Stability for the Compressible Quantum Magnetohydrodynamic Model Without Magnetic Diffusivity

ABSTRACT Physical experiments and numerical simulations have observed a remarkable phenomenon that the energy is dissipated at a rate that is independent of the ohmic resistivity in the magnetohydrodynamic systems (MHD). In other words, the viscosity for the magnetic field equation can be zero and the system may still be dissipative. To understand the mechanism of this phenomenon, we will focus on 3D quantum magnetohydrodynamic model for quantum plasmas without magnetic dissipation in the periodic box and examine the stability near a background magnetic field. Due to the lack of dissipation for density and magnetic field, this stability problem is not trivial. By exploiting the cancellation structure of the system and introducing several new unknown quantities, we prove the global well‐posedness of strong solutions in the framework of Soboles spaces. In addition, the time decay rates of the solutions are also obtained. To our knowledge, this is the first global well‐posedness result for the quantum magnetohydrodynamic model without magnetic dissipation.