Les theoremes de Leray et de Fujita-Kato pour le systeme de Boussinesq partiellement visqueux. The Leray and Fujita-Kato theorems for the Boussinesq system with partial viscosity
By Raphaël Danchin, Marius Paicu. 2008
128 citations recorded by the source corpus.
We are concerned with the so-called Boussinesq equations with partial viscosity. These equations consist of the ordinary incompressible Navier-Stokes equations with a forcing term which is transported {\it with no dissipation} by the velocity field. Such equations are simplified models for geophysics (in which case the forcing term is proportional either to the temperature, or to the salinity or to the density). In the present paper, we show that the standard theorems for incompressible Navier-Stokes equations may be extended to Boussinesq system despite the fact that there is no dissipation or decay at large time for the forcing term. More precisely, we state the global existence of finite energy weak solutions in any dimension, and global well-posedness in dimension $N\geq3$ for small data. In the two-dimensional case, the finite energy global solutions are shown to be unique for any data in $L^2(\R^2).$
Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/d0720b06048778e84ebe168bd2404793aed3d234
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Citations in the corpus
- Global well-posedness for the 2D Boussinesq system with anisotropic viscosity and without heat diffusion (2013, unknown): “The requirements on the initial data were weakened further in [10], where global well-posedness results were established in the whole space without any smoothness assumption, namely they only require u0, θ0 ∈ L(2)(R(2)) for both existence and uniqueness.”
- Global Well-Posedness of the 2D Boussinesq Equations with Vertical Dissipation (2015, background): “Some generalizations of the results in [3, 14], toward weakening the assumption on the initial data, were established in Hmidi– Keraani [13] and Danchin–Paicu [10]; in particular, no smoothness assumptions on the initial data are required in [10].”