Skip to main content
SpringerLink
Log in
Book cover

Handbook of Mathematical Analysis in Mechanics of Viscous Fluids pp 1–57Cite as

Existence and Uniqueness of Strong Stationary Solutions

  • Living reference work entry
  • First Online:

  • 309 Accesses

Abstract

This chapter contains a survey of results in the existence theory of strong solutions to the steady compressible Navier–Stokes system. In the first part, the compressible Navier–Stokes equations are studied in bounded domains, both for homogeneous (no inflow) and inhomogeneous (inflow) boundary conditions. The solutions are constructed in Sobolev spaces. The next part contains the results for unbounded domains, especially for the exterior domains. Here, not only the question of existence and uniqueness is considered, but also the asymptotic structure near infinity is studied. Due to the different nature of the problems, the two- and three-dimensional problems are treated separately.

This is a preview of subscription content, log in via an institution.

References

  1. M. Bause, J.G. Heywood, A. Novotný, M. Padula, An Iterative Scheme for Steady Compressible Viscous Flow, Modified to Treat Large Potential Forces. Mathematical Fluid Mechanics, Advances in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2001), pp. 27–45

    Google Scholar 

  2. M. Bause, J.G. Heywood, A. Novotný, M. Padula, On some approximation schemes for steady compressible viscous flow. J. Math. Fluid Mech. 5(3), 201–230 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. H. Beirão da Veiga, On a stationary transport equation. Ann. Univ. Ferrara Sez. VII (N.S.) 32, 79–91 (1986)

    Google Scholar 

  4. H. Beirão da Veiga, An Lp-theory for the n-dimensional, stationary, compressible Navier–Stokes equations, and the incompressible limit for compressible fluids. The equilibrium solutions. Commun. Math. Phys. 109, 229–248 (1987)

    Article  MATH  Google Scholar 

  5. H. Beirão da Veiga, Existence results in Sobolev spaces for a stationary transport equation. Ricerche Mat. 36(suppl), 173–184 (1987)

    MathSciNet  MATH  Google Scholar 

  6. C. Dou, F. Jiang, S. Jiang, Y.-F. Yang, Existence of strong solutions to the steady Navier–Stokes equations for a compressible heat-conductive fluid with large forces. J. Math. Pures Appl. 103(5), 1163–1197 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Dutto, A. Novotný, Physically reasonable solutions to steady compressible Navier–Stokes equations in the plane. Preprint University of Toulon (1998)

    Google Scholar 

  8. P. Dutto, A. Novotný, Physically reasonable solutions to steady compressible Navier–Stokes equations in 2d exterior domains with nonzero velocity at infinity. J. Math. Fluid. Mech. 3, 99–138 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Dutto, J.L. Impagliazzo, A. Novotný, Schauder estimates for steady compressible Navier–Stokes equations in bounded domains. Rend. Sem. Mat. Univ. Padova 98, 125–139 (1997)

    MathSciNet  MATH  Google Scholar 

  10. R. Farwig, Stationary solutions of compressible Navier–Stokes equations with slip boundary condition. Commun. Partial Differ. Equ. 14(11), 1579–1606 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  11. R. Farwig, A. Novotný, M. Pokorný, The fundamental solution of a modified Oseen problem. Z. Anal. Anwendungen 19, 713–728 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. G.P. Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations. Steady-State Problems. Springer Monographs in Mathematics, 2nd edn. (Springer, New York, 2011)

    Google Scholar 

  13. G.P. Galdi, A. Novotný, M. Padula, On the twodimensional steady-state problem of a viscous gas in an exterior domain. Pac. J. Math. 179, 65–100 (1997)

    Article  MATH  Google Scholar 

  14. Y. Guo, S. Jiang, C. Zhou, Steady viscous compressible channel ows. SIAM J. Math. Anal. 47(5), 3648–3670 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. J.G. Heywood, M. Padula, On the existence and uniqueness theory for steady compressible viscous flow, in Fundamental Directions in Mathematical Fluid Mechanics, ed. by G.P. Galdi, J.G. Heywood, R. Rannacher. Advances in Mathematical Fluid Mechanics (Birkhäuser, Basel, 2000), pp. 171–189

    Google Scholar 

  16. B.J. Jin, M. Padula, Steady flows of compressible fluids in a rigid container with upper free boundary. Math. Ann. 329(4), 723–770 (2004)

    MathSciNet  MATH  Google Scholar 

  17. R.B. Kellogg, J.R. Kweon, Compressible Navier–Stokes equations in a bounded domain with inflow boundary condition. SIAM J. Math. Anal. 28(1), 94–108 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. R.B. Kellogg, J.R. Kweon, Smooth solution of the compressible Navier–Stokes equations in an unbounded domain with inflow boundary condition. J. Math. Anal. Appl. 220, 657–675 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Koch, Partial differential equations and singular integrals, in: Dispersive Nonlinear Problems in Mathematical Physics. Quaderni di Matematica, vol. 15 (Department of Mathematics, Seconda Università di Napoli, Caserta, 2004), pp. 59–122

    Google Scholar 

  20. P. Konieczny, O. Kreml, On the 3D steady flow of a second grade fluid past an obstacle. J. Math. Fluid Mech. 14(2), 295–309 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Kračmar, A. Novotný, M. Pokorný, Estimates of Oseen kernels in weighted L p spaces. J. Math. Soc. Jpn 53(1), 59–111 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  22. J.R. Kweon, A regularity result of solution to the compressible Stokes equations on a convex polygon. Z. Angew. Math. Phys. 55(3), 435–450 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. J.R. Kweon, R.B. Kellogg, Regularity of solutions to the Navier–Stokes equations for compressible barotropic flows on a polygon. Arch. Ration. Mech. Anal. 163(1), 35–64 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  24. J.R. Kweon, R.B. Kellogg, Regularity of solutions to the Navier–Stokes system for compressible flows on a polygon. SIAM J. Math. Anal. 35(6), 1451–1485 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. J.R. Kweon, M. Song, Boundary geometry and regularity of solution to the compressible Navier–Stokes equations in bounded domains of Rn. ZAMM Z. Angew. Math. Mech. 86(6), 495–504 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Matsumura, T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids. Proc. Jpn Acad. Ser. A Math. Sci. 55(9), 337–342 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Matsumura, T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20(1), 67–104 (1980)

    MathSciNet  MATH  Google Scholar 

  28. A. Matsumura, T. Nishida, Exterior stationary problems for the equations of motion of compressible heat-conductive fluids, in Proceedings of Equadiff 89, ed. by C.M. Dafermos, et al. (Dekker, Amsterdam, 1989), pp. 473–479

    Google Scholar 

  29. P.B. Mucha, T. Piasecki, Compressible perturbation of Poiseuille type flow. J. Math. Pures Appl. (9) 102(2), 338–363 (2014)

    Google Scholar 

  30. P.B. Mucha, R. Rautmann, Convergence of Rothe’s scheme for the Navier–Stokes equations with slip conditions in 2D domains. ZAMM Z. Angew. Math. Mech. 86(9), 691–701 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. S.A. Nazarov, A. Novotný, K. Pileckas, On steady compressible Navier–Stokes equations in plane domains with corners. Math. Ann. 304(1), 121–150 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  32. A. Novotný, Existence and uniqueness of stationary solutions for viscous compressible heat-conductive fluid with great potential and small nonpotential forces, in International Conference on Differential Equations, Barcelona, 1991, vols. 1 and 2 (World Scientific, River Edge, 1993), pp. 795–799

    Google Scholar 

  33. A. Novotný, Steady flows of viscous compressible fluids in exterior domains under small perturbations of great potential forces. Math. Mod. Methods Appl. Sci. 3, 725–757 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Novotný, About steady transport equation. I. Lp-approach in domains with smooth boundaries. Comment. Math. Univ. Carol. 37(1), 43–89 (1996)

    MATH  Google Scholar 

  35. A. Novotný, Compactness of steady compressible isentropic Navier–Stokes equations via the decomposition method (the whole 3-D space), in Theory of the Navier–Stokes Equations, ed. by J. Málek, J. Nečas, M. Rokyta. Series on Advances in Mathematics for Applied Sciences, vol. 47 (World Scientific, River Edge, 1998), pp. 106–120

    Google Scholar 

  36. A. Novotný, M. Padula, Existence and uniqueness of stationary solutions of equations of a compressible viscous heat-conducting fluid for large potential and small nonpotential external forces. (Russian) Sibirsk. Mat. Zh. 34(5), 120–146 (1993). Translation in Siberian Math. J. 34(5), 898–922 (1993)

    Google Scholar 

  37. A. Novotný, M. Padula, L p-approach to steady flows of viscous compressible fluids in exterior domains. Arch. Ration. Mech. Anal. 126, 243–297 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  38. A. Novotný, M. Padula, Note on decay of solutions of steady Navier–Stokes equations in 3-D exterior domains. Differ. Integral Equ. 8(7), 1833–1842 (1995)

    MathSciNet  MATH  Google Scholar 

  39. A. Novotný, M. Padula, Physically reasonable solutions to steady compressible Navier–Stokes equations in 3D-exterior domains (v ∞ = 0). J. Math. Kyoto Univ. 36(2), 389–422 (1996)

    MathSciNet  MATH  Google Scholar 

  40. A. Novotný, M. Padula, Physically reasonable solutions to steady compressible Navier–Stokes equations in 3D-exterior domains (v ∞ ≠ 0). Math. Ann. 308, 439–489 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  41. A. Novotný, M. Padula, P. Penel, A remark on the well posedness of the problem of a steady flow of a viscous barotropic gas in a pipe. Commun. Partial Differ. Equ. 21(1–2), 23–34 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Novotný, P. Penel, An L p approach for steady flows of compressible heat conducting gas. Preprint University of Toulon (1997)

    Google Scholar 

  43. A. Novotný, P. Penel, V.A. Solonnikov, Investigation of the problem of viscous compressible fluid flow around a bounded body in Hölder spaces. (Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 245 (1997), Vopr. Kvant. Teor. Polya i Stat. Fiz. 14, 247–269, 287; translation in J. Math. Sci. (N.Y.) 100(2), 2166–2180 (2000)

    Google Scholar 

  44. A. Novotný, K. Pileckas, Steady compressible Navier–Stokes equations with large potential forces via a method of decomposition. Math. Methods Appl. Sci. 21(8), 665–684 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  45. A. Novotný, I. Straškraba, Introduction to the Mathematical Theory of Compressible Flow. Oxford Lecture Series in Mathematics and Its Applications, vol. 27 (Oxford University Press, Oxford, 2004)

    Google Scholar 

  46. M. Padula, Existence and continuous dependence for solutions to the equations of a one-dimensional model in gas dynamics. Meccanica 17, 128–135 (1981)

    Article  MATH  Google Scholar 

  47. M. Padula, Existence and uniqueness for viscous steady compressible motions, in Proc. Sem. Fis. Mat., Dinamica dei Fluidi e dei gaz ionizzati, Trieste (1982)

    Google Scholar 

  48. M. Padula, Uniqueness theorems for steady, compressible, heat-conducting fluids: bounded domains. I. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 74(6), 380–387 (1983)

    Google Scholar 

  49. M. Padula, Uniqueness theorems for steady, compressible, heat-conducting fluids: exterior domains. II. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 75(1–2), 56–60 (1984)

    Google Scholar 

  50. M. Padula, Existence and uniqueness for viscous, steady compressible motions. Arch. Ration. Mech. Anal. 77, 89–102 (1987)

    MathSciNet  MATH  Google Scholar 

  51. M. Padula, A representation formula for the steady solutions of a compressible fluid moving at low speed, in Proceedings of the Fourth International Workshop on Mathematical Aspects of Fluid and Plasma Dynamics, Kyoto, 1991. Transp. Theory Stat. Phys. 21(4–6), 593–613 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Padula, On the exterior steady problem for the equations of a viscous isothermal gas. Comment. Math. Univ. Carol. 34(2), 275–293 (1993)

    MathSciNet  MATH  Google Scholar 

  53. M. Padula, Mathematical properties of motions of viscous compressible fluids, in Progress in Theoretical and Computational Fluid Mechanics, Paseky, 1993, ed. by G.P. Galdi, J. Nečas. Pitman Research Notes in Mathematics Series, vol. 308 (Longman Scientific & Technical, Harlow, 1994), pp. 128–173

    Google Scholar 

  54. M. Padula, Steady Flows of Barotropic Viscous Fluids. Classical Problems in Mechanics, Quaderni di Matematica, vol. 1 (Department of Mathematics, Seconda Università di Napoli, Caserta, 1997), pp. 253–345

    Google Scholar 

  55. M. Padula, K. Pileckas, On the existence of steady motions of a viscous isothermal fluid in a pipe, in Navier–Stokes Equations and Related Nonlinear Problems, Funchal, 1994 (Plenum, New York, 1995), pp. 171–188

    MATH  Google Scholar 

  56. M. Padula, K. Pileckas, On the existence and asymptotical behaviour of a steady flow of a viscous barotropic gas in a pipe. Ann. Mat. Pura Appl. 172(4), 191–218 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  57. T. Piasecki, Steady compressible Navier–Stokes flow in a square. J. Math. Anal. Appl. 357, 447–467 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  58. T. Piasecki, On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain. J. Differ. Equ. 248, 2171–2198 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  59. T. Piasecki, M. Pokorný, Strong solutions to the Navier–Stokes-Fourier system with slip-inflow boundary conditions. ZAMM Z. Angew. Math. Mech. 94(12), 1035–1057 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  60. P.I. Plotnikov, J. Sokolowski, Shape derivative of drag functional. SIAM J. Control Optim. 48(7), 4680–4706 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  61. P. Plotnikov, J. Sokolowski, Compressible Navier–Stokes Equations. Theory and Shape Optimization. Instytut Matematyczny Polskiej Akademii Nauk. Monografie Matematyczne (New Series) [Mathematics Institute of the Polish Academy of Sciences. Mathematical Monographs (New Series)], vol. 73 (Birkhäuser/Springer/Basel AG, Basel, 2012)

    Google Scholar 

  62. P.I. Plotnikov, E.V. Ruban, J. Sokolowski, Inhomogeneous boundary value problems for compressible Navier–Stokes equations: well-posedness and sensitivity analysis. SIAM J. Math. Anal. 40(3), 1152–1200 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  63. M. Pokorný, Asymptotic behaviour of solutions of certain PDE’s describing the flow of fluids in unbounded domains. PhD. thesis, Charles University in Prague and University of Toulon (1999)

    Google Scholar 

  64. C.G. Simader, H. Sohr, The weak Dirichlet problem for Δ in L q in bounded and exterior domains. Stab. Anal. Cont. Media 2, 183–202 (1992)

    Google Scholar 

  65. C.G. Simader, H. Sohr, A new approach to the Helmholtz decomposition and the Neumann problem in L q-spaces for bounded and exterior domains, in Mathematical Problems Relating to the Navier–Stokes Equation, ed. by G.P. Galdi. Series on Advances in Mathematics for Applied Sciences, vol. 11 (World Scientific, River Edge, 1992), pp. 1–35

    Google Scholar 

  66. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, vol. 43. Monographs in Harmonic Analysis, III (Princeton University Press, Princeton, 1993)

    Google Scholar 

  67. A. Valli, Periodic and stationary solutions for compressible Navier–Stokes equations via a stability method. Ann. Scuola Normale Sup. Pisa 4(1), 607–646 (1983)

    MathSciNet  MATH  Google Scholar 

  68. A. Valli, On the existence of stationary solutions to compressible Navier–Stokes equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 4(1), 99–113 (1987)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author (PBM) has been partly supported by the National Science Centre grant 2014/14/M/ST1/00108 (Harmonia). The work of the first (OK) and the third author (MP) was partially supported by the Grant Agency of the Czech Republic, Grant No. 16-03230S.

Author information

Authors and Affiliations

  1. Institute of Mathematics, Czech Academy of Sciences, Praha, Czech Republic

    Ondřej Kreml

  2. Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland

    Piotr B. Mucha

  3. Faculty of Mathematics and Physics, Charles University, Praha, Czech Republic

    Milan Pokorný

Authors
  1. Ondřej Kreml
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Piotr B. Mucha
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Milan Pokorný
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ondřej Kreml .

Editor information

Editors and Affiliations

  1. Tokyo, Tokyo, Japan

    Yoshikazu Giga

  2. Toulon, Var, France

    Antonin Novotny

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this entry

Cite this entry

Kreml, O., Mucha, P.B., Pokorný, M. (2016). Existence and Uniqueness of Strong Stationary Solutions. In: Giga, Y., Novotny, A. (eds) Handbook of Mathematical Analysis in Mechanics of Viscous Fluids. Springer, Cham. https://doi.org/10.1007/978-3-319-10151-4_65-1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-10151-4_65-1

  • Received:

  • Accepted:

  • Published:

  • Publisher Name: Springer, Cham

  • Online ISBN: 978-3-319-10151-4

  • eBook Packages: Springer Reference MathematicsReference Module Computer Science and Engineering

Publish with us

Policies and ethics

Search

Navigation