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Upper bound of higher-order Sobolev norms for Zkaharov system in one space dimension

Nobutatsu Kobayashi

Abstract

We study the Cauchy problem for the Zakharov system in one space dimension with the Diriclet boundary conditions. We establish the global well-posedness and the growth of higher-order Sobolev norms of solutions to the Zakharov system by using the modified energy method.

Upper bound of higher-order Sobolev norms for Zkaharov system in one space dimension

Abstract

We study the Cauchy problem for the Zakharov system in one space dimension with the Diriclet boundary conditions. We establish the global well-posedness and the growth of higher-order Sobolev norms of solutions to the Zakharov system by using the modified energy method.
Paper Structure (5 sections, 16 theorems, 82 equations)

This paper contains 5 sections, 16 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Construction of approximate solutions
  3. Uniform boundedness of the sequence of approximate solutions
  4. Convergence of the sequence of approximate solutions
  5. Proof of the main theorem

Key Result

Theorem 1.1

Let $(u_0,v_0,w_0)\in (H^2\cap H_0^1)(I)\times H_0^1(I)\times H_0^1(I)$. Then there exists the unique solution $(u,v,w)$ of eq:(1.2) such that Moreover, the following properties hold:

Theorems & Definitions (28)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • proof
  • ...and 18 more