Local well-posedness for Benard convection without surface tension

TL;DR

This paper establishes a local well-posedness theory for three-dimensional Bénard convection with a free moving surface, neglecting surface tension, using nonlinear energy methods.

Contribution

It develops the first local well-posedness results for Bénard convection with a free surface without surface tension in a 3D setting.

Findings

Proves local existence and uniqueness of solutions.

Provides energy estimates for the free boundary problem.

Extends the mathematical understanding of fluid dynamics with free surfaces.

Abstract

We consider the B\'enard convection in a three-dimensional domain bounded below by a fixed flatten boundary and above by a free moving surface. The domain is horizontally periodic. The fluid dynamics are governed by the Boussinesq approximation and the effect of surface tension is neglected on the free surface. Here we develop a local well-posedness theory for the equations of general case in the framework of the nonlinear energy method. \end{abstract}