Critical velocity averaging lemmas

TL;DR

This paper introduces new velocity averaging lemmas for multi-dimensional hyperbolic-parabolic PDEs, enabling compactness results for deterministic and stochastic convection-diffusion equations, even with complex source terms.

Contribution

The paper presents novel velocity averaging lemmas that handle critical source terms including second-order derivatives and stochastic noise, advancing PDE analysis.

Findings

Established new compactness results for convection-diffusion equations

Extended averaging lemmas to stochastic PDEs with complex sources

Demonstrated applicability to hyperbolic-parabolic PDEs

Abstract

We prove new velocity averaging lemmas for multi-dimensional hyperbolic-parabolic partial differential equations. These theorems may be applied to establish several compactness results for both deterministic and stochastic convection-diffusion equations. Among the strengths of our theory is the criticality of the source term, which may include spatial derivatives of second order and stochastic noises.