Abstract
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 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].
Highlights
In this paper, we consider optimal decay rates for higher–order spatial derivatives of strong solutions to the following 3D compressible viscous quantum magnetohydrodynamic model: ρt + div(ρu) =,(ρu)t + div(ρu ⊗ u) − μ∆u − (μ + λ)∇divu + ∇P(ρ) − θ √ρ∇( ∆√ρρ ) = (∇ × B) × B, (1.1)Bt − ∇ × (u × B) = −∇ × (ν∇ × B), divB =, where t ≥ is time and x ∈ R is the spatial coordinate, and the symbol ⊗ is the Kronecker tensor product
We only review some results closely related for simplicity
We show that the high-frequency part of the fourth order spatial derivatives of velocity u and magnetic B converge to zero at the L –rate ( + t)−, which are faster than ones of themselves, and totally new as compared to Pu–Xu [27] and Xi–Pu–Guo [35]
Summary
We consider optimal decay rates for higher–order spatial derivatives of strong solutions to the following 3D compressible viscous quantum magnetohydrodynamic (vQMHD) model: ρt + div(ρu) = ,. Bt − ∇ × (u × B) = −∇ × (ν∇ × B), divB = , where t ≥ is time and x ∈ R is the spatial coordinate, and the symbol ⊗ is the Kronecker tensor product. The unknown functions ρ = ρ(x, t) is the density, u = (u , u , u )(x, t) denotes the velocity, and. B = (B , B , B )(x, t) represents the magnetic eld. The constant viscosity coe cients μ and λ satisfy the physical conditions: μ > , μ + λ ≥ , and ν > denotes the magnetic di usion coe cient.
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