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

TL;DR

This paper establishes the well-posedness of the viscous Boussinesq system in certain Besov spaces of negative order, expanding the understanding of initial data conditions for the system's solutions.

Contribution

It proves well-posedness of the Boussinesq system in novel Besov spaces with negative regularity, including logarithmically modified spaces, for small initial data.

Findings

Well-posedness in specific Besov spaces for small initial data.

Extension of well-posedness results to negative order Besov spaces.

Use of logarithmically modified Besov spaces to analyze the system.

Abstract

This paper is concerned with well-posedness of the Boussinesq system. We prove that the $n$ ($n\ge2$) dimensional Boussinesq system is well-psoed for small initial data $(\vec{u}_0,\theta_0)$ ($\nabla\cdot\vec{u}_0=0$) either in $({B}^{-1}_{\infty,1}\cap{B^{-1,1}_{\infty,\infty}})\times{B}^{-1}_{p,r}$ or in ${B^{-1,1}_{\infty,\infty}}\times{B}^{-1,\epsilon}_{p,\infty}$ if $r\in[1,\infty]$, $\epsilon>0$ and $p\in(\frac{n}{2},\infty)$, where $B^{s,\epsilon}_{p,q}$ ($s\in\mathbb{R}$, $1\leq p,q\leq\infty$, $\epsilon>0$) is the logarithmically modified Besov space to the standard Besov space $B^{s}_{p,q}$. We also prove that this system is well-posed for small initial data in $({B}^{-1}_{\infty,1}\cap{B^{-1,1}_{\infty,\infty}})\times({B}^{-1}_{\frac{n}{2},1}\cap{B^{-1,1}_{\frac{n}{2},\infty}})$.