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Pointwise convergence of the Klein-Gordon flow

Renato Lucà, Pablo Merino

Abstract

We consider the PDEs version of the Carleson problem in the context of the cubic nonlinear Klein-Gordon equation. This means that we aim to establish the lowest regularity class for which one has almost everywhere pointwise convergence of the solutions to the initial data, as $t \to 0$. We prove sharp results for initial data in Sobolev spaces and for their randomized counterparts.

Pointwise convergence of the Klein-Gordon flow

Abstract

We consider the PDEs version of the Carleson problem in the context of the cubic nonlinear Klein-Gordon equation. This means that we aim to establish the lowest regularity class for which one has almost everywhere pointwise convergence of the solutions to the initial data, as . We prove sharp results for initial data in Sobolev spaces and for their randomized counterparts.
Paper Structure (9 sections, 22 theorems, 154 equations)

This paper contains 9 sections, 22 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Strategy of the proof
  4. The Cowling convergence theorem
  5. The Maximal approach to nonlinear pointwise convergence
  6. Deterministic preliminaries
  7. Proof of Theorem \ref{['MainThm1']}
  8. Probabilistic preliminaries
  9. Proof of Theorem \ref{['MainThm2']}

Key Result

Theorem 1.1

Let $s > 1/2$ and $(u_0, u_1) \in H^{s}(\mathbb{T}^3) \times H^{s-1}(\mathbb{T}^3)$. Let $u$ the (local in time) solution of the Cauchy problem nlwcauchy. We have

Theorems & Definitions (36)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • Theorem 4.1
  • proof
  • Lemma 5.1
  • proof
  • Lemma 5.2
  • Lemma 6.1
  • Lemma 6.2
  • ...and 26 more