I can’t see why this is true under the assumption $u, u'\in L_t^\infty L_x^2$ .

[Correction added, thanks -T.]

23 April, 2015 at 10:22 am

Fan

In the last second of lines of p.124, is $L_t^\infty H_x^s$ really necessary as it is subsumed by $C_t^0 H_x^s$ ?

[Correction added, thanks -T.]

23 April, 2015 at 11:10 am

Fan

there is an extra semicolon at the end of equation (3.22).

[Correction added, thanks -T.]

24 April, 2015 at 2:08 pm

Fan

I’m confused about the diagram on p.139: why is the strichartz estimate able to bound $\|u\|_{L_t^10 W_x^{1,30/13}}$ by $\|F(u)\|_{L_t^2 W_x^{1,6/5}}$

[This step uses the Leibniz rule and Holder inequality, not the Strichartz estimate. -T.]

17 January, 2016 at 7:43 am

Sergio Mayorga

In page 4, it should be “Kovalevskaya” instead of “Kowalevski.” https://en.wikipedia.org/wiki/Sofia_Kovalevskaya

[There are several different transliterations of Kovalevskaya’s name, including Kowalevski which she herself used in publications, as noted on the above Wikipedia page. -T.]

26 January, 2016 at 6:54 pm

Anonymous

I think in the 6th display in page 167 it should be $\varepsilon^{2\alpha}\| u\|_{S^1(I\times \mathbb{R}^3)}^{3-2\alpha}$ , to account for the square in the $L^5_{t,x}$ norm in the third displayed equation in the same page.

[Corrected, thanks – T.]

28 January, 2016 at 2:25 pm

Anonymous

I think in page 169, in the last paragraph “We can thus use the global $H^1_x$ -wellposedness theory (from Exercise 3.35)”, the reference should be to Proposition 3.25 instead since we’re in the two dimensional case.

[Thanks; this errata has already been added to the web page. -T.]

7 February, 2016 at 8:40 pm

Fan

Errata to the errara: the errata on page 220 is actually on page 206.

[Corrected, thanks – T.]

7 February, 2016 at 9:06 pm

Fan

In the bibligraphy on p365, “disperives” in Ref [CS] should be “dispersives”.

[Erratum added, thanks – T.]

9 February, 2016 at 6:09 pm

Fan

Prof. Tao, I have a question about the proof of the periodic Schrodinger $X^{s,b}$ space estimate (Proposition 2.13). I’m trying to adapt the proof there to the cubic dispersion relation $\tau=\xi^3$ and to recover Bourgain’s result of bounding $L_{t,x}^4$ norm by $X^{0,1/3}$ norm, but I failed in estimating the sum over $k_1+k_2=k$ . Apparently to get that exponent I need $k_1^3+k_2^3$ to grow as $k_1^3$ when $k_1+k_2=k$ is fixed, but that is not the case: it only grows as $k_1^2$ due to cancellations. It doesn’t help either to replace (both) $+$ by $-$ . I’m wondering is it possible to extend the proof to this case, or do we have to use Bourgain’s original proof?

10 February, 2016 at 9:20 am

Terence Tao

I haven’t done the calculations in full, but I suspect that the argument of Tzvetkov reproduced in my text would only cover some of the cases needed in the cubic case (depending on the relative sizes of $k_1, k_2, k, M$ ) but not all. (In particular, the Fubini argument used there is a little lossy in the “high-high” interaction case when $k_1,k_2$ are large compared with $k$ .) There are other proofs of Bourgain’s estimate; for instance, I have one in Proposition 6.4 of http://www.ams.org/mathscinet-getitem?mr=1854113 .

10 February, 2016 at 12:49 pm

Fan

Thanks.

24 February, 2016 at 2:40 am

Anonymous

Dear Pr. Tao, I have another question/remark concerning the proof of Proposition 2.13 (p104). The dyadic decomposition of u, according to the corrected version, consists in decomposing u into $u_M$ where $u_M$ is localized in the spacetime frequency region $M \le < 2M$ . Shouldn't there be some kind of exception for $M=0$ ? Or perhaps, one should keep the decomposition as $2^M \le < 2^{M+1}$ , have $M$ range over Z and change M^{3/4} to $2^{M*3/4}$ below ?

[ $M$ should range over the powers of two $1, 2, 4, \dots$ ; note that since $\langle \tau - k^2 \rangle$ is always at least $1$ , the smaller values of $M$ are not needed. -T.]

27 February, 2016 at 7:16 pm

Finite time blowup for a supercritical defocusing nonlinear wave system | What's new

[…] the divergence-free nature of this tensor: See for instance the text of Shatah-Struwe, or my own PDE book, for more details. The energy-critical regularity results have also been extended to slightly […]

16 March, 2016 at 9:17 pm

Jason

In equation (6.36) on page 302, should the commutator [A_\alpha,A_\beta] be added to the right-hand side as well?

I didn’t see any reason why this commutator should vanish in the general case…

Thanks in advance!

[Corrected, thanks – T.]

16 April, 2016 at 6:44 am

Georg

Dear Prof. Tao,

Remark 1.2 seems to provide a counterexample to Lemma 1.3:

The function $u : I \rightarrow \mathbf R$ with open interval $I = (-\infty, 1)$ and $u(t)=\frac{1}{1-t}$ for all $t\in I$ is a strong solution $u\in C_{\rm loc}^0 (I\rightarrow \mathbf R)$ of the Cauchy problem $\partial_t u(t)=F(u(t))$ with initial value $u(0)=1$ and nonlinearity $F(u)=u^2\in C_{\rm loc}^0 (\mathbf R\rightarrow \mathbf R)$ . However, since $u$ is unbounded on $I$ ,
$u$ is not a weak solution of the Cauchy problem (according to your definition on page 6) since $u\notin L^\infty(I\rightarrow \mathbf R)$ .

The existence of such a counterexample to Lemma 1.3 may perhaps be ruled out by redefining the notion of a weak solution or by restricting Lemma 1.3 to Cauchy problems with compact domains.

Please let me know how to fix this problem.

Additional errata:

Page xiv, line 14: the spelling of the name Frechet should be identical to the one on page xv, which has been changed in the errata.

Page 1, line 17: “is still its infancy” should be “is still in its infancy”.

Page 4, line 5: $Y$ should be $\mathcal Y$ .

Page 5, line 4: “a open interval” should be “an open interval”.

Page 5, line 7: $u_t=u^2$ should be $\partial_t u =u^2$ .

Page 8, line 17: $C^0(I\rightarrow \Omega_{\epsilon})$ should be $C^0(I\rightarrow \mathcal D)$ as in the errata for line 8 on the same page.

Page 11, line 7: “Cauchy-Kowaleski theorem” should be “Cauchy-Kowalevski theorem” as on page 4.

Page 13, line 36: $|F(u(s))-F(v(s))|\le M|u(s)-v(s)|$ should be $\|F(u(s))-F(v(s))\|_{\mathcal D}\le M\|u(s)-v(s)\|_{\mathcal D}$ .

Page 22, line 13: “be such that such that” should be “be such that”.

16 April, 2016 at 10:22 am

Terence Tao

Thanks for the corrections! In the definition of weak solution, $L^\infty$ should be $L^\infty_{loc}$ .

13 May, 2016 at 4:11 pm

Gawin

Dear Prof. Tao,

In your errata, you said that in Gronwall’s inequality (Page 12, Theorem 1.10) , the condition that “$B$ is non-negative” can be weakened to “$B$ is real-valued”. But I don’t think that’s possible, at least from looking at your proof anyway. In the first equation in your proof, you multiplied $B(t)$ on both sides of the inequality, and you need $B$ to be non-negative to keep the direction of the inequality.

But please let me know if I am missing something.

13 May, 2016 at 4:50 pm

Terence Tao

This erratum concerns Theorem 1.12, rather than Theorem 1.10. (It may be though that the page numbering is off; I will check this when I have access to a physical copy of the book after the weekend.)

11 January, 2017 at 2:45 am

hsyn

I would like to notify that blow-up and global self-similar solutions of the semilinear dispersion equation,
$u_t=u_{xxx}\pm({|u|^{p-1}u})_{xx} \quad \mbox{in} \quad \mathbb{\textbf{R}} \times \mathbb{\textbf{R}_+}, \quad p>1$ , have been studied in the following link:

http://opus.bath.ac.uk/47058/

For the first critical exponent, $p=p_0=2$ , an admissible global similarity solution has been numerically observed for the ‘ $+$ ‘ case.

26 July, 2017 at 2:44 am

Anonymous

Dear Prof. Tao,

in the proof oh theorem 2.3 in page 74 is said that (2.26) follows from the inequality in the second display after using the Christ-Kiselev lemma.
In terms of that lemma this means that “ $X=L^{\tilde{r}'}_{x}$ ” , “ $Y=L^{r}$ ” and “ $K(t,s)=e^{i(t-s)\Delta /2}$ “, but I don’t get how this kernel could be bounded from “ $X$ ” to “ $Y$ “, given that the only estimates available are (2.21), (2.22) and (2.23), which regard conjugate exponents.

26 July, 2017 at 4:02 am

Terence Tao

This can be resolved by a standard regularisation argument, e.g. replace $K$ by $P_N K P_N$ for some suitable Littlewood-Paley operator $P_N$ (so that the Bernstein inequalities become available), apply Christ-Kiselev, and then send $N \to \infty$ . (In general, qualitative requirements such as continuity can usually be waived for the purpose of proving quantitative estimates by such arguments.)

27 July, 2017 at 5:52 am

Anonymous

Dear Prof. Tao,

Thank you very much for your time and the advise.

6 August, 2017 at 6:45 pm

Fan

Dear Prof. Tao,

Just a typographical remark: In Exercise 3.34 in my book, the RHS of the bound on the fourth line is mysteriously subscripted, which it should not.

[Actually, the subscripting is intentional: the exercises asks to prove the H^1 norm of u(t) is bounded by a (possibly nonlinear) function of the H^1 norm of the initial data. -T.]

31 August, 2017 at 7:00 am

Cattle

Dear Prof. Tao,

I guess it is a dumb question, but I still don’t know in P.64, why “space time Fourier transform” is

$\displaystyle \int_R \int_{R^d} u(t,x)e^{-i(t\tau + x\cdot\xi)} dt dx$

not

$\displaystyle \int_R \int_{R^d} u(t,x)e^{-i(t\tau + x\cdot\xi)} dx dt$ ?

I have such confusion is because I think $x\in R^d$ and $t\in R$ .

[This was a typo, now added to the errata – T.]

31 August, 2017 at 3:04 pm

Cattle

I see.
Thank you, Prof. Tao！
:)))

31 August, 2017 at 4:58 pm

Cattle

So on the same page,
in inversion formula is

“$d\xi d\tau$” not “$d\tau d\xi$”, right?

(Sorry I didn’t ask the same questions at the same time…)

[Correction added – T.]

1 September, 2017 at 3:15 am

Juha-Matti Perkkiö

I am wondering why in most textbooks and lecture notes (including these) the Grönwall lemma is stated only in a form where the majorizing ODE is linear, when the non-linear counterpart seems to be completely analogous, more natural, and quite elementary to reduce to the linear case by contradiction:

Let $F:\mathbb R \to \mathbb R$ be locally Lipschitz-continuous and $x,y:[0,T]\to \mathbb R$ absolutely continuous with $\dot x(t) \leq F(x(t))$ and $\dot y(t) = F(y(t))$ for every $t\in[0,T]$ . If $x(0)\leq y(0)$ , then $x(t)\leq y(t)$ for every $t\in[0,T]$ .

In fact this statement does not even show up when googling non-linear Grönwall lemma, while some special cases pop up. Am I missing something?

1 September, 2017 at 10:01 am

Terence Tao

This is Exercise 1.7 of my text (in fact it is literally the first exercise after Gronwall’s inequality is introduced); I view it more as a prototypical example of a comparison principle (which are extremely common and important in elliptic and parabolic PDE) than as a nonlinear Gronwall inequality, though one can certainly take the latter viewpoint also.

This comparison principle is certainly one of many common applications of Gronwall’s inequality; see the other exercises in that section for further examples. But in subjects such as PDE, it is often not as useful to state theorems in maximal generality as it is in the more algebraic parts of mathematics, as in applications one often has to tweak the result anyway to fit one’s particular setting (e.g. one may not have enough regularity, one may have additional forcing terms, one may be working in higher dimensions, etc..), and it is usually the _techniques_ or _principles_ that are more useful than the _results_. (Cf. Gowers’ “The two cultures of mathematics, or Klainerman’s “PDE as a unified subject“.)

13 September, 2017 at 12:09 am

Fan

Dear Prof. Tao,

I have a question about the second estimate of Lemma 2.11. On page 102 you mention that by composition it suffices to prove it for $0\leq b^\prime\leq b$ and for $b^\prime\leq b\leq 0$ . To conclude that the estimate holds for $b^\prime<0<b$ , I think we need to write $\eta=\eta_1\cdot\eta_2$ , where $\eta_1,\eta_2$ both Schwartz, and combine the estimates above. However, I am not sure whether we can do this or not.

In other word, my question is, given any Schwartz function $\eta$ , is it always possible to find two Schwartz functions $\eta_1,\eta_2$ such that $\eta(x)=\eta_1(x)\eta_2(x)$ for all $x$ ? How could we prove or disprove this? I was wondering whether you could help me with this.

Thank you in advance for your time and help.

13 September, 2017 at 7:15 am

Terence Tao

Well, one can prove things first for $C^\infty_c$ functions where the factorisation is trivial (and in which the bounds can be seen to depend only on the width of the support and on some finite $C^k$ norm – one can also appeal to the closed graph theorem if desired), and then decompose a Schwartz function as a rapidly decreasing sum of translated $C^\infty_c$ functions.

But it is also possible to factorise the Schwartz function directly, by taking $\eta_1(x) = (1+|x|^2)^{-\omega(x)}$ for some sufficiently slowly growing function $\omega(x)$ (which one can choose so that $\nabla^k \omega(x) = O_k( \omega(x) / \langle x \rangle^k )$ for all $k$ and $x$ ). There are a countable number of conditions on the growth of $\omega$ that need to be satisfied, but this can be accomplished by a diagonalisation argument (presumably there is also a compactness argument that is applicable here).

15 December, 2017 at 3:51 pm

Fan

In Exercise 4.2 on page 200, I think the formula of $P(t)$ on the book does not work. I think the correct formula here should be $P(t)=4D^3-3(Du(t)+u(t)D)$ . See also https://en.wikipedia.org/wiki/Lax_pair#Example_-_KdV. Thank you.

[Correction added -T.]

24 March, 2018 at 11:41 pm

Anonymous

Dear Prof. Tao,

In Exercise 3.61, I think all the occurrences $C_{t}^{1} l_{k}^{1}$ should be $C_{t}^{0} l_{k}^{1}$ , as in the paper [CKSTT13].

19 April, 2018 at 11:44 pm

Anonymous

Dear Prof. Tao,

page 215: In the third display on page 215, it seems that we need $\frac{3i\xi}{2\pi}$ instead of $3i\xi$ .
page 216: in the third and fourth displays on page 216, minus signs should be deleted.

2 May, 2018 at 1:33 pm

itaibn

Typos in the errata section:

Page 55 (i.e. the errata section for page 55): This page makes no references to the Schrödinger equation.

Page 57: $\mathbf{R}^3+1$ should be $\mathbf{R}^{3+1}$ .

Page 72: Unclear prescriptions over whether $\hat {u}_0 (x)$ should be replaced with $\bar {\hat {u}}_0 (x)$ or with $\hat {u}_0 (-x)$. I believe it is correct to replace $\hat {u}_0 (x)$ with $\bar {\hat {u}}_0 (x)$ in both places it occurs.

Page 74: Most of the corrections here are actually for page 77.

[Corrected, thanks -T.]

2 May, 2018 at 1:47 pm

itaibn

Page 100: Claims there are two instances where $f_{\tau}$ is written instead of $f_{\tau_0}$. I only see one.

[Corrected, thanks -T.]

3 May, 2018 at 11:53 am

itaibn

Page 77: $e^{i(t-s)\Delta}$ doesn’t appear in LaTeX.

Page 110: F(zu) should be in LaTeX.

Page 120: \mu=+1 and \mu=-1 don’t appear in LaTeX.

Page 125: $\mathbf {R}$ unbolded in the first correction.

Page 137: Add a space between “be” and “ $2 (d+2)/d$ “.

Page 140: I think you mean that $H^1_x$ should be replaced with $\dot {H}^1_x$ in both appearances of the text for the figure; the wording seems vague to me.

Page 141: “bet” should be “be $t$ “.

Page 144: Missing space in “be\”by\””.

Page 179: $u^(\lambda)$ should be $u^{(\lambda)}$ . Missing space in “firstdisplay”.

Page 180: The exponent is $\frac {i} {2}$ , and presumably should be changed to $- \frac {i} {2}$ .

Page 193: This correction is actually for page 199.

Page 282: Actually page 283.

Page 349: “maxximiser” should be “maximiser”.

Schrodinger should be Schrödinger throughout.

In the errata I see occasion mention of blackboard bold $\mathbb {R}$ and $\mathbb {Z}$ , whereas the text consistently uses the boldface $\mathbf {R}$ and $\mathbf {Z}$ .

[Corrected, thanks – T.]

4 November, 2018 at 10:58 pm

Anonymous

Page 33, Exercise 1.34: “[..] with a vector field $X_x$ on $\mathcal D$ in the obvious manner.”

Here, $D$ should be $G$ I believe. Great book!

4 November, 2018 at 11:04 pm

Anonymous

Oh, please ignore! The book is correct.

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