$C^{\alpha}$ regularity of weak solutions of non-homogenous ultraparabolic equations with drift terms

TL;DR

This paper proves that weak solutions to certain non-homogeneous ultraparabolic equations with drift terms are H"{o}lder continuous, extending classic parabolic regularity results using weak Poincaré inequalities and Moser iteration.

Contribution

It establishes H"{o}lder continuity for weak solutions of non-homogeneous ultraparabolic equations with drift, generalizing classical results for second-order parabolic equations.

Findings

Weak solutions are H"{o}lder continuous.

Uses weak Poincaré inequality and Moser iteration techniques.

Extends regularity results to non-homogeneous ultraparabolic equations.

Abstract

Consider a class of non-homogenous ultraparabolic differential equations with drift terms or lower order terms arising from some physical models, and we prove that weak solutions are H\"{o}lder continuous, which also generalizes the classic results of parabolic equations of second order. The main ingredients are a type of weak Poincar\'{e} inequality satisfied by non-negative weak sub-solutions and Moser iteration.