The energy conservation for the Navier-Stokes equations on the Lipschitz domains

TL;DR

This paper proves energy conservation for weak solutions of the Navier-Stokes equations on Lipschitz domains under Shinbrot's condition, using a novel approach that separates boundary effects from mollification without boundary layer assumptions.

Contribution

It introduces a new method to handle boundary effects in energy conservation proofs for Navier-Stokes equations on Lipschitz domains, removing the need for boundary layer assumptions.

Findings

Energy conservation holds under Shinbrot's condition on Lipschitz domains.

The method separates boundary effects from mollification, improving previous results.

Provides a unified approach for domains with or without boundary.

Abstract

In this paper, we consider the energy conservation of the Leray-Hopf weak solution $u$ to the Navier-Stokes equations on bounded domains $\Omega$ with Lipschitz boundary $\partial\Omega$. We prove that although the boundary effect appears, the Shinbrot's condition $u\in L^q_{loc}((0,T];L^p(\Omega))$ with $\frac{1}{p}+\frac{1}{q}=\frac{1}{2},p\geq 4$ still guarantees the validity of energy conservation of $u$, no boundary layer assumptions are required when dealing with domains with Lipschitz boundary. Compared to the existed methods, our critical strategies are that we first separate the mollification of weak solution from the boundary effect by considering non-standard local energy equality and transform the boundary effects into the estimates of the gradient of the cut-off functions, then by establishing a sharp $L^2L^2$ estimate for pressure $P$ and using the zero boundary condition, we obtain global energy equality by taking suitable cut-off functions. Our result provides a unified method to deal with domains with or without boundary and improves the corresponding results in \cite{C-L,Yu}.