# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021009

## 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

* 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.

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 - A, 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

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##### 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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