• Previous Article
    Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions
  • DCDS Home
  • This Issue
  • Next Article
    Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed
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

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  August 2021 Early access  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, 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. ChangB. 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. ChoeA. 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. UkaiT. 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. ChangB. 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. ChoeA. 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. UkaiT. 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

[1]

Laurence Cherfils, Stefania Gatti, Alain Miranville. Long time behavior of the Caginalp system with singular potentials and dynamic boundary conditions. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2261-2290. doi: 10.3934/cpaa.2012.11.2261

[2]

Weike Wang, Xin Xu. Large time behavior of solution for the full compressible navier-stokes-maxwell system. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2283-2313. doi: 10.3934/cpaa.2015.14.2283

[3]

Zhong Tan, Yong Wang, Xu Zhang. Large time behavior of solutions to the non-isentropic compressible Navier-Stokes-Poisson system in $\mathbb{R}^{3}$. Kinetic & Related Models, 2012, 5 (3) : 615-638. doi: 10.3934/krm.2012.5.615

[4]

Huicheng Yin, Lin Zhang. The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, Ⅱ: 3D Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2018, 38 (3) : 1063-1102. doi: 10.3934/dcds.2018045

[5]

Teng Wang, Yi Wang. Large-time behaviors of the solution to 3D compressible Navier-Stokes equations in half space with Navier boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021080

[6]

E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39

[7]

Matthew Paddick. The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2673-2709. doi: 10.3934/dcds.2016.36.2673

[8]

Zhong Tan, Leilei Tong. Asymptotic behavior of the compressible non-isentropic Navier-Stokes-Maxwell system in $\mathbb{R}^3$. Kinetic & Related Models, 2018, 11 (1) : 191-213. doi: 10.3934/krm.2018010

[9]

Petr Kučera. The time-periodic solutions of the Navier-Stokes equations with mixed boundary conditions. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 325-337. doi: 10.3934/dcdss.2010.3.325

[10]

Eduard Feireisl. Long time behavior and attractors for energetically insulated fluid systems. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1587-1609. doi: 10.3934/dcds.2010.27.1587

[11]

Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207

[12]

Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141

[13]

Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783

[14]

María Anguiano, Francisco Javier Suárez-Grau. Newtonian fluid flow in a thin porous medium with non-homogeneous slip boundary conditions. Networks & Heterogeneous Media, 2019, 14 (2) : 289-316. doi: 10.3934/nhm.2019012

[15]

Hannes Eberlein, Michael Růžička. Global weak solutions for an newtonian fluid interacting with a Koiter type shell under natural boundary conditions. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020419

[16]

Zhenhua Guo, Wenchao Dong, Jinjing Liu. Large-time behavior of solution to an inflow problem on the half space for a class of compressible non-Newtonian fluids. Communications on Pure & Applied Analysis, 2019, 18 (4) : 2133-2161. doi: 10.3934/cpaa.2019096

[17]

Jing Wang, Lining Tong. Stability of boundary layers for the inflow compressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2595-2613. doi: 10.3934/dcdsb.2012.17.2595

[18]

Ciprian G. Gal, M. Grasselli. On the asymptotic behavior of the Caginalp system with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 689-710. doi: 10.3934/cpaa.2009.8.689

[19]

Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks & Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465

[20]

Andrea Giorgini, Roger Temam, Xuan-Truong Vu. The Navier-Stokes-Cahn-Hilliard equations for mildly compressible binary fluid mixtures. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 337-366. doi: 10.3934/dcdsb.2020141

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (100)
  • HTML views (188)
  • Cited by (0)

[Back to Top]