Global Well-Posedness of 2D Second Grade Fluid Equations in Exterior Domain

TL;DR

This paper proves the global well-posedness of 2D second grade fluid equations in exterior domains with Dirichlet boundary conditions, establishing existence, uniqueness, and regularity of solutions for initial data in Sobolev spaces.

Contribution

It extends the understanding of second grade fluid equations by establishing global well-posedness and regularity results in exterior domains with Dirichlet boundary conditions.

Findings

Solutions exist globally for initial data in H^3

Solutions remain bounded in H^s for s ≥ 3 over time

Regularity of solutions persists for all time

Abstract

In this article, we consider 2D second grade fluid equations in exterior domain with Dirichlet boundary conditions. For initial data $\boldsymbol{u}_0 \in \boldsymbol{H}^3(\Omega)$, the system is shown to be global well-posed. Furthermore, for arbitrary $T > 0$ and $s \geq 3$, we prove that the solution belongs to $L^\infty([0, T]; \boldsymbol{H}^s(\Omega))$ provided that $\boldsymbol{u}_0$ is in $\boldsymbol{H}^s(\Omega)$.