Nonlinear Kinetic Diffusion Equations with $p$-Growth

TL;DR

This paper proves local boundedness of solutions to nonlinear kinetic diffusion equations with p-growth, introducing kinetic Gagliardo-Nirenberg inequalities to relate Lebesgue norms across transport and diffusion directions.

Contribution

It establishes the local boundedness of solutions and develops kinetic Gagliardo-Nirenberg inequalities for equations with p-growth, a novel analytical tool.

Findings

Proved local boundedness of solutions to nonlinear kinetic diffusion equations.

Developed kinetic Gagliardo-Nirenberg inequalities for p-growth equations.

Abstract

We establish the local boundedness of (sub-)solutions to nonlinear kinetic diffusion equations with $p$-growth, where the kinetic p-Laplace equation is a prototypical example. A key ingredient is the derivation of kinetic Gagliardo-Nirenberg inequalities, where the Lebesgue norm of a function is estimated in terms of its transport and diffusive directions controlled in different Lebesgue spaces.