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On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions
1. | Faculty of Arts and Science, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan |
2. | Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67 Praha 1, Czech Republic, Institute of Mathematics, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany |
3. | IMATH, EA 2134, Université de Toulon, BP 20132, 83957 La Garde, France |
We consider the barotropic Navier–Stokes system describing the motion of a compressible Newtonian fluid in a bounded domain with in and out flux boundary conditions. We show that if the boundary velocity coincides with that of a rigid motion, all solutions converge to an equilibrium state for large times.
References:
[1] |
G. Avalos and P. G. Geredeli,
Exponential stability of a nondissipative, compressible flow-structure PDE model, J. Evol. Equ., 20 (2020), 1-38.
doi: 10.1007/s00028-019-00513-9. |
[2] |
T. Chang, B. J. Jin and A. Novotný,
Compressible Navier-Stokes system with general inflow-outflow boundary data, SIAM J. Math. Anal., 51 (2019), 1238-1278.
doi: 10.1137/17M115089X. |
[3] |
H. J. Choe, A. Novotný and M. Yang,
Compressible Navier-Stokes system with hard sphere pressure law and general inflow-outflow boundary conditions, J. Differential Equations, 266 (2019), 3066-3099.
doi: 10.1016/j.jde.2018.08.049. |
[4] |
E. Feireisl and H. Petzeltová,
On the zero-velocity-limit solutions to the Navier-Stokes equations of compressible flow, Manuscr. Math., 97 (1998), 109-116.
doi: 10.1007/s002290050089. |
[5] |
E. Feireisl and H. Petzeltová,
Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Rational Mech. Anal., 150 (1999), 77-96.
doi: 10.1007/s002050050181. |
[6] |
E. Feireisl and D. Pražák, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS, Springfield, 2010. |
[7] |
P. G. Geredeli, A time domain approach for the exponential stability of a linearized compressible flow–structure pde system, Math. Meth. Appl. Sci., 2020. Early view. |
[8] |
V. Girinon,
Navier-Stokes equations with nonhomogeneous boundary conditions in a bounded three-dimensional domain, J. Math. Fluid Mech., 13 (2011), 309-339.
doi: 10.1007/s00021-009-0018-x. |
[9] |
Y.-S. Kwon and A. Novotný, Dissipative solutions to compressible Navier–Stokes equations with general inflow-outflow data: Existence, stability and weak strong uniqueness, J. Math. Fluid Mech., 2021 (in press). arXiv: 1905.02667. |
[10] |
B. Melinand and K. Zumbrun,
Existence and stability of steady compressible Navier-Stokes solutions on a finite interval with noncharacteristic boundary conditions, Phys. D, 394 (2019), 16-25.
doi: 10.1016/j.physd.2019.01.006. |
[11] |
A. Novotný and I. Straškraba,
Stabilization of weak solutions to compressible Navier-Stokes equations, J. Math. Kyoto Univ., 40 (2000), 217-245.
doi: 10.1215/kjm/1250517713. |
[12] |
A. Novotný and I. Straškraba,
Convergence to equilibria for compressible Navier-Stokes equations with large data, Annali Mat. Pura Appl., 179 (2001), 263-287.
doi: 10.1007/BF02505958. |
[13] |
Y. Shibata and K. Tanaka,
Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid, Comput. Math. Appl., 53 (2007), 605-623.
doi: 10.1016/j.camwa.2006.02.030. |
[14] |
S. Ukai, T. Yang and H. Zhao,
Convergence rate for the compressible Navier-Stokes equations with external force, J. Hyperbolic Differ. Equ., 3 (2006), 561-574.
doi: 10.1142/S0219891606000902. |
show all references
References:
[1] |
G. Avalos and P. G. Geredeli,
Exponential stability of a nondissipative, compressible flow-structure PDE model, J. Evol. Equ., 20 (2020), 1-38.
doi: 10.1007/s00028-019-00513-9. |
[2] |
T. Chang, B. J. Jin and A. Novotný,
Compressible Navier-Stokes system with general inflow-outflow boundary data, SIAM J. Math. Anal., 51 (2019), 1238-1278.
doi: 10.1137/17M115089X. |
[3] |
H. J. Choe, A. Novotný and M. Yang,
Compressible Navier-Stokes system with hard sphere pressure law and general inflow-outflow boundary conditions, J. Differential Equations, 266 (2019), 3066-3099.
doi: 10.1016/j.jde.2018.08.049. |
[4] |
E. Feireisl and H. Petzeltová,
On the zero-velocity-limit solutions to the Navier-Stokes equations of compressible flow, Manuscr. Math., 97 (1998), 109-116.
doi: 10.1007/s002290050089. |
[5] |
E. Feireisl and H. Petzeltová,
Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Rational Mech. Anal., 150 (1999), 77-96.
doi: 10.1007/s002050050181. |
[6] |
E. Feireisl and D. Pražák, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS, Springfield, 2010. |
[7] |
P. G. Geredeli, A time domain approach for the exponential stability of a linearized compressible flow–structure pde system, Math. Meth. Appl. Sci., 2020. Early view. |
[8] |
V. Girinon,
Navier-Stokes equations with nonhomogeneous boundary conditions in a bounded three-dimensional domain, J. Math. Fluid Mech., 13 (2011), 309-339.
doi: 10.1007/s00021-009-0018-x. |
[9] |
Y.-S. Kwon and A. Novotný, Dissipative solutions to compressible Navier–Stokes equations with general inflow-outflow data: Existence, stability and weak strong uniqueness, J. Math. Fluid Mech., 2021 (in press). arXiv: 1905.02667. |
[10] |
B. Melinand and K. Zumbrun,
Existence and stability of steady compressible Navier-Stokes solutions on a finite interval with noncharacteristic boundary conditions, Phys. D, 394 (2019), 16-25.
doi: 10.1016/j.physd.2019.01.006. |
[11] |
A. Novotný and I. Straškraba,
Stabilization of weak solutions to compressible Navier-Stokes equations, J. Math. Kyoto Univ., 40 (2000), 217-245.
doi: 10.1215/kjm/1250517713. |
[12] |
A. Novotný and I. Straškraba,
Convergence to equilibria for compressible Navier-Stokes equations with large data, Annali Mat. Pura Appl., 179 (2001), 263-287.
doi: 10.1007/BF02505958. |
[13] |
Y. Shibata and K. Tanaka,
Rate of convergence of non-stationary flow to the steady flow of compressible viscous fluid, Comput. Math. Appl., 53 (2007), 605-623.
doi: 10.1016/j.camwa.2006.02.030. |
[14] |
S. Ukai, T. Yang and H. Zhao,
Convergence rate for the compressible Navier-Stokes equations with external force, J. Hyperbolic Differ. Equ., 3 (2006), 561-574.
doi: 10.1142/S0219891606000902. |
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