Lower bound of decay rate for higher order derivatives of solution to the compressible quantum magnetohydrodynamic model
Abstract: The lower bounds of decay rates for global solution to the compressible viscous quantum magnetohydrodynamic model in three‐dimensional whole space under the H5 × H4 × H4 framework are investigated in this paper. We first show that the lower bound of decay rate for the solution converging to constant equilibrium state (1, 0, 0) in L2‐norm is false(1+tfalse)−34 when the initial data satisfy some low‐frequency assumption. Moreover, we prove that the lower bound of decay rate of k(k ∈ [1, 3]) order spatial deriva… Show more
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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 H 5 × H 4 × H 4 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 L 2 - rate ( 1 + t ) - 11 4 {L^2} - {\rm{rate}}\,{(1 + t)^{- {{11} \over 4}}} , which is same as one of the heat equation, and particularly faster than the L 2 - rate ( 1 + t ) - 5 4 {L^2} - {\rm{rate}}\,{(1 + t)^{- {5 \over 4}}} in Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and the L 2 - rate ( 1 + t ) - 9 4 {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 L 2 - rate ( 1 + t ) - 13 4 {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 L 2 - rate ( 1 + t ) - 13 4 {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].
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 H 5 × H 4 × H 4 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 L 2 - rate ( 1 + t ) - 11 4 {L^2} - {\rm{rate}}\,{(1 + t)^{- {{11} \over 4}}} , which is same as one of the heat equation, and particularly faster than the L 2 - rate ( 1 + t ) - 5 4 {L^2} - {\rm{rate}}\,{(1 + t)^{- {5 \over 4}}} in Pu–Xu [Z. Angew. Math. Phys., 68:1, 2017] and the L 2 - rate ( 1 + t ) - 9 4 {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 L 2 - rate ( 1 + t ) - 13 4 {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 L 2 - rate ( 1 + t ) - 13 4 {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].
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