Large Time Decay of Solutions to a Linear Nonautonomous System in Exterior Domains

Toshiaki Hishida

doi:10.3390/math9080841

Abstract

In this expository paper, we study Lq-Lr decay estimates of the evolution operator generated by a perturbed Stokes system in n-dimensional exterior domains when the coefficients are time-dependent and can be unbounded at spatial infinity. By following the approach developed by the present author for the physically relevant case where the rigid motion of the obstacle is time-dependent, we clarify that some decay properties of solutions to the same system in whole space Rn together with the energy relation imply the desired estimates in exterior domains provided n≥3.

Highlights

This paper studies the large time decay of solutions to the initial value problem for a linear nonautonomous system arising from fluid dynamics, a viscous incompressible flow past an obstacle
The well-posedness in Assumption (i), in other words, the generation of the evolution operator, is never obvious; this is a different issue from what we address in this paper
The stability or attainability of physically relevant basic (Navier–Stokes) flow V is a significant issue in mathematical fluid dynamics

Summary

Introduction

This paper studies the large time decay of solutions to the initial value problem for a linear nonautonomous system arising from fluid dynamics, a viscous incompressible flow past an obstacle. The desired uniformly boundedness of the perturbation is discussed by a bootstrap argument and by duality argument with the aid of Assumption (iii) on the energy relations, and the use of duality is why we need to simultaneously study adjoint evolution operator T (t, s)∗ with T (t, s) This step does not work when n = 2. We claim that some decay properties of solutions to the same system in whole space Rn together with the energy relation imply the desired estimates in exterior domains provided n ≥ 3, and that we need to find (10) through analysis of pressure to justify this statement. We close the paper with a conclusion in the final section

Result

Proof of Theorem 1

Conclusions