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August  2021, 41(8): 3615-3627. doi: 10.3934/dcds.2021009

On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions

Jan Březina 1,, , Eduard Feireisl 2, and  Antonín Novotný 3,

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

* Corresponding author: Jan Březina

Received  May 2020 Revised  October 2020 Published  January 2021

Fund Project: Jan Březina and Eduard Feireisl, The work of E.F. was partially supported by the Czech Sciences Foundation (GAČR), Grant Agreement 18-05974S. The Institute of Mathematics of the Academy of Sciences of the Czech Republic is supported by RVO:67985840.
Antonín Novotný, The work of A.N. was supported by Brain Pool program funded by the Ministry of Science and ICT through the National Research Foundation of Korea (NRF-2019H1D3A2A01101128).

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.

Keywords: compressible Newtonian fluid, Navier–Stokes system, in/out–flux boundary conditions, long–time behavior.
Mathematics Subject Classification: 35Q30, 37L15, 76N15.
Citation: Jan Březina, Eduard Feireisl, Antonín Novotný. On convergence to equilibria of flows of compressible viscous fluids under in/out–flux boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3615-3627. doi: 10.3934/dcds.2021009
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. Feireisl and D. Pražák, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS, Springfield, 2010. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. Feireisl and D. Pražák, Asymptotic Behavior of Dynamical Systems in Fluid Mechanics, AIMS, Springfield, 2010. Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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