Existence, Uniqueness, Regularity and Long-term Behavior for Dissipative Systems Modeling Electrohydrodynamics

TL;DR

This paper investigates the mathematical properties of a dissipative electrohydrodynamics model, proving existence, uniqueness, regularity, and long-term convergence of solutions in different dimensions.

Contribution

It establishes the existence, uniqueness, and regularity of solutions for the nonlinear nonlocal system in 2D and small data in 3D, using entropy methods.

Findings

Solutions exist and are unique in 2D for general data.

Solutions exist in 3D for small initial data.

Solutions converge to stationary states at a quantifiable rate.

Abstract

We study a dissipative system of nonlinear and nonlocal equations modeling the flow of electrohydrodynamics. The existence, uniqueness and regularity of solutions is proven for general $\mathbf{L}^2$ initial data in two space dimensions and for small data in data in three space dimensions. The existence in three dimensions is established by studying a linearization of a relative entropy functional. We also establish the convergence to the stationary solution with a rate.