A continuation criterion for the Landau equation with very soft and Coulomb potentials

TL;DR

This work analyzes the spatially inhomogeneous Landau equation with very soft and Coulomb potentials ($\gamma\in[-3,-2]$) and proves a continuation criterion that requires only finiteness of the mass density, a small-velocity moment, and a velocity-space $L^p$ norm with $p>3/(5+\gamma)$, with no energy density bound assumed. The authors develop a novel blend of local Moser iteration, scaling arguments, and a nonlinear Gaussian barrier to obtain local and global $L^{\infty}$ bounds under conditional assumptions, and they establish Gaussian velocity bounds to control high-velocity behavior. They also connect these inhomogeneous results to the homogeneous Landau equation, discuss the sharpness of the $L^p_v$ condition, and show how the energy density bound can be bypassed through interpolation with velocity moments. The methods yield a pathway to improved large-data regularity and continuation criteria for the Landau equation in the Coulomb regime, with potential implications for kinetic theory in plasmas.

Abstract

We consider the spatially inhomogeneous Landau equation in the case of very soft and Coulomb potentials, $γ\in [-3,-2]$. We show that solutions can be continued as long as the following three quantities remain finite, uniformly in $t$ and $x$: (1) the mass density, (2) the velocity moment of order $s$ for any small $s>0$, and (3) the $L^p_v$ norm for any $p>3/(5+γ)$. In particular, we do not require a bound on the energy density.

Theorems & Definitions (25)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• proof