Blow-up criterion for the density dependent inviscid Boussinesq equations

Abstract

Abstract In this work, we consider the density-dependent incompressible inviscid Boussinesq equations in R N ( N ≥ 2 ) $\mathbb{R}^{N}\ (N\geq 2)$ . By using the basic energy method, we first give the a priori estimates of smooth solutions and then get a blow-up criterion. This shows that the maximum norm of the gradient velocity field controls the breakdown of smooth solutions of the density-dependent inviscid Boussinesq equations. Our result extends the known blow-up criteria.

Keywords

• Inviscid Boussinesq equations
• Density-dependent
• Local existence
• Blow-up criterion