Upper bounds for parabolic equations and the Landau equation

TL;DR

This paper establishes upper bounds for solutions to certain parabolic equations and applies these results to derive $L^ty$ estimates for the Landau equation with soft potentials, depending only on initial mass, energy, and entropy.

Contribution

It introduces new upper bounds for parabolic equations with nonlinearities and applies them to obtain $L^ty$ estimates for the Landau equation, advancing understanding of its regularity.

Findings

Derived an $L^p$ weighted upper bound for parabolic equations.

Established $L^ty$ bounds for the Landau equation solutions.

Results depend only on initial mass, energy, and entropy.

Abstract

We consider a parabolic equation in nondivergence form, defined in the full space $[0,\infty) \times \mathbb R^d$, with a power nonlinearity as the right hand side. We obtain an upper bound for the solution in terms of a weighted control in $L^p$. This upper bound is applied to the homogeneous Landau equation with moderately soft potentials. We obtain an estimate in $L^\infty(\mathbb R^d)$ for the solution of the Landau equation, for positive time, which depends only on the mass, energy and entropy of the initial data.