CBMS Regional Conference Series in Mathematics 106. Softcover, 373 pages.
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.
p. xibottom: “certain many” should be “certainly many”.
p. xiiShaunglin should be Shuanglin.
p. xiv$End(V \to W)$ should be $Hom(V \to W)$ throughout the text (e.g. on pages 33, 34). Frechet should be Fréchet.
p. xvFrechet should be Fréchet.
p. 1“is still its infancy” should be “is still in its infancy”.
p. 2“A Study in Scarlet” should be “A Scandal in Bohemia”.
p. 3In 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 ${\mathcal Y}$“.
p. 4In 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 ${\mathcal Y}$.
p. 5“a open interval” should be “an open interval”. $u_t = u^2$ should be $\partial_t u = u^2$.
p. 6In the definition of weak solution, $L^\infty$ should be $L^\infty_{loc}$.
p. 8In the proof of Theorem 1.4, $C^0(I \to \Omega_\varepsilon)$ should be $C^0(I \to {\mathcal D})$. Similarly on line 17.
p. 10In the last line of Exercise 1.1, G should be F.
p. 11In Exercise 1.4, S(t) should be $S_{t_0}$. “Kowaleski” should be “Kowalevski”.
p. 12In Theorem 1.12, $B$ should take values in ${\mathbb R}$ (and the hypothesis that $B, u$ arenon-negative should be dropped.)
p. 13$|F(u(s))-F(v(s))| \leq M |u(s)-v(s)|$ should be $\|F(u(s))-F(v(s))\|_{\mathcal D} \leq M \|u(s) - v(s) \|_{\mathcal D}$.
p. 16When defining $B(t)$, note that we can assume this quantity to be finite as the claim is vacuously true otherwise.
p. 17In Corollary 1.1, “for all $t \in {\mathbb R}$ should be $n \in {\mathbb Z}$.
p. 18In 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 $dt$. In Exercise 1.10, $S_{t_0}(u_0) = u(t_0)$ should be $S_{t_0}(t)(u_0) = u(t)$ where $u(t)$ solves the equation with initial condition $u(t_0)=u_0$.
p. 19In Exercise 1.15, $M_n({\mathbb C})$ should be $\hbox{End}(H)$.
p. 20In 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”.
p. 22“be such that such that” should be “be such that”.
p. 25In 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 $\langle F(v), dH_j(v)\rangle \geq 0$“.
p. 27In 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).
p. 28In 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.
p. 29In (1.33), there should be a minus sign on the RHS. Just before (1.34), $H(u(t_0))$ should be $H(u(t))$.
p. 30In (1.35), the $m_i$ should be on the denominator.
p. 31In Exercise 1.27, add the hypothesis that J is skew-adjoint. Also, $\nabla_\omega H = - J \nabla H$ should be $\nabla_\omega H = J \nabla H$.
p. 32In the 10th line from the bottom, Louville should be Liouville.
p. 33In Exercise 1.37, $G : C^0_{loc} \ldots$ should be $G \in C^0_{loc}\ldots$.
p. 34In Exercise 1.41, “exists real numbers” should be “exist real numbers”, and $z_d$ should be $z_n$.
p. 36in Example 1.31, $C \geq 2$ should be $0 < C \leq 2$.
p. 46In Exercise 1.46, $u$ and $u_0$ should be $x$ and $x_0$.
p. 40In 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.
p. 41In (1.42), $N(\|u\|_D)$ should be $\|N(u)\|_D$.
p. 45In 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”.
p. 46In 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 $e^{2\sigma t}$.
p. 47After the fourth display, “${\mathcal N}$ is bounded” should be “$N_k$ is bounded”.
p. 48In Exercise 1.51, $\varepsilon$ should equal $1 / (2k C_0 C_1)^{1/(k-1)}$ rather than $1/(2kC_0C_1)$. In (1.56), there should be summations over $\xi_1,\xi_2$.
p. 53In 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 $-P(u(t))$.
p. 54In Exercise 1.59, “Exercise 1.27” should be “Example 1.27”. In the last line (above the footnotes), ${\Bbb Z}^n$ should be ${\Bbb Z}^d$.
p. 57In 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.
p. 58In the “Conversely” portion of Exercise 2.2, one must assume the Lorenz gauge condition $\partial^\alpha A_\alpha = 0$.
p. 59In 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 $e^{-i m t |v|^2/ 2\hbar}$.
p. 60In the last display, $x_n$ should be $x_d$.
p. 61In 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 $- \frac{d-1}{2}-2$.
p. 62In the second paragraph of Section 2.1, $P:{\mathbb R}^d \to {\mathbb R}$ should be $h: {\mathbb R}^d \to {\mathbb R}$.
p. 63In the 8th line from the bottom, “propagator” should be “propagators”, and there is a semicolon missing in the preceding display.
p. 64In the definition of the spacetime Fourier transform, $dt dx$ should be $dx dt$. Similarly, in the inversion formula, $d\tau d\xi$ should be $d\xi d\tau$.
p. 65After 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”.
p. 66In Exercise 2.18, $\delta(\tau-|\xi|)$ should be $\delta(|\tau|-|\xi|)$. In the second to last display, the closing right parenthesis should be deleted.
p. 67In Exercise 2.19, the normalisation $\hbar = 1$ is missing. In the two-sheeted hyperboloid, $|\xi|2$ should be $|\xi|^2$.
p. 67bottom: “forall” should be “for all”.
p. 68In the hint for Exercise 2.24, $e^{ixz+itz^2}$ should be $e^{ixz-itz^2}$.
p. 70In the second display, $x/t^{-1/3}$ should be $x/t^{1/3}$.
p. 71Two lines before (2.19), $\lambda^n$ should be $\lambda^d$. In the first display, $|t|^{n-1}$ should be $|t|^{d-1}$.
p. 72In 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”.
p. 73In Exercise 2.33, $x_n$ should be $x_d$.
p. 74In (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”.
p. 75In the proof of Theorem 2.3, $q,q' \neq 2$ should be $q,r,\tilde q, \tilde r \neq 2$.
p. 76In Figure 1, the role of $1/r$ and $1/q$ should be interchanged. “Applying Holder’s inequality” should be “Applying Holder’s inequality twice”.
p. 77On 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 $N \to \infty$.)”.
p. 78In Figure 2, the role of $1/r$ and $1/q$ should be interchanged.
p. 80In 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 $\lim_{t \to \pm \infty} \|u(t) \|_{L^p_x}$.
p. 81In 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$|x_2|, \cdots, |x_n|\ll 1/\varepsilon$“.
p. 81-82In Exercise 2.46, the hypothesis $r \leq \infty$ should be replaced with $r 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.
p. 83two lines above (2.33), “transation” should be “translation”.
p. 84In 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”.
p. 85Before (2.40), $4 \pi$ should be $8 \pi$. In (2.40), $4\pi$ should be $2\pi$.
p. 87In 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”.
p. 92In the equation just below (2.54), $T^{00}(t,x)$ should be $T^{00}(0,x)$. In (2.54), $u(t.x)$ should be $u(t,x)$.
p. 94In the first display, $\int^{S_{d-1}}$ should be$\int_{S^{d-1}}$. In the second and third display, $t^{-1-d}$ should be $t^{1-d}$.
p. 94?: In the paragraph after (2.56), $x_j \partial_{x_j} - x_k \partial_{x_j}$ should be $x_j \partial_{x_k} - x_k \partial_{x_j}$.
p. 96?: In Exercise 2.64, (2.32) should be (2.44) (with $F=0$).
p. 99in 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 $\eta \in {\mathcal S}_t({\mathbf R})$.
p. 100In 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 $f_{\tau_0}$.
p. 101In 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 $\tau + \tau_0 - h(\xi)$.
p. 102The 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.
p. 102In 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 $m(\xi) \hat f(\xi)$.
p. 103In 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.
p. 104In 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.
p. 105In 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.
p. 106In 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 ${\mathbb T}^2$ should be ${\mathbb T}$.
p. 106In the hint for Exercise 2.75, add that the case where $d \geq 3$, $N \leq 2M$ and both functions are frequency supported where $|\xi| \leq 2M$ can also be handled using the linear $L_x^2 \to L_{t,x}^4$ estimates, which one can get from interpolating between the $L_x^2 \to L_{t,x}^\infty$ and $L^2 \to L_{t,x}^{\frac{2(d+2)}{d}}$ estimates.July 10, 2026
p. 107In 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”.
p. 109“defocusing, absent, or focusing” should be “focusing, absent, or defocusing”.
p. 110In the second paragraph, $F(zu)$ should equal $|z|^{p-1} z F(u)$ rather than $|z|^p F(u)$.
p. 112In 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”.
p. 113Before (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”.
p. 114In (3.10), $e^{it|v|^2/2}$ should be $e^{-it|v|^2/2}$.
p. 116In (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 “$-i \mu |\alpha|^{p-1}$“.
p. 117In (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 $e^{-i\mu|\alpha|^{p-1} t}$.
p. 119In the end of the first main paragraph, “if Principle 3.1” should be “of Principle 3.1”.
p. 120In 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”.
p. 122In the first paragraph, “show existence of solution” should be “show existence of a solution”. In Proposition 3.2, the final inequality should have exponents $p-1$ instead of $p$.
p. 123In 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.
p. 124the 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 $B(0,R)$.”
p. 125In 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”.
p. 129In the second-to-last line of the main text, “in one usually needs” should just be “one usually needs”.
p. 130In 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. For Remark 3.9, strictly speaking there is a technical issue establishing the Schwartz well-posedness purely from Proposition 3.8 due to the dependence of the local time of existence on k, but this can be fixed by using (a version of) the persistence of regularity result in Proposition 3.11.
p. 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.
p. 132In 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 $p-1$. In the final display of proof of Proposition 3.8, the iplied constants should depend on $s$ rather than $k$.
p. 133In Remark 3.12, the phrase “by Sobolev embedding” should be placed in parentheses and moved to before “and hence in”.
p. 134In Remark 3.14, “a critical controlling norms” should be “a critical controlling norm”.
p. 135In 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.
p. 136“$1 1/q'$” should be “$p/q < 1/q'$”.
p. 137In 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.
p. 138In (3.28), the $H^1$ norm should be on ${\Bbb R}^d$, not on $I \times {\Bbb R}^d$.
p. 139In the second to last display in the proof of Proposition 3.19, the exponent $5/2p$ should be $5(p-1)/2$.
p. 140In Figure 5, $H^1$ should be $\dot H^1$ in both appearances in the caption.
p. 141In the formula of Exercise 3.16, the $t_0$ in the LHS should be $t$.
p. 142In Exercise 3.18, “n” should be “d” throughout (for consistency with the rest of the text).
p. 144In the line before the first formula, “by by” should be “by”.
p. 145In Proposition 3.23, “some time interval” should be “the time interval”.
p. 146In 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.
p. 147A period is missing after Footnote 28.
p. 148second 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”.
p. 150“subcritical” should be “sub-critical”
p. 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 $\partial_j (\frac{1}{2} Im( \overline{\partial_{kk} u(t,x)} \partial_j u(t,x))$.
p. 152In exercise 3.35, the first appearance of “defocusing” should be omitted.
p. 153In the formula of Exercise 3.39, the $H^{k-1}$ norm shouldbe taken for $\partial_t u(t)$ but not $u(t)$.
p. 154fourth to last line: $\mu=-1$ should be $\mu=+1$.
p. 155In the paragraph before (3.36), “Morawetz inequalities for the NLS and NLW” should be “Morawetz inequalities for the Schrödinger and wave equations”.
p. 156After (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).
p. 157In (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 $\delta_{jk} - \frac{(x_j-y_j)(x_k-y_k)}{|x-y|^2}$.
p. 158In the first display, $\pi$ should be $2\pi$.
p. 159In 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.
p. 160After 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.
p. 161In 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 $\frac{2}{p+1}$.
p. 162In line 4 and 7, $p_d$ should be $p_{L^2_x}$.
p. 166$L^q_x$ should be $L^q_{t,x}$; similarly on (3.51) in page 167.
p. 167In 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 $\varepsilon^{2\alpha} \|u\|_{S^1(I \times {\mathbb R}^3)}^{3-2\alpha}$.
p. 168In 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 $u$“.
p. 169In 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.
p. 170In 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).
p. 171After (3.52), “small some suitable norms” should be “small in some suitable norms”.
p. 173In (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”.
p. 174In 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”.
p. 175In 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.
p. 176first line, “sufficiently small depending on t” should be “sufficiently small depending on $\psi$“.
p. 177In Exercise 3.55, the energy should be normalized by subtracting $\varepsilon^{p+1}$ from $(|u^{(\varepsilon)}|^2 + \varepsilon^2)^{(p+1)/2}$.
p. 178In the 9th line of the third paragraph, $e^{it \tau+\theta(t)}$ should be $e^{i (t \tau+\theta(t))}$.
p. 179In 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.
p. 180In 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.
p. 182In (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 $1+\sqrt{2}$.
p. 183After (3.73), Exercise 3.38 should be Exercise 3.35 and Exercise 3.39.
p. 184Before the first display, $|v|^2 \omega_\varepsilon$ should be $|\omega_\varepsilon|^2 \omega_\varepsilon$. In the last display, one should replace “p” by “3”.
p. 186In the quote, “Law” should not be capitalised.
p. 189After (3.74), “wellposednes” should be “wellposedness”.
p. 190In the penultimate display, the slash should be a period.
p. 191In 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.
p. 192In 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 $\|u(t)\|_{H^s_x}\lesssim_s \langle t \rangle^{(1-s) / (\frac{1}{2}-\frac{2(1-s)}{s})}$.
p. 198top: the reflection symmetry claimed for the KdV equation is incorrect and should be deleted.
p. 199In (4.7), $-5u^4$ should be $+5u^4$. In the bottom middle box, a right-parenthesis is missing. In footnote 4, the nonlinearity should have a minus sign.
p. 200In Exercise 4.2, $P(t)$ should be $4D^3 - 3(Du(t) + u(t) D)$, and $P(t) f$ should be $4 \partial_{xxx} f - 3 (\partial_x (u(t) f) + u(t) \partial_x f)$.
p. 204In the fourth equation of the second display, the final $L^\infty_x$ norm should be $L^2_x$.
p. 206In (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 $L^4_x L^\infty_t$.
p. 208Superfluous ) parenthesis on (4.18) and on the preceding equation, as well as the display two equations down.
p. 210In Exercise 4.13, $(\varepsilon t)^{-3/4}$ should be $(\varepsilon t)^{-3/8}$. The reference [Tzv] should be [Tzv2].
p. 235In the definition of the local energy $E_\Omega[u[t_0]]$, all occurrences of $t$ should be $t_0$.
p. 236In (5.5), the limit superior should be to $T_*$ rather than $T_*^+$.
p. 238In the last line of Proposition 5.6, insert “is the linear solution” before “with initial data”.
p. 240The 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 $\epsilon_2$).
p. 247In the third line of Theorem 5.1, $H^1 \times L^2$ should be $H^1$.
p. 249In the fourth display, $\eta^2$ should be $\eta^4$.
p. 254In the sixth to last line, “unexceptional” should be “exceptional”.
p. 261In the last paragraph above the exercises, $J - O_{E,\eta}(1)$ should be $\gtrsim_{E,\eta} J$.
p. 275In the first line after the display in Exercise 5.21, “$N(t)=1$” should be “$N(0)=1$“.
p. 280In (6.3), u should be $\phi$ (two occurrences). In equation (6.5), the $\frac{1}{2}$ should be outside the integral.
p. 281In the display after (6.7), a factor $\frac{1}{2}$ is missing from the right-hand side.
p. 283In Exercise 6.2(iii), one of the superscripts $\alpha$ should instead be a subscript.
p. 285In Exercise 6.6, the $Y \nabla_Y X$ term in the zero torsion property should just be $\nabla_Y X$.
p. 287In the last display of Exercise 6.13, $\psi$ should be $\Psi$.
p. 302In (6.35), $\tilde \phi_\phi$ should be $\tilde \phi+\phi$. In (6.36), $\partial_\beta A$ should be $\partial_\beta A_\alpha + [A_\alpha,A_\beta]$.
p. 334In (A.7), the condition “for $1 < p < \infty$” should be added.
p. 337In the upper bound on $\|P_N f\|_{L^p_x({\bf R}^d)$, the constants should also depend on $q, \eta$ and not just $d,p,s$.
p. 339second display: $++$ should be $+$. In the right-hand side of the fifth display, $L^2$ should be $H^s$. (The latter correction should also apply to the second line of the fourth display.)
p. 340equation (A.20): $N^{-2k}$ should be $N^{-k}$. In the last display, the $L^\infty_x$ norm should be $L^2_x$.
p. 341last 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.
p. 343Exercise 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 $\|F(f)-F(g)\|_{H^s_x({\mathbb R}^d)} \leq O_{\|f\|_{L^\infty_x({\mathbb R}^d)},\|g\|_{L^\infty_x({\mathbb R}^d)},F,V,s,d}(\|f-g\|_{H^s_x({\mathbb R}^d)})$ $+ O_{\|f\|_{H^s_x({\mathbb R}^d)},\|g\|_{H^s_x({\mathbb R}^d)}, \|f\|_{L^\infty_x({\mathbb R}^d)}, \|g\|_{L^\infty_x({\mathbb R}^d)},F,V,s,d}(\|f-g\|_{L^\infty_x({\mathbb R}^d)})$. In Exercise A.15, one needs to subtract the mean $\frac{1}{|I|} \int_I f$ from the first left-hand side as well as the second (and in the second the denominator should be $|I|$ rather than $I$.
p. 344Exercise A.18: The hypothesis that $u$ is spherically symmetric is missing. In the final part of the exercise, replace the hypothesis $R \geq 1$ by $R>0$.
p. 347The 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 $\frac{2}{p-1}(\frac{2}{p-1}+d)$.
p. 348Before (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 $\frac{2(p+1)}{d(p-1)} \|Q\|_{L^{p+1}}^{-p-1} \| \nabla Q \|_{L^2}^2$.
p. 349In 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 $x,N$)”.
p. 351In 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).
p. 352In the second display, $1/p+1$ should be $1/(p+1)$ for clarity. In the third display, $\sup_{n \to \infty}$ should be $\sup_{n \geq 1}$.
p. 353Proposition B.7: “Let Q be non-negative solution” should be “Let Q be a non-negative solution”.
p. 354Proposition 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 $t \xi - x_t$. $u_{t_n}(x_{t_n}) \leq 0$ should be $u_{t_n}(x_{t_n}) < 0$, and Exercise B.6 should be Exercise B.7. “from below” should be “from above”.
p. 355the second appearance of $u$ should be $u_y$.
p. 359In Exercise B.2, $O_k$ should be $O_{k,d}$, and the condition $|x| \geq 1$ should be added.
p. 360In the hint for Exercise B.3, $Q$ and $x \cdot \nabla_x Q$ should be $\overline{Q}$ and $x \cdot \nabla_x \overline{Q}$. Also, the suggested approach only works under the hypothesis that $\beta$ is non-negative; to rule out $\beta < 0$ requires some Agmon-Kato spectral theory and is beyond the scope of this text. In Exercise B.7, $\xi \cdot \nabla u(x)$ should be positive rather than negative. In Exercise B.5, $u$ should be defined on the closure of the unit ball.
p. 362A 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.
p. 365In reference [CS], “disperives” should be “dispersives”.