Construction of a right inverse for the divergence in non-cylindrical time dependent domains

TL;DR

This paper develops a stable right inverse for the divergence operator in non-cylindrical, time-evolving domains with H"older regularity, enabling improved pressure estimates in Navier--Stokes equations.

Contribution

It constructs a Bogovskij-type inverse operator for divergence in non-cylindrical domains with Sobolev space estimates, extending known results to more general time-dependent geometries.

Findings

Constructed a divergence inverse operator with Sobolev estimates.

Connected the results to existing theory for Lipschitz domains.

Applied the inverse to improve pressure estimates in Navier--Stokes solutions.

Abstract

We construct a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be H\"older regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed H\"older regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier--Stokes equations in time dependent domains.