On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces

Lucas C. F. Ferreira; Elder J. Villamizar-Roa

doi:10.3934/dcds.2010.27.171

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2010, Volume 27, Issue 1: 171-183. Doi: 10.3934/dcds.2010.27.171

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On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces

Lucas C. F. Ferreira^1, and
Elder J. Villamizar-Roa^2,

1.
Universidade Estadual de Campinas, Departamento de Matemática, Campinas, São Paulo, CEP: 13083-859
2.
Universidad Nacional de Colombia-Medellín, Escuela de Matemáticas, Medellìn, A.A. 3840

Received: January 2009

Revised: December 2009

Published: March 2010

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Abstract

We study the Navier-Stokes system with initial data belonging to sum of two weak-$L^{p}$ spaces, which contains the sum of homogeneous function with different degrees. The domain $\Omega$ can be either an exterior domain, the half-space, the whole space or a bounded domain with dimension $n\geq 2$. We obtain the existence of local mild solutions in the same class of initial data and moreover we show results about uniqueness, regularity and continuous dependence of solutions with respect to the initial data. To obtain our results we prove a new Hölder-type inequality on the sum of Lorentz spaces.

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Mathematics Subject Classification: Primary: 35Q30; Secondary: 76D03, 76D05.

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