Space-time decay rates of a two-phase flow model with magnetic field in R^3
DOI:
https://doi.org/10.58997/ejde.2023.41Keywords:
Compressible Euler equations; Two-phase flow model; Space-time decay rate; Weighted Sobolev space.Abstract
We investigate the space-time decay rates of strong solution to a two-phase flow model with magnetic field in the whole space \(\mathbb{R}^3 \). Based on the temporal decay results by Xiao [24] we show that for any integer \(\ell\geq 3\), the space-time decay rate of \(k(0\leq k \leq \ell)\)-order spatial derivative of the strong solution in the weighted Lebesgue space \( L_\gamma^2 \) is \(t^{-\frac{3}{4}-\frac{k}{2}+\gamma}\). Moreover, we prove that the space-time decay rate of \(k(0\leq k \leq \ell-2)\)-order spatial derivative of the difference between two velocities of the fluid in the weighted Lebesgue space \( L_\gamma^2 \) is \(t^{-\frac{5}{4}-\frac{k}{2}+\gamma}\), which is faster than ones of the two velocities themselves.
For more information see https://ejde.math.txstate.edu/Volumes/2023/41/abstr.html
References
C. Baranger, L. Desvillettes; Coupling Euler and Vlasov equations in the context of sprays: the local-in-time, classical solutions. J. Hyperbolic Differ. Equ., 3 (2006) (1): 1{26.
S. Berres, R. Burger, K. H. Karlsen, E. M. Tory; Strongly degenerate parabolic-hyperbolic systems modeling polydisperse sedimentation with compression. SIAM J. Appl. Math., 64 (2003) (1):41{80.
Q. Chen, Z. Tan; Time decay of solutions to the compressible Euler equations with damping. Kinet. Relat. Models, 7 (2014) (4):605{619.
Y. Choi; Global classical solutions and large-time behavior of the two-phase fluid model. SIAM J. Math. Anal., 48 (2016) (5): 3090{3122.
Y. Choi, B. Kwon; The Cauchy problem for the pressureless Euler/isentropic Navier-Stokes equations. J. Differential Equations, 261 (2016) (1):654{711.
R. Duan, S. Liu; Cauchy problem on the Vlasov-Fokker-Planck equation coupled with the compressible Euler equations through the friction force. Kinet. Relat. Models, 6 (2013) (4):687{700.
L. C. Evans; Partial differential equations, volume 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.
J. Fan, W. Yu; Strong solution to the compressible magnetohydrodynamic equations with vacuum. Nonlinear Anal. Real World Appl., 10 (2009) (1): 392{409.
J. Gao, Z. Lyu, and Z. Yao; Lower bound and space-time decay rates of higher order derivative of solution for the compressible navier-stokes and hall-mhd equations. arXiv preprint arXiv:1909.13269, 2019.
T. K. Karper, A. Mellet, and K. Trivisa. Existence of weak solutions to kinetic flocking models. SIAM J. Math. Anal., 45 (2013) (1):215{243.
I. Kukavica; Space-time decay for solutions of the Navier-Stokes equations. volume 50, pages 205-222. 2001. Dedicated to Professors Ciprian Foias and Roger Temam (Bloomington, IN, 2000).
I. Kukavica; On the weighted decay for solutions of the Navier-Stokes system. Nonlinear Anal., 70 (2009) (6): 2466{2470.
I. Kukavica, J. J. Torres; Weighted bounds for the velocity and the vorticity for the Navier-Stokes equations. Nonlinearity, 19 (2006) (2):293{303.
I. Kukavica, J. J. Torres; Weighted Lp decay for solutions of the Navier-Stokes equations. Comm. Partial Differential Equations, 32 (20097) (4-6): 819-831.
A. Mellet, A. Vasseur; Global weak solutions for a Vlasov-Fokker-Planck/Navier-Stokes system of equations. Math. Models Methods Appl. Sci., 17 (2007) (7):1039{1063.
L. Qin, C. Xiao, Y. Zhang; Optimal decay rates for higher-order derivatives of solutions to 3D compressible Navier-Stokes-Poisson equations with external force. Electron. J. Differential Equations., 20222 (2022) 64: 1-18.
V. Sohinger, R. M. Strain; The Boltzmann equation, Besov spaces, and optimal time decay rates in Rnx. Adv. Math., 261 (2014) :274{332.
S. Takahashi; A weighted equation approach to decay rate estimates for the Navier-Stokes equations. Nonlinear Anal., 37 (1999) (6, Ser. A: Theory Methods): 751{789.
Z. Tan, Y. Wang, F. Xu; Large-time behavior of the full compressible Euler-Poisson system without the temperature damping. Discrete Contin. Dyn. Syst., 36 (2016) (3): 1583-1601.
A. I.Vol'pert, S. I. Hudjaev; The Cauchy problem for composite systems of nonlinear differential equations. Mat. Sb. (N.S.), 87 (1972) (129):504{528.
S. Weng; Space-time decay estimates for the incompressible viscous resistive MHD and Hall-MHD equations. J. Funct. Anal., 270 (2016) (6): 2168{-2187.
G. Wu, Y. Zhang, L. Zou; Optimal large-time behavior of the two-phase
uid model in the whole space. SIAM J. Math. Anal., 52 (2020) (6):5748{5774.
G. Wu, Y. Zhang, W. Zou; Optimal time-decay rates for the 3D compressible magnetohydrodynamic flows with discontinuous initial data and large oscillations. J. Lond. Math. Soc. (2), 103 (2021) (3):817-845.
C. Xiao; Global existence and large-time behavior to a two-phase flow model with magnetic eld. J. Math. Fluid Mech., 24 (2022) (3): Paper No. 75, 25.
F. Xu, X. Zhang, Y. Wu, L. Caccetta; Global well-posedness of the non-isentropic full compressible magnetohydrodynamic equations. Acta Math. Sin. (Engl. Ser.), 32 (2016) (2):227-250.
Y. Zhang, J. Wang, C. Xiao, L. Ma; Global existence and time decay rates of the two-phase fluid system in R3. Z. Angew. Math. Phys., 72 (2021) (5): Paper No. 180, 28.
E. Zuazua; Large time asymptotics for heat and dissipative wave equations. Manuscript available at http://www. uam. es/enrique.zuazua, 2003.
Downloads
Published
License
Copyright (c) 2023 Qin Ye, Yinghui Zhang
This work is licensed under a Creative Commons Attribution 4.0 International License.
