Transport equations in H\"older space by vanishing viscosity and applications

TL;DR

This paper establishes sharp H"older continuity results for transport equations using vanishing viscosity methods, and extends these results to related parabolic and Burgers' equations, ensuring existence and uniqueness under certain conditions.

Contribution

It introduces a novel vanishing viscosity approach to achieve sharp H"older regularity and proves existence and uniqueness of solutions under specific structural assumptions.

Findings

Sharp H"older continuity for transport equations

Extension of results to parabolic and Burgers' equations

Existence and uniqueness of solutions under structural hypotheses

Abstract

We obtain a sharp limit H\"older continuity of the solution for the transport equations thanks to a vanishing viscosity analysis. We also derive the same control for parabolic equations and for inviscid Burgers' equation. Eventually, under a structural hypothesis on the coefficients, we provide existence and uniqueness of a H\"older continuous solution.