Strong solutions to the Navier-Stokes equations on thin 3D domains

TL;DR

This paper establishes the existence of strong solutions to the Navier-Stokes equations in thin three-dimensional domains, using energy and Poincaré inequalities, with specific assumptions on initial data and forcing terms.

Contribution

It provides a new proof of strong solutions existence in thin domains without mean value operators, under stronger regularity assumptions on initial velocity and forcing.

Findings

Existence of strong solutions in thin 3D domains.

Proof relies on energy and Poincaré inequalities.

Requires derivatives of initial data and forcing in Lebesgue spaces.

Abstract

We prove the existence of strong solutions to Navier-Stokes equations in three dimensional thin domains. Our proof is based on the energy and the Poincar\'e inequalities as well as contraction principle argument and is free of the mean value operator. The price we pay for the simplicity of the proof are stronger assumptions on the initial velocity and the forcing term. We need to assume that their derivatives with respect to time belong to certain Lebesgue space.