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Local-in-time strong solutions of the homogeneous Landau-Coulomb equation with $L^p$ initial datum

William Golding, Amélie Loher

Abstract

We consider the homogeneous Landau equation with Coulomb potential and general initial data $f_{in} \in L^p$, where $p$ is arbitrarily close to $3/2$. We show the local-in-time existence and uniqueness of smooth solutions for such initial data. The constraint $p > 3/2$ has appeared in several related works and appears to be the minimal integrability assumption achievable with current techniques. We adapt recent ODE methods and conditional regularity results appearing in [arXiv:2303.02281] to deduce new short time $L^p \to L^\infty$ smoothing estimates. These estimates enable us to construct local-in-time smooth solutions for large $L^p$ initial data, and allow us to show directly conditional regularity results for solutions verifying \emph{unweighted} Prodi-Serrin type conditions. As a consequence, we obtain additional stability and uniqueness results for the solutions we construct.

Local-in-time strong solutions of the homogeneous Landau-Coulomb equation with $L^p$ initial datum

Abstract

We consider the homogeneous Landau equation with Coulomb potential and general initial data , where is arbitrarily close to . We show the local-in-time existence and uniqueness of smooth solutions for such initial data. The constraint has appeared in several related works and appears to be the minimal integrability assumption achievable with current techniques. We adapt recent ODE methods and conditional regularity results appearing in [arXiv:2303.02281] to deduce new short time smoothing estimates. These estimates enable us to construct local-in-time smooth solutions for large initial data, and allow us to show directly conditional regularity results for solutions verifying \emph{unweighted} Prodi-Serrin type conditions. As a consequence, we obtain additional stability and uniqueness results for the solutions we construct.
Paper Structure (10 sections, 11 theorems, 144 equations)

This paper contains 10 sections, 11 theorems, 144 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Outline of the Paper
  4. Preliminaries
  5. Propagation of Lp norms
  6. Quantitative Short Time Smoothing
  7. Short time existence
  8. Proof of Theorem \ref{['thm:main']}
  9. Continuation Criterion
  10. Uniqueness

Key Result

Theorem 1.1

Fix $p > \frac{3}{2}$ and let $m >\frac{9}{2}\frac{p-1}{p - \frac{3}{2}}$. For any given $M$, $H$ and $f_{in} \in L^1_m \cap L^p({\mathbb R}^3)$ satisfying there exists $0 < T = T(p, m, M, H)$ such that equation eq:landau admits a smooth solution on $(0,T) \times {\mathbb R}^3$ with initial datum $f_{in}$ in the sense that $\lim_{t \rightarrow 0^+} \left\|f(t) - f_{in}\right\|_{L^p} = 0$. This so

Theorems & Definitions (14)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 2.1: FromHendersonSnelsonTarfulea
  • Lemma 2.2: From BedrossianGualdaniSnelson, GoldingGualdaniZamponi, Silvestre
  • Lemma 2.3: From CarrapatosoDesvillettesHe
  • Lemma 2.4
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • ...and 4 more