Optimal time-decay rates for the 3D compressible Magnetohydrodynamic flows with discontinuous initial data and large oscillations

This paper is concerned with time-decay rates of the weak solutions to the 3D compressible magnetohydrodynamic flows with discontinuous initial data and large oscillations. The global existence of weak solutions to the Cauchy problem of the 3D compressible magnetohydrodynamic flows has been established by Suen-Hoff (Arch. Ration. Mech. Anal. 205 (2012) 27-58) and Suen (J. Differential Equations 268 (2020) 2622-2671) (also see Liu, Yu and Zhang J. Differential Equations 254 (2013) 229-255), under the condition that the initial energy is suitably small, the initial density is positive and essentially bounded and the gradients of initial velocity and magnetic field are bounded in L2. However, to our best knowledge, so far there is no result on time-decay rates of such solutions. The main novelty of this paper is to give a positive response to this problem. More precisely, we obtain the optimal time-decay rates of the solutions in Lr-norm with 2 <= r <=infinity and the first-order derivatives of the velocity and magnetic field in L2-norm when the L1-norm of the initial perturbation is bounded. Moreover, we also show the lower bounds on the rates of decay.

Automated summary

This paper investigates the time-decay rates of weak solutions to 3D compressible magnetohydrodynamic flows with discontinuous initial data and large oscillations. The study establishes the global existence of weak solutions to the Cauchy problem under certain conditions, and provides optimal time-decay rates of solutions in Lr-norm with 2 <= r <= infinity and first-order derivatives of velocity and magnetic field in L2-norm. Lower bounds on the rates of decay are also presented.