Blow-up criterion of smooth solutions for the Boussinesq equations

Abstract

This paper is dedicated to the study the Boussinesq equations with fractional dissipation in n-dimensions (n≥2). We obtain a blow-up criterion for the local in time classical solution to the Boussinesq equations with arbitrarily small fractional powers of the Laplacian. Therefore our results significantly improve the previous works and are more closer to the resolution of the well-known regularity criterion issue on the inviscid Boussinesq equations.