Mathematical Aspects of Nonlinear Dispersive Equations (AM-163)

Jean Bourgain
Carlos E. Kenig
S. Klainerman
Pages: 296
https://www.jstor.org/stable/j.ctt7s1f9

1. Front Matter
(pp. i-iv)
(pp. v-vi)
3. Preface
(pp. vii-viii)
4. Chapter One On Strichartz’s Inequalities and the Nonlinear Schrödinger Equation on Irrational Tori
(pp. 1-20)
J. Bourgain

Strichartz’s inequalities and the Cauchy problem for the nonlinear Schrӧdinger equation are considerably less understood when the spatial domain is a compact manifoldM, compared with the Euclidean situationM= ℝd. In the latter case, at least the theory of Strichartz inequalities (i.e., moment inequalities for the linear evolution, of the form$\parallel e^{it\Delta} \phi \parallel _{L_{x,t}^p} \leq C\parallel \phi \parallel_{L_x^2})$is basically completely understood and is closely related to the theory of oscillatory integral operators. LetM= 𝕋dbe a flat torus. IfMis the usual torus, i.e.,$(e^{it\Delta} \phi)(x) = \sum\limits_{n \in \mathbb{Z}^d } {\hat{\phi}}(n)e^{2\pi i(nx + |n|^2 t)}\quad (|n|^2 = n_1^2 + \cdots + n_d^2), \caption{(1.0.1)}$a partial Strichartz theory was developed in [B1], leading to the almost exact counterparts of...

5. Chapter Two Diffusion Bound for a Nonlinear Schrödinger Equation
(pp. 21-42)
J. Bourgain and W.-M. Wang

We study diffusion for a nonlinear lattice Schrödinger equation. This problem falls within the same general category of bounds on the higher Sobolev norms (H¹ and beyond) for the continuum nonlinear Hamiltonian PDE in a compact domain, e.g., a circle or a finite interval with Dirichlet boundary conditions; see, e.g., [B]. (Recall that typicallyL² norm is conserved. SoH¹ is the first nontrivial norm to consider.)

As in previous papers [BW1,2], the nonlinear random Schrödinger equation is the medium where we make the construction, the random variables (potential) being the needed parameters. Here we work with a slightly tempered...

6. Chapter Three Instability of Finite Difference Schemes for Hyperbolic Conservation Laws
(pp. 43-54)
A. Bressan, P. Baiti and H. K. Jenssen

Asystem of conservation lawsin one space dimension takes the form$u_{t} + f(u)_{x} = 0. \caption{(3.1.1)}$The components of the vectoru= (u₁,...,un) ∈ ℝnare theconserved quantities, while the components of the functionf= (f₁,...,fn): ℝn↦ ℝnare the correspondingfluxes. For smooth solutions, (3.1.1) is equivalent to the quasi-linear system$u_{t} + A(u)u_{x} = 0, \caption{(3.1.2)}$whereA(u) ≐Df(u) is then×nJacobian matrix of the flux functionf. We recall that the system isstrictly hyperbolicif this Jacobian matrixA(u) hasnreal distinct eigenvalues, λ₁(u) < ··· < λn(u) for everyu∈ ℝn. In this case,A(u) admits...

7. Chapter Four Nonlinear Elliptic Equations with Measures Revisited
(pp. 55-110)
H. Brezis, M. Marcus and A. C. Ponce

Let Ω ⊂ ℝNbe a bounded domain with smooth boundary. Letg: ℝ → ℝ be a continuous, nondecreasing function such thatg(0) = 0. In this paper we are concerned with the problem$\left\{ \begin{array}{rl} - \Delta u + g(u) = \mu & \text{in}\ \Omega,\\ u = 0 & \text{on}\ \partial \Omega, \\ \end{array} \right. \caption{(4.0.1)}$where μ is a measure. The study of (4.0.1) when μ ∈L¹(Ω) was initiated by Brezis-Strauss [BS]; their main result asserts that foreveryμ ∈L¹ andevery gas above, problem (4.0.1) admits a unique weak solution (see Theorem 4.B.2 in Appendix 4B below). The right concept of weak solution is the following:$\left\{ \begin{array}{l} u \in L^1 (\Omega), g(u) \in L^1 (\Omega )\ \text{and}\\ - \int_{\Omega} u\Delta \zeta + \int_{\Omega } g(u)\zeta = \int_{\Omega} \zeta\, d\mu \quad \forall \zeta \in C^2 (\bar{\Omega}), \zeta = 0\ \text{on}\ \partial \Omega. \\ \end{array} \right. \caption{(4.0.2)}$It will be convenient to write$C_0(\bar{\Omega}) = \{ \zeta \in C(\bar{\Omega});\zeta = 0\ \text{on}\ \partial \Omega \}$and$C_0^2 (\bar{\Omega}) = \{ \zeta \in C^2 (\bar{\Omega});\zeta = 0\ \text{on}\ \partial \Omega \}$, and...

8. Chapter Five Global Solutions for the Nonlinear Schrödinger Equation on Three-Dimensional Compact Manifolds
(pp. 111-130)
N. Burq, P. Gérard and N. Tzvetkov

Let (M,g) be a Riemannian compact manifold of dimensiond. In this paper we address global wellposedness of the Cauchy problem for the following nonlinear Schrödinger equation (NLS),$i\partial_t u + \Delta u = F(u),\quad u_{|t = 0} = u_0. \caption{(5.1.1)}$In (5.1.1),uis a complex valued function on ℝ ×Mandu₀ ∈Hs(M) forslarge enough. The nonlinear interactionFis supposed to be of the form$F = \frac{\partial V}{\partial \bar{z}}$withV∈ C(ℂ; ℝ) satisfying$V(e^{i\theta } z) = V(z),\quad \theta \in \mathbb{R}, z \in \mathbb{C}, \caption{(5.1.2)}$and, for some α > 1,$|\partial_z^{k_1} \partial_{\bar{z}}^{k_2}\, V(z)| \leq C_{k_1,k_2}\,(1 + |z|)^{1 + \alpha - k_1 - k_2}. \caption{(5.1.3)}$The number α involved in the second condition onVcorresponds to the “degree” of the nonlinearityF(u) in (5.1.1). Moreover, we make the following defocusing assumption:...

9. Chapter Six Power series solution of a nonlinear Schrödinger equation
(pp. 131-156)
M. Christ

The Cauchy problem for the one-dimensional periodic cubic nonlinear Schrödinger equation is

$\left\{ \begin{array}{l} iu_t + u_{xx} + \omega |u|^2 u = 0 \\ u(0,x) = u_0 (x), \\ \end{array} \right.\caption{(NLS)}$wherex∈ 𝕋 = ℝ/2πℤ,t∈ ℝ, and the parameter ω equals ±1. Bourgain [2] has shown this problem to be wellposed in the Sobolev spaceHsfor alls≥ 0, in the sense of uniformly continuous dependence on the initial datum. InH⁰ it is wellposed globally in time, and as is typical in this subject, the uniqueness aspect of wellposedness is formulated in a certain auxiliary space more restricted thanC⁰([0,T],Hs(𝕋)), in which existence is also established. Fors< 0 it...

10. Chapter Seven Eulerian-Lagrangian Formalism and Vortex Reconnection
(pp. 157-170)
P. Constantin

Incompressible Newtonian fluids are described by the Navier-Stokes equations, the viscous regularization of the friction-free, incompressible Euler equations$D_t u + \nabla p = 0,\quad \nabla \cdot u = 0. \caption{(7.1)}$The velocityu=u(x,t) = (u₁,u₂,u₃) is a function of$x \in \mathbf{R}^{3}$and$t \in \mathbf{R}$. The material derivative associated to the velocityuis$D_t = D_t (u, \nabla) = \partial_t + u \cdot \nabla. \caption{(7.2)}$The Euler equations are conservative, meaning that no dissipation of energy occurs during smooth evolution. The total energy is proportional to theL² norm of velocity. The Onsager conjecture ([22], [17]) states that conservation of energy happens if and only if the solutions are smoother than required by the classical Kolmogorov turbulence theory (roughly speaking, Hölder...

11. Chapter Eight Long Time Existence for Small Data Semilinear Klein-Gordon Equations on Spheres
(pp. 171-180)
J.-M. Delort and J. Szeftel

Consider (M,g) a Riemannian manifold and denote by Δg(resp. ∇g) the Laplacian (resp. the gradient) associated tog. Letm> 0 be given and consider a local solution u to the Cauchy problem\begin{align*} (\partial_t^2 - \Delta_g + m^2)u &= f(x,u,\partial_t u,\nabla_g u) \\ u|_{t = 0} &= \epsilon u_0 \\ \partial_t u|_{t = 0} & = \epsilon u_1,\\ \end{align*} \caption{(8.1)}p wherefis a real polynomial in (u, ∂tu, ∇gu), vanishing at some orderp≥ 2 at 0, withCcoefficients inx, where the data (u₀,u₁) are in$C_0^\infty (M)$, real valued, and where ϵ > 0 goes to zero. Denote by ]T*(ϵ),T*(ϵ)[ (withT*(ϵ) < 0 <T*(ϵ)) the maximal interval of existence of a smooth solution to (8.1). We aim at giving lower bounds...

12. Chapter Nine Local and Global Wellposedness of Periodic KP-I Equations
(pp. 181-212)
A. D. Ionescu and C. E. Kenig

Let 𝕋 = ℝ/(2πℤ). In this paper we consider the Kadomstev-Petviashvili initial value problems$\left\{ \begin{array}{l} \partial_t u + P(\partial_x)u - \partial_{x}^{ - 1} \partial_{y}^2 u + u\partial_x u = 0; \\ u(0) = \phi, \\ \end{array} \right. \caption{(9.1.1)}$on 𝕋 × 𝕋 and ℝ × 𝕋, where$p(\partial_x) = \partial_x^3$(the third-order KP-I) or$P(\partial_x) = - \partial_x^5$(the fifth-order KP-I). Our goal is to prove local and global wellposedness theorems for the third-order KP-I initial value problem on 𝕋 × 𝕋 and ℝ × 𝕋, and for the fifth-order KP-I initial value problem on ℝ × 𝕋.

KP-I equations, as well as KP-II equations in which the sign of the term$\partial_x^{-1} \partial_y^2 u$in (9.1.1) is + instead of − arise naturally in physical contexts as models for the...

13. Chapter Ten The Cauchy Problem for the Navier-Stokes Equations with Spatially Almost Periodic Initial Data
(pp. 213-222)
Y. Giga, A. Mahalov and B. Nicolaenko

We consider the Cauchy problem for the Navier-Stokes equations (n≥ 2):$\partial_t u - \nu \Delta u + (u,\nabla)u + \nabla p = 0, \quad \text{div}\,u = 0\quad \text{in}\quad {\mathbf{R}}^n \times (0,T), \caption{(10.1.1)}$$u|_{t = 0} = u_0 \quad (\text{div}\,u_0 = 0)\quad \text{in}\quad {\mathbf{R}}^n, \caption{(10.1.2)}$when the initial datau₀ is spatially nondecreasing, in particular almost periodic. We use a standard convention of notation;u(x,t) = (u¹(x,t),...,un(x,t)) represents the unknown velocity field whilep(x,t) represents the unknown pressure field; ν > 0 denotes the kinematic viscosity and$\partial_t u = \partial u/\partial t,(u,\nabla) = \sum\nolimits_{i = 1}^n u^i \partial _{x_i},\partial_{x_i } = \partial /\partial x_i,\nabla p = (\partial_{x_1} p, \ldots,\partial_{x_n} p)$withx= (x₁,...,xn).

It is by now well known ([CK], [Ca], [CaM], [GIM], [KT]) that the problem (10.1.1)–(10.1.2) admits a local in time-classical solution for any bounded initial data. It is unique under an extra assumption for pressure. It is also well known [GMS]...

14. Chapter Eleven Longtime Decay Estimates for the Schrödinger Equation on Manifolds
(pp. 223-254)
I. Rodnianski and T. Tao

Let (M,g) = (R³,g) be a compact perturbation of Euclidean space¹ R³, thusMis R³ endowed with a smooth metricgwhich equals the Euclidean metric outside of a Euclidean ballB(0,R₀) := {x∈ R³ : |x| ≤R₀} for some fixedR₀. We consider smooth solutions to the Schrödinger equation$u_t = -iHu, \caption{(11.1)}$where for each timet,u(t) :M→ C is a Schwartz function onM, andHis the Hamiltonian operator$H := - \frac{1}{2}\Delta_M,$where ΔM:= ∇jjis the Laplace-Beltrami operator (with ∇jdenoting covariant differentiation with respect to the Levi-Civita connection, in contrast with the Euclidean...

15. Chapter Twelve Dispersive Estimates for Schrödinger operators: A survey
(pp. 255-286)
W. Schlag

The purpose of this note is to give a survey of some recent work on dispersive estimates for the Schrödinger flow$e^{itH} P_c,\quad H = - \Delta + V\ \text{on}\ \mathbb{R}^d, d \geq 1, \caption{(12.1)}$wherePcis the projection onto the continuous spectrum ofH.Vis a real-valued potential that is assumed to satisfy some decay condition at infinity. This decay is typically expressed in terms of the point-wise decay |V(x)| ≤Cx, for allx∈ ℝdand some β > 0. Throughout this paper,$\langle x \rangle = (1 + |x|^2 )^{\frac{1}{2}}$. Occasionally, we will use an integrability conditionVLp(ℝd) (or a weighted variant thereof) instead of a point-wise condition. These decay conditions will...

16. Contributors
(pp. 287-290)
17. Index
(pp. 291-300)