Stability of the 2D MHD system with fractional vertical dissipation and magnetic damping

Abstract

Due to the importance in physical applications and mathematics, the stability problems of partially dissipative hydrodynamic system have attracted considerable attention in recent years and achieved significant progress. However, research on the stability analysis for the partial fractional dissipative system is relatively limited. The present study demonstrates the stability of a two-dimensional (2D) incompressible magnetohydrodynamic (MHD) system with fractional vertical dissipation and magnetic damping within the Sobolev space H²(R²). In view of the presence of fractional order operators, classical tools and techniques designed for the fully dissipated MHD system no longer work; therefore, we develop new anisotropic inequalities that involve fractional derivatives to address this issue. Our result extends and improves the work in [Nonlinearity, 2021, 34(4): 2527].

Keywords

MHD system / partial and fractional dissipation / magnetic damping / stability

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