The Schauder estimates for higher-order parabolic systems with time irregular coefficients

TL;DR

This paper establishes Schauder estimates for higher-order parabolic systems with coefficients that are only measurable in time and Hölder continuous in space, covering divergence and non-divergence types in various domains.

Contribution

It provides the first Schauder estimates for such systems with minimal regularity assumptions on the coefficients in the time variable.

Findings

Schauder estimates proven for divergence and non-divergence systems

Existence results for divergence systems in cylindrical domains

Applicable to systems with coefficients measurable in time and Hölder in space

Abstract

We prove Schauder estimates for solutions to both divergence and non-divergence type higher-order parabolic systems in the whole space and the half space. We also provide an existence result for divergence type systems in a cylindrical domain. All coefficients are assumed to be only measurable in the time variable and H\"{o}lder continuous in the spatial variables.