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

Abstract

This paper is devoted to the study of the Cauchy problem for the Boussinesq system with partial viscosity in dimension $N\ge 3.$ 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.

24 pages. Physica D, in press

Cited by in corpus (15)

• Global regularity for the 2D anisotropic Boussinesq Equations with vertical dissipation
• Global well-posedness for axisymmetric Boussinesq system with horizontal viscosity
• The regularized 3D Boussinesq equations with fractional Laplacian and no diffusion
• Periodic solutions for Boussinesq systems in weak-Morrey spaces
• Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system
• Global Existence Results for the Anisotropic Boussinesq System in Dimension Two
• Large time decay properties of solutions to a viscous Boussinesq system in a half space
• Global persistence of geometrical structures for the boussinesq equation with no diffusion
• Generalized gravitational fields and well-posedness of the Boussinesq systems on non-compact Riemannian Manifolds
• Large time decay and growth for solutions of a viscous Boussinesq system
• Uniqueness theorems for the Boussinesq system
• On the regularity of weak solutions of the Boussinesq equations in Besov spaces Dedicated to Enrique Zuazua on the occasion of his sixtieth birthday
• The 2D Boussinesq equations with logarithmically supercritical velocities
• Well-posedness and exponential stability for Boussinesq systems on real hyperbolic Manifolds and application
• Analytical smoothing effect of solution for the boussinesq equations