A Runge-type approximation theorem for the 3D unsteady Stokes system

TL;DR

This paper proves a Runge-type approximation theorem for the 3D unsteady Stokes system, showing local solutions can be approximated globally with controlled growth, and discusses the necessity of certain growth conditions at infinity.

Contribution

It establishes a Runge-type approximation theorem for the 3D unsteady Stokes system and analyzes growth conditions at infinity for solutions.

Findings

Local smooth solutions can be approximated by global solutions with controlled growth.

Certain growth conditions at infinity are necessary for approximation.

The approximation holds in the $L^ abla$ norm with arbitrary small error.

Abstract

We investigate Runge-type approximation theorems for solutions to the 3D unsteady Stokes system. More precisely, we establish that on any compact set with connected complement, local smooth solutions to the 3D unsteady Stokes system can be approximated with an arbitrary small positive error in $L^\infty$ norm by a global solution of the 3D unsteady Stokes system, where the velocity grows at most exponentially at spatial infinity and the pressure grows polynomially. Additionally, by considering a parasitic solution to the Stokes system, we establish that some growths at infinity are indeed necessary.