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Regularizing effects in a linear kinetic equation for cubic interactions

Miguel Escobedo

Abstract

We describe regularizing effects in the linearization of a kinetic equation that arises in study of a system of nonlinear waves satisfying the Schrödinger equation in terms of weak turbulence and condensate. The problem is first considered in spaces of bounded functions with weights, where existence of solutions and some first regularity properties are proved. After a suitable change of variables the equation is written in terms of a pseudo differential operator. Homogeneity of the equation and classical arguments of freezing of coefficients may then be used to prove regularizing effect in local Sobolev type spaces.

Regularizing effects in a linear kinetic equation for cubic interactions

Abstract

We describe regularizing effects in the linearization of a kinetic equation that arises in study of a system of nonlinear waves satisfying the Schrödinger equation in terms of weak turbulence and condensate. The problem is first considered in spaces of bounded functions with weights, where existence of solutions and some first regularity properties are proved. After a suitable change of variables the equation is written in terms of a pseudo differential operator. Homogeneity of the equation and classical arguments of freezing of coefficients may then be used to prove regularizing effect in local Sobolev type spaces.
Paper Structure (21 sections, 21 theorems, 464 equations)

This paper contains 21 sections, 21 theorems, 464 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['S7cor2']}.
  3. The pseudo differential operator $P$.
  4. Freezing the coefficient in $P$.
  5. Regularising result in $\xi, w$ variables.
  6. Localization.
  7. Proof of Theoremm. \ref{['S2Th3.1']}, (i).
  8. Estimate of $g_3$.
  9. The term $g_4$.
  10. Estimate of $g_2$.
  11. Proof of Theorem. \ref{['S2Th3.1']}, (ii).
  12. Estimate of $g_1$.
  13. Estimate of $g_3$.
  14. Estimate of $g_4$ and $g_2$.
  15. Proof of Theorem. \ref{['S2Th3.1']}, (iii).
  16. ...and 6 more sections

Key Result

Theorem 1.1

Suppose that $\nu \in C ((0, T); X _{\theta, \rho })\cap L^\infty((0, T); X _{ \theta, \rho })$ for some $T>0$, $\theta\ge 0$, $\rho >0$ such that $\theta+\rho \in (0, 3/2)$ and consider the function $v$ defined as Then, for all $t\in (0, T)$ and $\theta'$, $\rho '$ satisfying (thetaP),

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • ...and 35 more