Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-04-30T05:28:01.672Z Has data issue: false hasContentIssue false
  • English
  • Français

Local existence of strong solutions and weak–strong uniqueness for the compressible Navier–Stokes system on moving domains

Part of: Compressible fluids and gas dynamics, general Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:  01 April 2019

Ondřej Kreml ,
Šárka Nečasová  and
Tomasz Piasecki
Ondřej Kreml
Affiliation:
Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha, Czech Republic ([email protected]; [email protected])
Šárka Nečasová
Affiliation:
Institute of Mathematics, Czech Academy of Sciences, Žitná 25, 115 67 Praha, Czech Republic ([email protected]; [email protected])
Tomasz Piasecki
Affiliation:
Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the compressible Navier–Stokes system on time-dependent domains with prescribed motion of the boundary. For both the no-slip boundary conditions as well as slip boundary conditions we prove local-in-time existence of strong solutions. These results are obtained using a transformation of the problem to a fixed domain and an existence theorem for Navier–Stokes like systems with lower order terms and perturbed boundary conditions. We also show the weak–strong uniqueness principle for slip boundary conditions which remained so far open question.

Keywords

compressible Navier–Stokes equationstime-dependent domainstrong solutionlocal existenceweak–strong uniqueness

MSC classification

Primary: 35Q30: Navier-Stokes equations
Secondary: 76N10: Existence, uniqueness, and regularity theory
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 5 , October 2020 , pp. 2255 - 2300
Check for updates
Copyright
Copyright © Royal Society of Edinburgh 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Carrillo, J., Jüngel, A., Markowich, P. A., Toscani, G. and Unterreiter, A.. Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities. Monatshefte Math. 133 (2001), 1–82.CrossRefGoogle Scholar
2Dintelmann, E., Geissert, M. and Hieber, M.. Strong Lp-solutions to the Navier-Stokes flow past moving obstacles: the case of several obstacles and time dependent velocity. Trans. Amer. Math. Soc. 361 (2009), 653–669.CrossRefGoogle Scholar
3DiPerna, R. J. and Lions, P.-L.. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98 (1989), 511–547.CrossRefGoogle Scholar
4Doboszczak, S.. Relative entropy and a weak-strong uniqueness principle for the compressible Navier-Stokes equations on moving domains. Appl. Math. Lett. 57 (2016), 60–68.CrossRefGoogle Scholar
5Enomoto, Y. and Shibata, Y.. On the R-sectoriality and the initial boundary value problem for the viscous compressible fluid flow. Funkcialaj Ekvacioj 56 (2013), 441–505.CrossRefGoogle Scholar
6Feireisl, E.. Compressible Navier-Stokes equations with a non - monotone pressure law. J. Differ. Equ. 184 (2002), 97–108.CrossRefGoogle Scholar
7Feireisl, E.. Dynamics of viscous compressible fluids (Oxford: Oxford University Press, 2004).Google Scholar
8Feireisl, E.. On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J. 53 (2004), 1707–1740.CrossRefGoogle Scholar
9Feireisl, E., Novotný, A. and Petzeltová, H.. On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech. 3 (2001), 358–392.CrossRefGoogle Scholar
10Feireisl, E., Neustupa, J. and Stebel, J.. Convergence of a Brinkman-type penalization for compressible fluid flows. J. Differ. Equ. 250 (2011), 596–606.CrossRefGoogle Scholar
11Feireisl, E., Novotný, A. and Sun, Y.. Suitable weak solutions to the Navier-Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60 (2011), 611–631.CrossRefGoogle Scholar
12Feireisl, E., Jin, B. J. and Novotný, A.. Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier-Stokes system. J. Math. Fluid Mech. 14 (2012), 717–730.CrossRefGoogle Scholar
13Feireisl, E., Kreml, O., Nečasová, Š., Neustupa, J. and Stebel, J.. Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time dependent domains. J. Differ. Equ. 254 (2013), 125–140.CrossRefGoogle Scholar
14Geissert, M., Götze, K. and Hieber, M.. Lp-theory for strong solutions to fluid-rigid body interaction in Newtonian and generalized Newtonian fluids. Trans. Amer. Math. Soc. 365 (2013), 1393–1439.CrossRefGoogle Scholar
15Germain, P.. Weak-strong uniqueness for the isentropic compressible Navier-Stokes system. J. Math. Fluid Mech. 13 (2010), 137–146.CrossRefGoogle Scholar
16Hieber, M. and Murata, M.. The Lp-approach to the fluid-rigid body interaction problem for compressible fluids. Evol. Equ. Control Theory 4 (2015), 69–87.CrossRefGoogle Scholar
17Hoff, D.. Local solutions of a compressible flow problem with Navier boundary conditions in general three-dimensional domains. SIAM J. Math. Anal. 44 (2012), 633–650.CrossRefGoogle Scholar
18Kreml, O., Mácha, V., Nečasová, Ṧ. and Wróblewska-Kamińska, A.. Weak solutions to the full Navier-Stokes-Fourier system with slip boundary conditions in time dependent domain accepted to Journal de Mathématiques Pures et Appliquées, arXiv:1511.04915.Google Scholar
19Kreml, O., Mácha, V., Nečasová, Š. and Wróblewska-Kamińska, A.. Flow of heat conducting fluid in a time dependent domain Preprint, arXiv:1711.10042, submitted.Google Scholar
20Lions, P.-L.. Mathematical topics in fluid dynamics, vol. 2, Compressible models (Oxford: Oxford Science Publication, 1998).Google Scholar
21Masmoudi, N.. Incompressible inviscid limit of the compressible Navier-Stokes system. Ann. Inst. H. Poincaré, Anal. Non Linéaire 18 (2001), 199–224.CrossRefGoogle Scholar
22Matsumura, A. and Nishida, T.. The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Kyoto Univ. 20 (1980), 67–104.CrossRefGoogle Scholar
23Matsumura, A. and Nishida, T.. Initial-boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids. Comm. Math. Phys. 89 (1983), 445–464.CrossRefGoogle Scholar
24Mucha, P. B. and Zaja̧czkowski, W. M.. On local-in-time existence for the Dirichlet problem for equations of compressible viscous fluids. Ann. Polon. Math. 78 (2002), 227–239.CrossRefGoogle Scholar
25Mucha, P. B. and Zaja̧czkowski, W. M.. On a Lp-estimate for the linearized compressible Navier-Stokes equations with the Dirichlet boundary conditions. J. Differ. Equ. 186 (2002), 377–393.CrossRefGoogle Scholar
26Mucha, P. B. and Zaja̧czkowski, W. M.. Global existence of solutions of the Dirichlet problem for the compressible Navier-Stokes equations. ZAMM Z. Angew. Math. Mech. 84 (2004), 417–424.CrossRefGoogle Scholar
27Murata, M.. On a maximal L p−L q approach to the compressible viscous fluid flow with slip boundary condition. Nonlinear Anal. 106 (2014), 86–109.CrossRefGoogle Scholar
28Piasecki, T. and Pokorný, M.. Strong solutions to the stationary compressible Navier-Stokes-Fourier system with slip-inflow boundary conditions. ZAMM Zeitschrift für Angewandte Mathematik und Mech. 94 (2014), 1035–1057.CrossRefGoogle Scholar
29Saint-Raymond, L.. Hydrodynamic limits: some improvements of the relative entropy method. Ann. Inst. H. Poincaré, Anal. Non Linéaire 26 (2009), 705–744.CrossRefGoogle Scholar
30Shibata, Y. and Murata, M.. On the global well-posedness for the compressible Navier-Stokes equations with slip boundary condition. J. Differ. Equ. 260 (2016), 5761–5795.CrossRefGoogle Scholar
31Valli, A.. An existence theorem for compressible viscous fluids. Ann. Math. Pura Appl. (IV) 130 (1982), 197–213; Ann. Math. Pura Appl. (IV), 132:399–400, 1982.CrossRefGoogle Scholar
32Valli, A.. Periodic and stationary solutions for compressible Navier-Stokes equations via a stability method. Ann. Sc. Norm. Super. Pisa (IV) 10 (1983), 607–647.Google Scholar
33Wang, S. and Jiang, S.. The convergence of the Navier-Stokes-Poisson system to the compressible Euler equation. J. Math. Kyoto Univ. 26 (1986), 323–331.Google Scholar
34Zaja̧czkowski, W. M.. On nonstationary motion of a compressible barotropic viscous fluid bounded by a free surface. Dissertationes Math. (Rozprawy Mat.) 324 (1993).Google Scholar
35Zaja̧czkowski, W. M.. On nonstationary motion of a compressible barotropic viscous capillary fluid bounded by a free surface. SIAM J. Math. Anal. 25 (1994), 1–84.CrossRefGoogle Scholar
36Zaja̧czkowski, W. M.. On nonstationary motion of a compressible barotropic viscous fluid with boundary slip condition. J. Appl. Anal. 4 (1998), 167–204.CrossRefGoogle Scholar