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= ℝ^d. 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= 𝕋^dbe 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...

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...

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,...,u_n) ∈ ℝⁿare theconserved quantities, while the components of the functionf= (f₁,...,f_n): 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) < ··· < λ_n(u) for everyu∈ ℝⁿ. In this case,A(u) admits...

Let Ω ⊂ ℝ^Nbe 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...

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 ∈H^s(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:...

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 spaceH^sfor 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],H^s(𝕋)), in which existence is also established. Fors< 0 it...

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...

Consider (M,g) a Riemannian manifold and denote by Δ_g(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, withC^∞coefficients 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...

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...

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,...,x_n).

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]...

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 Δ_M:= ∇^j∇_jis the Laplace-Beltrami operator (with ∇^jdenoting covariant differentiation with respect to the Levi-Civita connection, in contrast with the Euclidean...

The purpose of this note is to give a survey of some recent work on dispersive estimates for the Schrödinger flow\[e^{itH} P_c,\quad H = - \Delta + V\ \text{on}\ \mathbb{R}^d, d \geq 1, \caption{(12.1)}\]whereP_cis 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)| ≤C〈x〉^-β, for allx∈ ℝ^dand some β > 0. Throughout this paper,$\langle x \rangle = (1 + |x|^2 )^{\frac{1}{2}}$. Occasionally, we will use an integrability conditionV∈L^p(ℝ^d) (or a weighted variant thereof) instead of a point-wise condition. These decay conditions will...