Skip to main content
SpringerLink
Log in
Book cover

Fluids Under Pressure pp 349–416Cite as

Birkhäuser

The Role of Pressure in the Theory of Weak Solutions to the Navier-Stokes Equations

  • Chapter
  • First Online:

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

Abstract

Sections 4.1 and 4.2 contain an introduction, notation and definitions and basic properties of used function spaces and operators. A pressure, associated with a weak solution to the Navier-Stokes equations for incompressible fluid, is constructed in Sect. 4.3. The interior regularity of the pressure in regions, where the velocity satisfies Serrin’s integrability conditions, is studied in Sect. 4.4. Finally, Sect. 4.5 is devoted to criteria of regularity for weak solutions to the Navier-Stokes equations, formulated in terms of the pressure.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Bibliography

  1. H. Al Baba, Ch. Amrouche, M. Escobedo: L p–theory for the time dependent Navier-Stokes problem with Navier-type boundary conditions. Proc. of the 13th International Conference Zaragoza-Pau on Mathematics and its Applications, Monografías Matemáticas García de Galdeano 40, Prensas Univ. Zaragoza, Zaragoza 2016, pp. 1–8.

    Google Scholar 

  2. Ch. Amrouche, N. Seloula: L p–theory for the Navier–Stokes equations with pressure boundary conditions. Discr. Contin. Dyn. Syst. Ser. S6 (2013), no. 5, 1113–1137.

    MathSciNet  MATH  Google Scholar 

  3. Ch. Amrouche, A. Rejaiba: L p-theory for Stokes and Navier–Stokes equations with Navier boundary condition. J. Differential Equations256 (2014), 1515–1547.

    MathSciNet  MATH  Google Scholar 

  4. Ch. Amrouche, A. Rejaiba: Navier-Stokes equations with Navier boundary condition. Math. Meth. Appl. Sci.39 (2016), 5091–5112.

    MathSciNet  MATH  Google Scholar 

  5. H. Beirão da Veiga: Concerning the regularity of the solutions to the Navier-Stokes equations via the truncation method, Part II. In Équations aux dérivées partielles et applications, Gauthier-Villars, Paris 1998, pp. 127–138.

    Google Scholar 

  6. H. Beirão da Veiga: A sufficient condition on the pressure for the regularity of weak solutions to the Navier-Stokes equations. J. Math. Fluid Mech.2 (2000), 96–106.

    MathSciNet  MATH  Google Scholar 

  7. L. Berselli: Sufficient conditions for the regularity of the solutions of the Navier-Stokes equations. Math. Meth. Appl. Sci.22 (1999), 1079–1085.

    MathSciNet  MATH  Google Scholar 

  8. L. Berselli, G. P. Galdi: Regularity criterions involving the pressure for the weak solutions to the Navier-Stokes equations. Proc. Amer. Math. Soc.130 (2002), no. 12, 3585–3595.

    MathSciNet  MATH  Google Scholar 

  9. S. Bosia, M. Conti, V. Pata: A regularity criterion for the Navier-Stokes equations in terms of the pressure gradient. Cent. Eur. J. Math.12 (2014), 1015–1025.

    MathSciNet  MATH  Google Scholar 

  10. W. Borchers, H. Sohr: On the equations rot v = g and div u = f with zero boundary conditions. Hokkaido Math. J.19 (1990), 67–87.

    MathSciNet  MATH  Google Scholar 

  11. P. Butzer, H. Berens: Semi-Groups of Operators and Approximation. Springer, Berlin-Heidelberg-New York 1967.

    MATH  Google Scholar 

  12. L. Caffarelli, R. Kohn, L. Nirenberg: Partial regularity of suitable weak solutions of the Navier-Stokes equations. Commun. Pure Appl. Math.35 (1982), 771–831.

    MathSciNet  MATH  Google Scholar 

  13. Z. Cai, J. Fan, J. Zhai: Regularity criteria in weak spaces for 3-dimensional Navier-Stokes equations in terms of the pressure. Differential Integral Equations23 (2010), no. 11–12, 1023–1033.

    MathSciNet  MATH  Google Scholar 

  14. D. Chae, J. Lee: Regularity criterion in terms of pressure for the Navier-Stokes equations. Nonlinear Analysis46 (2001), 727–735.

    MathSciNet  MATH  Google Scholar 

  15. D. Chamorro, P.-G. Lemarié-Rieusset, K. Mayoufi: The role of pressure in the partial regularity theory for weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal.228 (2018), 237–277.

    MathSciNet  MATH  Google Scholar 

  16. G. Q. Chen, D. Osborne, Z. Qian: The Navier-Stokes equations with the kinematic and vorticity boundary conditions on non–flat boundaries. Acta Math. Sci.29B (2009), no. 4, 919–948.

    MathSciNet  MATH  Google Scholar 

  17. G. Q. Chen, Z. Qian: A study of the Navier-Stokes equations with the kinematic and Navier boundary conditions. Indiana Univ. Math. J.59 (2010), no. 2, 721–760.

    MathSciNet  MATH  Google Scholar 

  18. Q. Chen, Z. Zhang: Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations. Proc. Amer. Math. Soc.136 (2007), no. 6, 1829–1837.

    MathSciNet  MATH  Google Scholar 

  19. L. Escauriaza, G. Seregin, V. Šverák: L 3,∞–solutions of the Navier–Stokes equations and backward uniqueness. Russian Math. Surveys58 (2003), 2, 211–250.

    Google Scholar 

  20. J. Fan, T. Ozawa: Regularity criterion for weak solutions to the Navier-Stokes equations in terms of pressure. J. Inequal. Appl. 2008, article ID 412678, 6 pages, DOI: 10.1155/2008/412678.

    Google Scholar 

  21. J. Fan, S. Jiang, G. Ni: On regularity criteria for the n-dimensional Navier–Stokes equations in terms of the pressure. J. Differential Equations244 (2008), 2963–2979.

    MathSciNet  MATH  Google Scholar 

  22. R. Farwig, H. Kozono, H. Sohr: The Helmholtz decomposition in arbitrary unbounded domains - a theory beyond L 2. Proc. of Equadiff 11, ed. M. Fila et al, Comenius University Press, ISBN 978-80-227-2624-5, Bratislava 2005, pp. 77–85.

    Google Scholar 

  23. R. Farwig, H. Kozono, H. Sohr: Energy–based regularity criteria for the Navier-Stokes equations. J. Math. Fluid Mech.8 (2008), no. 3, 428–442.

    MathSciNet  MATH  Google Scholar 

  24. R. Farwig, V. Rosteck: Maximal regularity of the Stokes system with Navier boundary condition in general unbounded domains. To appear in J. Math. Soc. Japan.

    Google Scholar 

  25. M. Feistauer, T. Neustupa: On non-stationary viscous incompressible flow through a cascade of profiles. Math. Meth. Appl. Sci.29 (2006), no. 16, 1907–1941.

    MathSciNet  MATH  Google Scholar 

  26. M. Feistauer, T. Neustupa: On the existence of a weak solution of viscous incompressible flow past a cascade of profiles with an arbitrarily large inflow. J. Math. Fluid Mech.15 (2013), no. 4, 701–715.

    MathSciNet  MATH  Google Scholar 

  27. D. Fujiwara, H. Morimoto: An L r-theorem of the Helmholtz decomposition of vector fields. J. Fac. Sci. Univ. Tokyo Sec. IA Math.24 (1977), no. 3, 685–700.

    MathSciNet  MATH  Google Scholar 

  28. G. P. Galdi, C. G. Simader, H. Sohr: On the Stokes problem in Lipschitz domains. Ann. Mat. Pura Appl. (IV)167 (1994), 147–163.

    MathSciNet  MATH  Google Scholar 

  29. G. P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. 2nd edition, Springer 2011.

    Google Scholar 

  30. G. P. Galdi: An Introduction to the Navier-Stokes initial–boundary value problem. In Fundamental Directions in Mathematical Fluid Mechanics, ed. G. P. Galdi, J. Heywood, R. Rannacher, series “Advances in Mathematical Fluid Mechanics”. Birkhäuser, Basel 2000, pp. 1–98.

    Google Scholar 

  31. G. P. Galdi, J. Neustupa: Steady-state Navier–Stokes flow around a moving body. In Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, eds. Y. Giga and A. Novotný, Springer 2018, pp. 341–417.

    Google Scholar 

  32. F. Gazzola: A note on the evolution of Navier-Stokes equations with a pressure-dependent viscosity. Z. Angew. Math. Phys.48 (1997), 760–773.

    MathSciNet  MATH  Google Scholar 

  33. F. Gazzola, P. Secchi: Some results about stationary Navier-Stokes equations with a pressure-dependent viscosity. In Navier-Stokes equations: theory and numerical methods (Varenna 1997), ed. R. Salvi, Pitman Res. Notes Math. Ser., Vol. 388, Longman, Harlow 1998, pp. 31–37.

    Google Scholar 

  34. J. Geng, Z. Shen: The Neumann problem and Helmholtz decomposition in convex domains. J. Funct. Anal.259 (8), 2010, 2147–2164.

    MathSciNet  MATH  Google Scholar 

  35. Y. Giga: Solutions for semilinear parabolic equations in L p and regularity of weak solutions of the Navier-Stokes system. J. Differential Equations62 (1986), no. 2, 186–212.

    MathSciNet  MATH  Google Scholar 

  36. Y. Giga, H. Sohr: Abstract L p-estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains. J. Funct. Anal.102 (1991), 72–94.

    MathSciNet  MATH  Google Scholar 

  37. K. Kang, J. Lee: On regularity criteria in conjunction with the pressure of the Navier–Stokes equations. Internat. Math. Res. Notes Vol. 2006, Issue 9, Article ID 80762, 25 pp.

    Google Scholar 

  38. K. Kang, J. Lee: Erratum: On regularity criteria in conjunction with the pressure of the Navier–Stokes equations. Internat. Math. Res. Notes Vol. 2010, Issue 9, 1772–1774.

    MATH  Google Scholar 

  39. S. Kaniel: A sufficient condition for smoothness of solutions of the Navier-Stokes equations. Israel J. Math.6 (1969), 354–358.

    MathSciNet  MATH  Google Scholar 

  40. H. Kozono, T. Yanagisawa: L r-Helmholtz decomposition and its application to the Navier-Stokes equations. In Lectures on the Analysis of Nonlinear Partial Differential Equations, Part 3, Morningside Lect. Math. 3, Int. Press, Somerville, MA, 2013, pp. 237–290.

    Google Scholar 

  41. S. Kračmar, J. Neustupa: A weak solvability of a steady variational inequality of the Navier–Stokes type with mixed boundary conditions. Nonlinear Analysis47 (2001), no. 6, 4169–4180.

    MathSciNet  MATH  Google Scholar 

  42. S. Kračmar, J. Neustupa: Modeling of the unsteady flow through a channel with an artificial outflow condition by the Navier–Stokes variational inequality. Math. Nachr.291 (2018), no. 11–12, 1–14.

    MathSciNet  MATH  Google Scholar 

  43. P. Kučera, Z. Skalák: Solutions to the Navier?Stokes equations with mixed boundary conditions. Acta Appl. Math.54 (1998), no. 3, 275–288.

    MathSciNet  MATH  Google Scholar 

  44. P. Kučera: Basic properties of the non-steady Navier-Stokes equations with mixed boundary conditions in a bounded domain. Ann. Univ. Ferrara Sez. VII Sci. Mat.55 (2009), 289–308.

    MathSciNet  MATH  Google Scholar 

  45. P. Kučera, J. Neustupa: On robustness of a strong solution to the Navier–Stokes equations with Navier’s boundary conditions in the L 3–norm. Nonlinearity30 (2017), no. 4, 1564–1583.

    MathSciNet  MATH  Google Scholar 

  46. P. Kučera, S. Nečasová, J. Neustupa: A pressure associated with a weak solution to the Navier-Stokes equations with Navier’s boundary conditions.

    Google Scholar 

  47. O. A. Ladyzhenskaya, G. Seregin: On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations. J. Math. Fluid Mech.1 (1999), 356–387.

    MathSciNet  MATH  Google Scholar 

  48. M. Lanzendörfer, J. Stebel: On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities. Appl. Math.56 (2011), no. 3, 265–285.

    MathSciNet  MATH  Google Scholar 

  49. F. Lin: A new proof of the Caffarelli-Kohn-Nirenberg theorem. Commun. Pure Appl. Math.51 (1998), 241–257.

    MathSciNet  MATH  Google Scholar 

  50. J. L. Lions: Quelques méthodes de résolution des problèmes âux limites non linéaire. Dunod, Gauthier-Villars, Paris 1969.

    MATH  Google Scholar 

  51. J. Málek, J. Nečas, K. R. Rajagopal: Global analysis of the flows of fluids with pressure-dependent viscosities. Arch. Rat. Mech. Anal.165 (2002), no. 3, 243–269.

    MathSciNet  MATH  Google Scholar 

  52. J. Málek, K. R. Rajagopal: Mathematical properties of the solutions to the equations governing the flow of fluids with pressure and shear rate dependent viscosities. In Handbook of Mathematical Fluid Dynamics, Vol. 4, Chap. 7, Elsevier/North-Holland, Amsterdam 2007, pp. 407–444.

    Google Scholar 

  53. S. Marušić: On the Navier–Stokes system with pressure boundary condition. Ann. Univ. Ferrara Sez. VII Sci. Mat.53 (2007), 319–331.

    MathSciNet  MATH  Google Scholar 

  54. G. Nardi: Schauder estimate for solutions of Poisson’s equation with Neumann boundary condition. Enseign. Math.60 (2014), no. 2, 421–435.

    MathSciNet  MATH  Google Scholar 

  55. J. Nečas, J. Neustupa: New conditions for local regularity of a suitable weak solution to the Navier-Stokes equations. J. Math. Fluid Mech.4 (2002), 237–256.

    MathSciNet  MATH  Google Scholar 

  56. Š. Nečasová, J. Wolf: On the existence of global strong solutions to the equations modelling a motion of a rigid body around a viscous fluid. Discr. Contin. Dyn. Syst.38 (2016), no. 3, 1539–1562.

    MATH  Google Scholar 

  57. J. Neustupa, P. Penel: Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations. In Mathematical Fluid Mechanics, Recent Results and Open Questions, ed. J. Neustupa and P. Penel, Birkhäuser, Basel 2001, pp. 237–268.

    Google Scholar 

  58. J. Neustupa, P. Penel: A weak solution to the Navier-Stokes system with Navier’s boundary condition in a time-varying domain. In Recent Developments of Mathematical Fluid Mechanics, Series: Advances in Mathematical Fluid Mechanics, ed. H. Amann, Y. Giga, H. Kozono, H. Okamoto, M. Yamazaki, Birhäuser-Verlag, Springer, Basel 2016, pp. 375–400.

    Google Scholar 

  59. J. Neustupa: The boundary regularity of a weak solution of the Navier-Stokes equation and connection with the interior regularity of pressure. Appl. Math.6 (2003), 547–558.

    MathSciNet  MATH  Google Scholar 

  60. J. Neustupa: A geometric improvement of a velocity–pressure local regularity criterion for a suitable weak solution to the Navier-Stokes equations. Mathematica Bohem.139 (2014), no. 4, 685–698.

    MathSciNet  MATH  Google Scholar 

  61. J. Neustupa, H. Al Baba: The interior regularity of pressure associated with a weak solution to the Navier-Stokes equations with the Navier-type boundary conditions. J. Math. Anal. Appl.463 (2018), no. 1, 222–234.

    MathSciNet  MATH  Google Scholar 

  62. J. Neustupa: A contribution to the theory of regularity of a weak solution to the Navier-Stokes equations via one component of velocity and other related quantities. J. Math. Fluid Mech.20 (2018), no. 3, 1249–1267.

    MathSciNet  MATH  Google Scholar 

  63. T. Ohyama: Interior regularity of weak solutions of the time dependent Navier-Stokes equations. Proc. Japan. Acad.36 (1960), 273–277.

    MathSciNet  MATH  Google Scholar 

  64. W. Rudin: Functional Analysis. Mc Graw-Hill Inc., New York 1973.

    MATH  Google Scholar 

  65. J. Saal: Stokes and Navier-Stokes equations with Robin boundary conditions in a half-space. J. Math. Fluid Mech.8 (2006), 211–241.

    MathSciNet  MATH  Google Scholar 

  66. V. Scheffer: Partial regularity of solutions to the Navier-Stokes equations. Pacific J. Math.66 (1976), 535–552.

    MathSciNet  MATH  Google Scholar 

  67. G. Seregin, V. Šverák: Navier-Stokes equations with lower bounds on the pressure. Arch. Rat. Mech. Anal.163 (2002), 65–86.

    MathSciNet  MATH  Google Scholar 

  68. J. Serrin: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Rat. Mech. Anal.9 (1962) 187–195.

    MathSciNet  MATH  Google Scholar 

  69. R. Shimada: On the L p − L q maximal regularity for Stokes equations with Robin boundary condition in a bounded domain. Math. Meth. Appl. Sci.30 (2007), 257–289.

    MathSciNet  MATH  Google Scholar 

  70. Ch. 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, Series: Adv. Math. Appl. Sci. 11, ed. G. P. Galdi, World Sci. Publ., River Edge, NJ, 1992, pp. 1–35.

    Google Scholar 

  71. J. Simon: On the existence of pressure for solutions of the variational Navier-Stokes equations. J. Math. Fluid Mech.1 (1999), no. 3, 225–234.

    MathSciNet  MATH  Google Scholar 

  72. Z. Skalák, P. Kučera: Regularity of pressure in the neighbourhood of regular points of weak solutions of the Navier-Stokes equations. Appl. Math.48 (2003), no. 6, 573–586.

    MathSciNet  MATH  Google Scholar 

  73. H. Sohr, W. von Wahl: On the regularity of the pressure of weak solutions of Navier-Stokes equations. Arch. Math.46 (1986), 428–439.

    MathSciNet  MATH  Google Scholar 

  74. H. Sohr: Zur Regularitätstheorie der instationären Gleichungen von Navier-Stokes. Math. Z.184 (1983), 359–375.

    MathSciNet  MATH  Google Scholar 

  75. H. Sohr: The Navier-Stokes Equations. An Elementary Functional Analytic Approach. Birkhäuser Advanced Texts, Basel-Boston-Berlin 2001.

    Google Scholar 

  76. M. Struwe: On a Serrin-type regularity criterion for the Navier-Stokes equations in terms of the pressure. J. Math. Fluid Mech.9 (2007), 235–242.

    MathSciNet  MATH  Google Scholar 

  77. T. Suzuki: Regularity criteria of weak solutions in terms of pressure in Lorentz spaces to the Navier-Stokes equations. J. Math. Fluid Mech.14 (2012), no. 4, 653–660.

    MathSciNet  MATH  Google Scholar 

  78. Y. Taniuchi: On generalized energy inequality of Navier-Stokes equations. Manuscripta Math.94 (1997), 365–384.

    MathSciNet  MATH  Google Scholar 

  79. R. Temam: Navier-Stokes Equations. North-Holland, Amsterdam-New York-Oxford 1977.

    MATH  Google Scholar 

  80. J. Wolf: A direct proof of the Caffarelli-Kohn-Nirenberg theorem. In Parabolic and Navier-Stokes equations,Banach Center Publ.81, Part 2, Polish Acad. of Sci., Inst. Math., Warsaw 2008, pp. 533–552.

    Google Scholar 

  81. J. Wolf: A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations. In Advances in Mathematical Fluid Mechanics, ed. R. Rannacher, A. Sequeira, Springer, Berlin, 2010, pp. 613–630.

    Google Scholar 

  82. J. Wolf: On the local regularity of suitable weak solutions to the generalized Navier-Stokes equations. Ann. Univ. Ferrara Sez. VII Sci. Mat.61 (2015), 149–171.

    MathSciNet  MATH  Google Scholar 

  83. J. Wolf: On the local pressure of the Navier-Stokes equations and related systems. Adv. Differential Equations22 (2017), no. 5–6, 305–338.

    MathSciNet  MATH  Google Scholar 

  84. Y. L. Xiao, Z. P. Xin: On the vanishing viscosity limit for the 3D Navier-Stokes equations with a slip boundary condition. Commun. Pure Appl. Math.60 (2007), 1027–1055.

    MathSciNet  MATH  Google Scholar 

  85. Y. Zhou: On regularity criteria in terms of pressure for the Navier-Stokes equations in \(\mathbb {R}^3\). Proc. Amer. Math. Soc.134 (2006), 149–156.

    Google Scholar 

  86. Y. Zhou: On a regularity criterion in terms of the gradient of pressure for the Navier-Stokes equations in \(\mathbb {R}^N\). Z. Angew. Math. Phys.57 (2006), 384–392.

    Google Scholar 

Download references

Acknowledgements

The author has been supported by the Mathematical Institute of the Czech Academy of Sciences (RVO 67985840) and by the Czech Science Foundation (grant No. 16-03230S).

Author information

Authors and Affiliations

  1. Institute of Mathematics CAS, Prague, Czech Republic

    Jiří Neustupa

Authors
  1. Jiří Neustupa
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiří Neustupa .

Editor information

Editors and Affiliations

  1. Department of Technical Mathematics, Czech Technical University, Prague, Czech Republic

    Tomáš Bodnár

  2. Department of Mechanical Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Giovanni P. Galdi

  3. Academy of Sciences of the Czech Republic, Prague, Czech Republic

    Šárka Nečasová

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Neustupa, J. (2020). The Role of Pressure in the Theory of Weak Solutions to the Navier-Stokes Equations. In: Bodnár, T., Galdi, G., Nečasová, Š. (eds) Fluids Under Pressure. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-39639-8_4

Download citation

Publish with us

Policies and ethics

Search

Navigation