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Global well-posedness and stability of the inhomogeneous kinetic wave equation near vacuum

Ioakeim Ampatzoglou

Abstract

In this paper, we prove global in time existence, uniqueness and stability of mild solutions near vacuum for the 4-wave inhomogeneous kinetic wave equation, for Laplacian dispersion relation in dimension $d=2,3$. We also show that for non-negative initial data, the solution remains non-negative. This is achieved by connecting the inhomogeneous kinetic wave equation to the cubic part of a quantum Boltzmann-type equation with moderately hard potential and no collisional averaging.

Global well-posedness and stability of the inhomogeneous kinetic wave equation near vacuum

Abstract

In this paper, we prove global in time existence, uniqueness and stability of mild solutions near vacuum for the 4-wave inhomogeneous kinetic wave equation, for Laplacian dispersion relation in dimension . We also show that for non-negative initial data, the solution remains non-negative. This is achieved by connecting the inhomogeneous kinetic wave equation to the cubic part of a quantum Boltzmann-type equation with moderately hard potential and no collisional averaging.
Paper Structure (13 sections, 7 theorems, 82 equations)

This paper contains 13 sections, 7 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. The inhomogeneous kinetic wave equation and functional spaces
  3. Backround
  4. Connection with the quantum Boltzmann equation
  5. Statement of the main results
  6. Strategy of the proofs
  7. Gain and loss operators and the notion of a mild solution
  8. Gain and loss operators
  9. The transport operator
  10. Notion of a mild solution
  11. A-priori estimates
  12. Well-posedness and stability for arbitrary data near vacuum
  13. Existence of a non-negative solution

Key Result

Theorem 1.1

Let $\alpha,\beta>0$ and $0<R\le\frac{\alpha^{1/4}}{4\sqrt{6}K_{d,\beta}^{1/2}}$, where $K_{d,\beta}>0$ is the constant given in K_beta text. Let $f_0\in\mathcal{M}_{\alpha,\beta}$ with $\|f_0\|\leq R$. Then equation KWE has a unique mild solution $f$ satisfying the bound Additionally, if $f_0,g_0\in\mathcal{M}_{\alpha,\beta}$ with $\|f_0\|, \|g_0\|\leq R$, and $f,g$ are the corresponding mild so

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 7 more