Last edited: Mar 18, 2022

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 45: In footnote 19, “the spectrum of L being contained entirely in the interior of the left half-plane” should be “the spectrum of L + L^* being contained entirely in the negative real axis”.
  • 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 68: In the hint for Exercise 2.24, e^{ixz+itz^2} should be e^{ixz-itz^2}.
  • 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 94?: In the paragraph after (2.56), x_j \partial_{x_j} - x_k \partial_{x_j} should be x_j \partial_{x_k} - x_k \partial_{x_j}.
  • Page 96?: In Exercise 2.64, (2.32) should be (2.44) (with F=0).
  • 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 177: In Exercise 3.55, the energy should be normalized by subtracting \varepsilon^{p+1} from (|u^{(\varepsilon)}|^2 + \varepsilon^2)^{(p+1)/2}.
  • 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 Xintian, Liu Xiao Chuan, Katie Marsden, Georg Meyer, Jason Murphy, Isaac Neal, Timothy Nguyen, Guilio Pasqualetti, Guillermo Reyley, Tristan Roy, Shuanglin Shao, Paul Smith, Elias Stein, Monica Visan, Haokun Xu, Chengbo Wang, Jason Zhao, Fan Zheng, Shijun Zheng, and Zuchong Zhi for corrections!