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Probabilistic well-posedness for the nonlinear wave equation on $B_2\times\mathbb{T}$

Aynur Bulut

Abstract

We establish probabilistic well-posedness results for the subcubic nonlinear wave equation, posed on the domain $B_2\times\mathbb{T}$, with randomly chosen initial data having radial symmetry in the $B_2$ variable, and with vanishing Dirichlet boundary conditions on $\partial B_2\times\mathbb{T}$.

Probabilistic well-posedness for the nonlinear wave equation on $B_2\times\mathbb{T}$

Abstract

We establish probabilistic well-posedness results for the subcubic nonlinear wave equation, posed on the domain , with randomly chosen initial data having radial symmetry in the variable, and with vanishing Dirichlet boundary conditions on .

Paper Structure

This paper contains 8 sections, 6 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and function spaces
  3. Function spaces
  4. Strichartz estimates
  5. Probabilistic estimates of the linear evolution
  6. Estimates of the nonlinearity
  7. Local well-posedness: contraction in $X^{s,b}$ spaces
  8. The choice of $s$ and $p$ in the proof of Theorem $\ref{['ref10']}$

Key Result

Theorem \oldthetheorem

Fix $0<\gamma<2$ and let $\alpha\in\mathbb{R}$ be such that Let $(F,G)\in H_x^\alpha(B_2\times\mathbb{T})\times H_x^{\alpha-1}(B_2\times\mathbb{T})$ be a pair of real-valued functions whose Fourier series representations are as in eq-fg. For $\omega\in \Omega$, let $(F_\omega,G_\omega)$ be the randomized pair defined in ref2--ref3. Moreover, for $(n,n')$ and for some $s\in [\frac{1}{2},1]$ (depen

Theorems & Definitions (12)

  • Theorem \oldthetheorem: Local well-posedness for (NLW)
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof : Proof of Lemma $\ref{['ref31']}$
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • ...and 2 more