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The Zakharov system in dimension $d \geqslant 4$

Timothy Candy, Sebastian Herr, Kenji Nakanishi

Abstract

The sharp range of Sobolev spaces is determined in which the Cauchy problem for the classical Zakharov system is well-posed, which includes existence of solutions, uniqueness, persistence of initial regularity, and real-analytic dependence on the initial data. In addition, under a condition on the data for the Schrödinger equation at the lowest admissible regularity, global well-posedness and scattering is proved. The results cover energy-critical and energy-supercritical dimensions $d \geqslant 4$.

The Zakharov system in dimension $d \geqslant 4$

Abstract

The sharp range of Sobolev spaces is determined in which the Cauchy problem for the classical Zakharov system is well-posed, which includes existence of solutions, uniqueness, persistence of initial regularity, and real-analytic dependence on the initial data. In addition, under a condition on the data for the Schrödinger equation at the lowest admissible regularity, global well-posedness and scattering is proved. The results cover energy-critical and energy-supercritical dimensions .

Paper Structure

This paper contains 23 sections, 20 theorems, 308 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Notation and Preliminaries
  4. Fourier multipliers
  5. Function spaces
  6. Duhamel formulae and energy inequalities.
  7. A product estimate for fractional derivatives
  8. Decomposability of norms
  9. Bilinear Estimates for Schrödinger nonlinearity
  10. Bilinear estimates for the wave nonlinearity
  11. Simplified small data global and large data local theory
  12. Local bilinear estimates in the endpoint case
  13. Well-posedness results
  14. Global well-posedness for the model problem
  15. Local and small data global results for the Zakharov system
  16. ...and 8 more sections

Key Result

Theorem 1.1

The Zakharov system eq:Zakharov with initial condition eq:ic is locally well-posed with a real-analytic flow map, if and only if $(s,\ell)\in \mathbb{R}^2$ satisfies eqn:cond on s l.

Figures (1)

  • Figure 1: In dimension $d=4$: Local well-posedness and small data global well-posedness within grey region, ill-posedness otherwise.

Theorems & Definitions (41)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: Nested embeddings
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 31 more