Optimal decay rate for higher–order derivatives of solution to the 3D compressible quantum magnetohydrodynamic model

Abstract

We investigate optimal decay rates for higher–order spatial derivatives of strong solutions to the 3D Cauchy problem of the compressible viscous quantum magnetohydrodynamic model in the H5 × H4 × H4 framework, and the main novelty of this work is three–fold: First, we show that fourth order spatial derivative of the solution converges to zero at the L2-rate (1+t)-114{L^2} - {\rm{rate}}\,{(1 + t)^{- {{11} \over 4}}} , which is same as one of the heat equation, and particularly faster than the L2-rate (1+t)-54{L^2} - {\rm{rate}}\,{(1 + t)^{- {5 \over 4}}} in Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and the L2-rate (1+t)-94{L^2} - {\rm{rate}}\,{(1 + t)^{- {9 \over 4}}} , in Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Second, we prove that fifth–order spatial derivative of density ρ converges to zero at the L2-rate (1+t)-134{L^2} - {\rm{rate}}\,{(1 + t)^{- {{13} \over 4}}} , which is same as that of the heat equation, and particularly faster than ones of Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019]. Third, we show that the high-frequency part of the fourth order spatial derivatives of the velocity u and magnetic B converge to zero at the L2-rate (1+t)-134{L^2} - {\rm{rate}}\,{(1 + t)^{- {{13} \over 4}}} , which are faster than ones of themselves, and totally new as compared to Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and Xi–Pu–Guo [Z. Angew. Math. Phys., 70:1, 2019].

Keywords

• compressible quantum magnetohydrodynamic model
• optimal decay rate
• 35b40
• 35q35
• 76w05