Conditions for uniform regularity of compressible MHD system in small Alfvén and Mach numbers with tangential magnetic fields to the physical boundary

This paper investigates the uniform regularity of solutions to the compressible magnetohydrodynamics (MHD) system in the half-space $R_{+}^3$ in small Alfvén and Mach Numbers. The study focuses on the case where the magnetic field is tangential to the physical boundary, satisfying the perfect conducting boundary condition, while the velocity field adheres to a Navier-slip boundary condition. Under the condition that bounds the normal derivatives of the velocity and magnetic field uniformly in ϵ (the small Alfvén and Mach Numbers), we establish uniform estimates for the solutions in high-order conormal Sobolev norms. The results distinguish the previous works primarily addressing cases where the magnetic field is transversal to the boundary.