On the analyticity and Gevrey regularity of solutions to the three-dimensional inviscid Boussinesq equations in a half space (Q2076206)

In this paper, the author considers the three-dimensional inviscid Boussinesq equations \[ \begin{rcases} u_t+u\cdot\nabla u+\nabla p=\theta e_3,\\ \theta_t+u\cdot\nabla\theta=0,\\ \nabla u=0, \end{rcases}\quad\text{for }x_3>0, \] \[ u\cdot n=0 \quad\qquad\qquad\qquad\qquad\text{for }x_3=0 \] in the half space \(\mathbb{R}^3_+\), where \(x\in\Omega=\{x\in\mathbb{R}^3;x_3>0\}\), \(t\geq0\), and where \(n=(0,0,-1)\) is the outward unit normal vector, \(e_3=(0,0,1)\) is the unit vector, \(p\) is the scalar pressure, and \(\theta\) is the scalar density. The unknown function \(u(x,t)\) represents here the velocity vector field. The Boussinesq equations are very important to model many physical problems such as, for example, the propagation of long waves in shallow water, the propagation of one-dimensional nonlinear lattice-waves, the propagation of vibrations in a nonlinear string, or the propagation of ionic sound waves in a plasma. They have already been the subject of many investigations and many results have already been established. However, the inviscid Boussinesq equations are much harder to study since there is no dissipation in the system and the global regularity, even in 2D, still remains a challenging open problem. Here, the author is interested in the analyticity of the solution \(u(x,t)\) and the persistence of its Gevrey regularity. In addition, he obtains lower bounds on the radius of the Gevrey regularity. The results obtained are very important since they are the first results regarding the analyticity and Gevrey regularity of solutions to the 3D inviscid Boussinesq equations in a domain with boundary conditions.