Long-time existence and sharp decay for two-dimensional incompressible magnetohydrodynamics with partial damping

Source abstract

This paper investigates the long-time behaviour of a two-dimensional incompressible magnetohydrodynamics system with partial linear damping. The model couples the Euler equations for the velocity field u with the induction equations for the magnetic field b, while attaching distinct zeroth-order damping coefficients (0, ν) to u and (0, κ) to b. The damping terms lack spatial derivatives, they serve solely to dissipate energy without offering any smoothing effect. In this setting, we establish both an explicit lifespan and sharp decay estimates. Specifically, when the initial data have size ɛ ≪ 1 in the appropriate Sobolev space, the solution exists at least up to time [0, Cγ−1ɛ−2], and the perturbations (ũ,b̃) obey the optimal algebraic decay rates predicted by the linearised dynamics.