Long time dynamics of electroconvection in bounded domains

Abdo, Elie; Ignatova, Mihaela

doi:10.1090/tran/9344

Long time dynamics of electroconvection in bounded domains
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by Elie Abdo and Mihaela Ignatova;

Trans. Amer. Math. Soc. 378 (2025), 2187-2245

DOI: https://doi.org/10.1090/tran/9344

Published electronically: December 27, 2024

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Abstract:

We discuss nonlinear nonlocal equations with fractional diffusion describing electroconvection phenomena in incompressible viscous fluids. We prove 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.

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