Well-posedness of the Viscous Boussinesq System in Besov Spaces of Negative Order Near Index $s=-1$

Abstract

This paper is concerned with well-posedness of the Boussinesq system. We prove that the $n$ ($n\ge 2$) dimensional Boussinesq system is well-psoed for small initial data $({\vec{u}}_0,{\theta }_0)$ ($\nabla \cdot {\vec{u}}_0=0$) either in $(B_{\infty ,1}^{-1}\cap B_{\infty ,\infty }^{-1,1})\times B_{p,r}^{-1}$ or in $B_{\infty ,\infty }^{-1,1}\times B_{p,\infty }^{-1,\epsilon }$ if $r\in [1,\infty ]$, $\epsilon >0$ and $p\in (\frac{n}{2},\infty )$, where $B_{p,q}^{s,\epsilon }$ ($s\in \mathbb{R}$, $1\le p,q\le \infty$, $\epsilon >0$) is the logarithmically modified Besov space to the standard Besov space $B_{p,q}^s$. We also prove that this system is well-posed for small initial data in $(B_{\infty ,1}^{-1}\cap B_{\infty ,\infty }^{-1,1})\times (B_{\frac{n}{2},1}^{-1}\cap B_{\frac{n}{2},\infty }^{-1,1})$.

18 pages

References in corpus (3)

• Global well-posedness for the Navier-Stokes-Boussinesq system with axisymmetric data
• Global regularity for the 2D anisotropic Boussinesq Equations with vertical dissipation
• Global Well-posedness for The 2D Boussinesq System Without Heat Diffusion and With Either Anisotropic Viscosity or Inviscid Voigt-$\alpha$ Regularization

Cited by in corpus (1)

• Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system