Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces

Details Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces Existence and uniqueness results for the Boussinesq system with data in Lorentz spaces

2008-06-25

arxiv.org/abs/0806.4084v1

Mathematics

Analysis of PDEs

24 pages. Physica D, in press

Scientific paper

10.1016/j.physd.2008.03.034

This paper is devoted to the study of the Cauchy problem for the Boussinesq system with partial viscosity in dimension $N\geq3.$ First we prove a global existence result for data in Lorentz spaces satisfying a smallness condition which is at the scaling of the equations. Second, we get a uniqueness result in Besov spaces with {\it negative} indices of regularity (despite the fact that there is no smoothing effect on the temperature). The proof relies on a priori estimates with loss of regularity for the nonstationary Stokes system with convection. As a corollary, we obtain a global existence and uniqueness result for small data in Lorentz spaces.

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