Local existence of solutions and blow-up criteria for the Boussinesq equations in Lei-Lin-Gevrey Spaces

Abstract

This paper studies the local well-posedness and the behavior at potential finite blow-up times of mild solutions to the three-dimensional fractional Boussinesq equations in Lei-Lin and Lei-Lin-Gevrey spaces ${\mathcal{X}}_{a,\sigma }^s({\mathbb{R}}^3)$. By combining fixed point arguments with Fourier estimates adapted to these spaces, we obtain local existence and uniqueness results for data in ${\mathcal{X}}_{a,\sigma }^s({\mathbb{R}}^3)$, including the usual Lei-Lin case $a=0$. We also establish several blow-up criteria for maximal mild solutions and derive lower bounds for the growth of the corresponding norms as the maximal time is approached. In the strict Lei-Lin-Gevrey regime, and when the dissipative exponents coincide, these estimates yield an exponential type blow-up criterion.

25 pages. Comments are welcome !