Long time dynamics of a fractional dissipative model of electroconvection in bounded domains

TL;DR

This paper studies a fractional dissipative model of electroconvection in bounded domains, establishing well-posedness, regularity, and long-term behavior, including existence of attractors and analyticity of solutions.

Contribution

It provides new results on global well-posedness, regularity, and long-term dynamics for a nonlocal fractional model of electroconvection in bounded domains.

Findings

Existence and uniqueness of exponentially decaying solutions for $H^1$ initial data.

Existence of a finite-dimensional global attractor with body forces.

Solutions are globally analytic in time under periodic boundary conditions.

Abstract

We consider a nonlocal nonlinear model with fractional diffusion motivated by studies of electroconvection phenomena in incompressible viscous fluids. We address the global well-posedness, global regularity and long time dynamics of the model in bounded smooth domains with Dirichlet boundary conditions. We prove the existence and uniqueness of exponentially decaying in time solutions for $H^1$ initial data regardless of the fractional dissipative regularity. In the presence of time independent body forces in the fluid, we prove the existence of a compact finite dimensional global attractor. In the case of periodic boundary conditions, we prove that the unique smooth solution is globally analytic in time, and belongs to a Gevrey class of functions that depends on the dissipative regularity of the model.