$\mathcal{A}$-caloric approximation and partial regularity for parabolic systems with Orlicz growth

TL;DR

This paper introduces an $ ext{A}$-caloric approximation lemma tailored for Orlicz spaces and uses it to prove partial regularity for certain parabolic systems with growth conditions governed by an N-function.

Contribution

It develops a new $ ext{A}$-caloric approximation lemma compatible with Orlicz growth and applies it to establish partial regularity for parabolic systems with N-function growth.

Findings

Established a new $ ext{A}$-caloric approximation lemma for Orlicz spaces.

Proved partial regularity results for parabolic systems with N-function growth conditions.

Demonstrated the applicability of the lemma to systems with growth bounded by derivatives of an N-function.

Abstract

We prove a new $\mathcal{A}$-caloric approximation lemma compatible with an Orlicz setting. With this result, we establish a partial regularity result for parabolic systems of the type $$ u_{t}- {\rm div} \,a(Du)=0. $$ Here the growth of $a$ is bounded by the derivative of an $N$-function $\varphi$. The primary assumption for $\varphi$ is that $t\varphi''(t)$ and $\varphi'(t)$ are uniformly comparable on $(0,\infty)$.