Spatial asymptotics of mild solutions to the time-dependent Oseen system

Paul Deuring

doi:10.3934/cpaa.2021044

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2021, Volume 20, Issue 5: 1833-1849. Doi: 10.3934/cpaa.2021044

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Spatial asymptotics of mild solutions to the time-dependent Oseen system

Paul Deuring

Univ. du Littoral Côte d'Opale, EA 2797 – LMPA – Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville, F-62228 Calais, France

Received: March 31, 2020

Revised: December 31, 2020

Early access: March 2021

Published: May 2021

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Abstract

We consider mild solutions to the 3D time-dependent Oseen system with homogeneous Dirichlet boundary conditions, under weak assumptions on the data. Such solutions are defined via the semigroup generated by the Oseen operator in $ L^q. $ They turn out to be also $ L^q $-weak solutions to the Oseen system. On the basis of known results about spatial asymptotics of the latter type of solutions, we then derive pointwise estimates of the spatial decay of mild solutions. The rate of decay depends in particular on $ L^p $-integrability in time of the external force.

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Mathematics Subject Classification: Primary: 76D07, 35B40; Secondary: 35Q35.

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