The H\"older continuity of a class of 3-dimension ultraparabolic equations

TL;DR

This paper proves H"older continuity for weak solutions of a class of 3D ultraparabolic equations with measurable coefficients, extending previous results on kinetic Fokker-Planck equations.

Contribution

It establishes $C^eta$ regularity for solutions of a broader class of ultraparabolic equations with measurable coefficients.

Findings

Proved $C^eta$ regularity for weak solutions.

Generalized results to a wider class of ultraparabolic equations.

Extended previous work on kinetic Fokker-Planck equations.

Abstract

We obtained the $C^\alpha$ continuity for weak solutions of a class of ultraparabolic equations with measurable coefficients of the form $${\partial_t u} = \partial_x(a(x,y,t)\partial_x u) +b_0(x,y,t)\partial_x u+b(x,y,t)\partial_y u,$$ which generalized our recent results on KFP equations.