Stability for the 2D incompressible MHD system with fractional dissipation and linear damping

Abstract

This paper is devoted to the stability analysis of a 2D incompressible magnetohydrodynamic (MHD) system with fractional dissipation and linear damping on the whole space

. The considered system generalizes a previously studied partially integer-order dissipative MHD model to the fractional setting, where the dissipation terms are characterized by directional fractional Laplacians

Λ 1 2 α ${{\Lambda }}_{1}^{2\alpha }$

and

Λ 2 2 β ${{\Lambda }}_{2}^{2\beta }$

with fractional exponents α , β ∈ (0, 1]. These fractional Laplace operators go beyond the scope of the standard directional integer-order Laplacians, i.e., α = β = 1, and present new mathematical challenges. To this end, we first develop a set of new anisotropic interpolation inequalities involving fractional derivatives, which play a crucial role in dealing with the nonlinear terms of the system. Then by constructing a suitable energy functional and applying the bootstrapping argument, we establish the global stability of the system for small initial data in the Sobolev space

H 2 ( R 2 ) ${H}^{2}({\mathbb{R}}^{2})$

.