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) (see [3]). 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 a special \(2\frac{1}{2}\) -D compressible MHD flow where the magnetic field is vertical 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 \(H^3\) . In addition, we also obtain the exponential decay for this partially dissipative system. In contrast to prior studies [25, 26, 43], our analysis eliminates the requirement for a positively defined background magnetic field-a critical relaxation of conventional assumptions. To the best of our knowledge, this work establishes the first global well-posedness framework for compressible magnetohydrodynamic flows without magnetic dissipation with initial perturbations near trivial equilibrium states, marking a fundamental advancement in low-amplitude regime analysis.
