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, Schrodinger, 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 xii: Shaunglin should be Shuanglin.
  • 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”.
  • Page 8: In the proof of Theorem 1.4, C^0(I \to \Omega_\varepsilon) should be C^0(I \to {\mathcal D}).
  • 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}.
  • Page 17: In Corollary 1.1, “for all t \in {\bf R} should be n \in {\bf Z}.
  • 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)).
  • 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.
  • 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 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 48: In Exercise 1.51, \varepsilon should equal 1 / (2k C_0 C_1)^{1/(k-1)} rather than 1/(2kC_0C_1).
  • Page 54: In the “Conversely” portion of Exercise 2.2, one must assume the Lorenz gauge condition \partial^\alpha A_\alpha = 0.
  • Page 55: For the Schrodinger equation, the phase velocity is half the group velocity rather than twice the group velocity (i.e. v/2 instead of 2v). 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 59: Exercise 2.4 the second line should be L:= ih(D), 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 65: After Principle 2.1, \hbar \xi/v should be \hbar \xi/m.

    • In the last paragraph, “thi principle” should be “this principle”.

  • Page 71: Two lines before (2.19), \lambda^n should be \lambda^d.
  • Page 72: In Exercise 2.28, there should be a complex conjugate for \hat{u_0} in the second display. The Laplacian \Delta in the third display should be \frac{\Delta}{2}. v(0,x) should equal \frac{1}{(2\pi)^d/2} \hat{u_0}(-x) rather than \frac{1}{(2\pi)^d/2} \hat{u_0}(x). In Exercise 2.30, “Airy function” should be “Airy equation”.
  • Page 74: In (2.26), ds should be dt'.
  • 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.
  • Page 77: In the second display, e^{-i s \Delta/2} should be e^{i<br /> (t-s) \Delta/2}.
  • 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″.
  • Page 81: In Exercise 2.43, the space-time domain “
    |t|\ll 1/\varepsilon, |x_1-t|\ll \varepsilon and |x_2|, \cdots,<br /> |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 Schrodinger-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 85: At and before (2.40), 4 \pi should be 8 \pi.
  • Page 92: In the equation just below (2.54), T^{00}(t,x) should be T^{00}(0,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.
  • Page 100: In the first line, “s'\le s and b'\le b” should be “s'\ge s and b'\ge b“. In the last two displays, f_\tau should be
    f_{\tau_0}.
  • Page 102: In the last line of fourth display, the X^{s,b} norm
    should be X^{0,b}.
  • 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}.
  • 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.
  • 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: 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 paragraph after equation (3.5) anticlockwise should be clockwise.
  • 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), “(-\mu, \tau^2) for NLW” should be “(-\mu, -\tau^2) for NLW”. After (3.8), “defocusing” should be “focusing”.
  • 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 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.
  • Page 124: the second line after the proof of Proposition 3.3, “one
    and nonlinearities” should be “and nonlinearities one”.
  • Page 125: In the second line of Definition 3.4, “u_0^*\in R^d“
    should be “u_0^*\in H^s_x (R^d)“.
  • 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)}.
  • 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.
  • 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.
  • Page 141: In the formula of Exercise 3.16, the t_0 in the LHS should be
    t.
  • 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.
  • Page 148: second paragraph after Principle 2.34, last line “n>6″ should be “d>6″
  • 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 should
    be taken for \partial_t u(t) but not u(t).
  • Page 154, fourth to last line: \mu=-1 should be \mu=+1.
  • 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 159: 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-critical
    should be H^{1/2}_x-critical. In the third formula of this page, the minus sign should not occur.
  • 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: W^{1,10/3} should be W^{1,10/3}_x.
  • Page 168: In the second formula of this page, the denominator should
    be 2d rather than 4d. In the statement and proof of Proposition 3.32, R^3 should be R^2 (three occurrences).
  • 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 173: In (3.55), (3.56) and the second line before (3.55), four
    occurrences of the exponent 2 should be p-1.
  • Page 174: In the first paragraph, (3.55) should be (3.56). In the second formula, 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″.
  • 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“.
  • 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 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 first
    display and “\max (\omega^{p}, \lambda^4 \omega^2)“, respectively.
  • 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.
  • 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 191: In the fourth display, m(\xi-\eta N) should be m(\frac{\xi-\eta}{ N}).
  • 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 193: In (4.7), -5u^4 should be +5u^4.
  • Page 208: Superfluous ) parenthesis on (4.18) and on the preceding equation, as well as the display two equations down.
  • Page 220: 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 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 282: 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.
  • 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 final display, N^s M^{-s} should be N^{2s} M^{-2s}, and L^2 should be H^s.
  • Page 340, equation (A.20): N^{-2k} should be N^{-k}.
  • 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.)
  • Page 344, Exercise A.18: The hypothesis that u is spherically symmetric is missing.
  • Page 348: Before (B.3): “a positive and finite” should be “positive and finite”.
  • 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.

Many thanks to Danny Goodman, Zaher Hani, Soonsik Kwon, Liu Xiao Chuan, Timothy Nguyen, Tristan Roy, Shuanglin Shao, Paul Smith, and Chengbo Wang for corrections!