Skip to main content
SpringerLink
Log in

An Introduction to the Navier-Stokes Initial-Boundary Value Problem

  • Chapter
Fundamental Directions in Mathematical Fluid Mechanics

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

Abstract

The equations of motion of an incompressible, Newtonian fluid — usually called Navier-Stokes equations — have been written almost one hundred eighty years ago. In fact, they were proposed in 1822 by the French engineer C.M.L.H. Navier upon the basis of a suitable molecular model. It is interesting to observe, however, that the law of interaction between the molecules postulated by Navier were shortly recognized to be totally inconsistent from the physical point of view for several materials and, in particular, for liquids. It was only more than twenty years later that the same equations were rederived by the twenty-six year old G. H. Stokes 1845 in a quite general way, by means of the theory of continua.

Rechts auf zwei hohen Felsen befinden sich Schlösser, unten breitet sich die Stadt

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

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  • Amann, H., 1999, On the Strong Solvability of the Navier-Stokes Equations, J. Math. Fluid Mech., in press

    Google Scholar 

  • Beiräo da Veiga, H., 1987, Existence and Asymptotic Behavior for Strong Solutions of the Navier-Stokes Equations in the Whole Space, Indiana Univ. Math. J., 36, 149

    Article  MathSciNet  Google Scholar 

  • Beiräo da Veiga, H., 1995a, Concerning the Regularity Problem for the Solutions of the Navier-Stokes Equations, Compte Rend. Acad. Sci. Paris, 321, 405

    Google Scholar 

  • Beiräo da Veiga, H., 1995b, A New Regularity Class for the Navier-Stokes Equations in ℝn, Chin. Ann. of Math, 16B, 407

    Google Scholar 

  • Beiräo da Veiga, H., 1996, Remarks on the Smoothness of the L∞(0,T;L 3) Solutions of the 3-D Navier-Stokes Equations, Preprint, University of Pisa

    Google Scholar 

  • Berselli L.C., 1999, Sufficient Conditions for the Regularity of the Solutions of the Navier-Stokes Equations, Math. Meth. Appl. Sci., 22, 1079

    Article  MathSciNet  MATH  Google Scholar 

  • Borchers, W. and Miyakawa, T., 1988, Decay for the Navier-Stokes Flow in Halfspaces, Math. Ann., 282, 139

    Article  MathSciNet  MATH  Google Scholar 

  • Caffarelli, L., Kohn, R. and Nirenberg, L., 1982, Partial Regularity of Suitable Weak Solutions of the Navier-Stokes Equations, Comm. Pure Appl. Math., 35, 771

    Article  MathSciNet  MATH  Google Scholar 

  • Calderon, P.C., 1990a, Existence of Weak Solutions for the Navier-Stokes Equations with Initial Data in L p, Trans. American Math. Soc., 318, 179

    MathSciNet  MATH  Google Scholar 

  • Calderon, P.C., 1990b, Addendum to the paper “Existence of Weak Solutions for the Navier-Stokes Equations with Initial Data in L p”, Trans. American Math. Soc., 318, 201

    MathSciNet  MATH  Google Scholar 

  • Calderon, P.C., 1993, Initial Values of Solutions of the Navier-Stokes Equations, Proc. American Math. Soc., 117, 761

    MathSciNet  MATH  Google Scholar 

  • Cannone M., 1997, A generalization of a theorem by Kato on Navier-Stokes equations, Rev. Mat. Iberoamer., 13 515.

    Article  MathSciNet  MATH  Google Scholar 

  • Chemin J.Y., 1999, Theorems of Unicity for the Threedimensional Navier-Stokes System, J. Anal. Math., 77 27

    Article  MathSciNet  MATH  Google Scholar 

  • Fabes, E.B., Jones, B.F. and Rivière, N.M., 1972, The Initial Value Problem for the Navier-Stokes Equations with Data in L p, Arch. Ratl Mech. Anal., 45, 222

    Article  MATH  Google Scholar 

  • Fabes, E.B., Lewis, J. and Rivière, N.M., 1977a, Singular Integrals and Hydrodynamical Potentials, American J. Math., 99, 601

    Article  MATH  Google Scholar 

  • Fabes, E.B., Lewis, J. and Rivière, N.M., 1977b, Boundary Value Problems for the Navier-Stokes Equations, American J. Math., 99, 626

    Article  MATH  Google Scholar 

  • Foiaş, C., 1961, Une Remarque sur l’Unicité des Solutions des Equations de Navier-Stokes en dimension n, Bull. Soc. Math. France, 89, 1

    MathSciNet  MATH  Google Scholar 

  • Foiaş, C. and Temam, R., 1979, Some Analytical and Geometric Properties of the Solutions of the Navier-Stokes Equations, J. Math. Pure Appl., 58, 339

    MATH  Google Scholar 

  • Galdi, G.P., 1994, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol I: Linearized Theory, Springer Tracts in Natural Phylosophy, Vol.38. Springer-Verlag.

    MATH  Google Scholar 

  • Galdi, G.P. and Maremonti, P., 1986, Monotonic Decreasing and Asymptotic Behavior of the Kinetic Energy for Weak Solutions of the Navier-Stokes Equations in Exterior Domains, Arch. Ratl Mech. Anal., 94, 253

    Article  MathSciNet  MATH  Google Scholar 

  • Galdi, G.P. and Maremonti, P., 1988, Sulla Regolarità delle Soluzioni Deboli al Sistema di Navier-Stokes in Domini Arbitrari, Ann. Univ. Femara, Sci. Mat, 34, 59

    MathSciNet  MATH  Google Scholar 

  • Giga, Y., 1986, Solutions for Semilinear Parabolic Equations in L P and Regularity of Weak Solutions of the Navier-Stokes System, J. Differential Eq., 62, 182

    Article  MathSciNet  Google Scholar 

  • Golovkin, K.K., 1964, On the Non-Uniqueness of Solutions of Certain Boundary-Value Problems for the Equations of Hydromechanics, Ž. Vyčisl. Mat. i. Mat. Fiz., 4, 773 (in Russian)

    MathSciNet  Google Scholar 

  • Heywood, J.G., 1980, The Navier-Stokes Equations: On the Existence, Regularity and Decay of Solutions, Indiana Univ. Math. J., 29, 639

    Article  MathSciNet  MATH  Google Scholar 

  • Heywood, J.G., 1988, Epochs of Irregularity for Weak Solutions of the Navier-Stokes Equations in Unbounded Domains, Tôhoku Math. J., 40, 293

    Article  MathSciNet  MATH  Google Scholar 

  • Hille E. and Phillips, R.S., 1957, Functional Analysis and Semigroups, American Math. Soc. Colloq. Publ. 31

    Google Scholar 

  • Hopf, E., 1950/1951, Über die Anfangswertaufgabe für die Hydrodynamischen Grundgleichungen, Math. Nachr., 4, 213

    Article  MathSciNet  Google Scholar 

  • Kaniel, S., 1969, A Sufficient Condition for Smoothness of Solutions of Navier-Stokes Equations, Israel J. Math., 6, 354

    Article  MathSciNet  MATH  Google Scholar 

  • Kaniel, S. and Shinbrot M., 1967, Smoothness of Weak Solutions of the Navier-Stokes Equations, Arch. Ratl Mech. Anal., 24, 302

    MathSciNet  MATH  Google Scholar 

  • Kato, H., 1977/78, On a Regularity of E. Hopf’s Weak Solutions for the Navier-Stokes Equations, Math. Rep. Kyushu Univ., 1, 15

    Google Scholar 

  • Kato, H., 1986, Regularity of Weak Solutions of the Navier-Stokes Equations, Math. Rep. Kyushu Univ., 15, 1

    Google Scholar 

  • Kato, H., 1989, Weak Solutions of the Navier-Stokes Initial Value Problem, Math. Rep. Kyushu Univ., 17, 57

    MathSciNet  MATH  Google Scholar 

  • Kato, H., 1993, On the Regularity and Uniqueness of Weak Solutions for the Navier-Stokes Equations, Memoirs of the Faculty Sci. Kyushu Univ. Ser. A, 47, 183

    Article  MATH  Google Scholar 

  • Kato T., 1984, Strong L p-Solutions to the Navier-Stokes Equations in ℝm, with Applications to Weak Solutions, Math. Z., 187, 471

    Article  MathSciNet  MATH  Google Scholar 

  • Kiselev, A.A. and Ladyzhenskaya, O.A., 1957, On Existence and Uniqueness of the Solution of the Nonstationary Problem for a Viscous Incompressible Fluid, Izv. Akad. Nauk SSSR, 21, 655

    MATH  Google Scholar 

  • Kozono H., 1998, Uniqueness and regularity of weak solutions to the Navier-Stokes equations, Lecture Notes in Num. and Appl. Anal. 16, 161

    MathSciNet  Google Scholar 

  • Kozono, H. and Sohr, H., 1996a, Remark on Uniqueness of Weak Solutions to the Navier-Stokes Equations, Analysis, 16, 255

    MathSciNet  MATH  Google Scholar 

  • Kozono, H. and Sohr, H., 1996b, Regularity Criterion on Weak Solutions to the Navier-Stokes Equations, Preprint, Universität Paderborn

    Google Scholar 

  • Kozono, H. and Yamazaki, M., 1998, Exterior problem for the stationary Navier-Stokes equations in the Lorentz space, Mathematische Annalen, 310, 279

    Article  MathSciNet  MATH  Google Scholar 

  • Ladyzhenskaya, O.A., 1958, Solution “in the Large” of the Nonstationary Boundary-Value Problem for the Navier-Stokes System for the Case of Two Space Variables, Doklady Akad. Nauk. SSSR, 123, 427 (in Russian)

    Google Scholar 

  • Ladyzhenskaya, O.A., 1967, Uniqueness and Smoothness of Generalized Solutions of the Navier-Stokes Equations, Zap. Naucn Sem. Leningrad Otdel Mat. Inst. Steklov, 5, 169 (in Russian)

    MATH  Google Scholar 

  • Ladyzhenskaya, O.A., 1969, Example of Nonuniqueness in the Hopf Class of Weak Solutions for the Navier-Stokes Equations, Izv. Akad. Nauk SSSR., Ser. Mat., 33 (1), 229 (in Russian)

    Google Scholar 

  • Ladyzhenskaya, O.A., and Seregin, G.A., 1999, On Partial Regularity of Suitable Weak Solutions to the Three-Dimensional Navier-Stokes Equations, J. Math. Fluid Mech., in press

    Google Scholar 

  • Ladyzhenskaya, O.A., Ural’ceva, N.N. and Solonnikov V.A., 1968, Linear and Quasilinear Equations of Parabolic Type, Transl. Math. Monographs, 23, American Math. Soc., Providence R.I.

    Google Scholar 

  • Leray, J., 1934a, Essai sur les Mouvements Plans d’un Liquide Visqueux que Limitent des Parois, J. Math. Pures Appl., 13, 331

    Google Scholar 

  • Leray, J., 1934b, Sur les Mouvements d’un Liquide Visqueux Emplissant l’Espace, Acta Math., 63, 193

    Article  MathSciNet  MATH  Google Scholar 

  • Lighthill, J., 1956, Nature, 178, 343

    Article  Google Scholar 

  • Lin, F.H., 1998, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math., 51, 241.

    Article  MathSciNet  MATH  Google Scholar 

  • Lions, J.L., 1960, Sur la Regularité et l’Unicité des Solutions Turbulentes des Équations de Navier-Stokes, Rend. Sem. Mat. Università di Padova, 30, 1

    Google Scholar 

  • Lions, J.L., 1969, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires. Paris: Gauthier-Villars

    MATH  Google Scholar 

  • Lions, J.L. and Prodi, G., 1959, Un Théorème d’Existence et Unicité dans les Équations de Navier-Stokes en Dimension 2, Compte Rend. Acad. Sci. Paris., 248, 3519

    MathSciNet  MATH  Google Scholar 

  • Maremonti, P., 1987, Partial Regularity of a Generalized Solution to the Navier-Stokes Equations in Exterior Domains, Commun. Math. Phys., 110, 75

    Article  MathSciNet  MATH  Google Scholar 

  • Masuda, K., 1984, Weak Solutions of the Navier-Stokes Equations, Tohoku Math. J., 36, 623

    Article  MathSciNet  MATH  Google Scholar 

  • Monniaux S., 1999, Uniqueness of Mild Solutions of the Navier-Stokes Equation and Maximal L p-regularity, C.R. Acad. Sci-Math, 328, 663

    MathSciNet  MATH  Google Scholar 

  • Miyakawa, T., and Sohr, H., 1988, On Energy inequality, Smoothness and Large Time Behavior in L 2 for Weak Solutions of the Navier-Stokes Equations in Exterior Domains, Math. Z., 199, 455

    Article  MathSciNet  MATH  Google Scholar 

  • Navier, C.L.M.H., 1827, Mémoire sur les Lois du Mouvement des Fluides, Mem. Acad. Sci. Inst. de France, 6 (2), 389

    Google Scholar 

  • Neustupa, J., 1999, Partial Regularity of Weak Solutions to the Navier-Stokes Equations in the Class L ∞(0,T; L 3(Ω)3), J. Math. Fluid Mech., in press

    Google Scholar 

  • Nečas, J., Růžička, M. and Šverák, V., 1996, On Leray’s Self-similar Solutions of the Navier-Stokes Equations, Acta Math., 176, 283

    Article  MathSciNet  MATH  Google Scholar 

  • Ohyama, T., 1960, Interior Regularity of Weak Solutions of the Time Dependent Navier-Stokes Equation, Proc. Japan Acad., 36, 273

    Article  MathSciNet  MATH  Google Scholar 

  • Prodi, G., 1959, Un Teorema di Unicità per le Equazioni di Navier-Stokes, Ann. Mat. Pura Appl., 48, 173

    Article  MathSciNet  MATH  Google Scholar 

  • Rautmann, R., 1983, On Optimum Regularity of Navier-Stokes Solution at Time t = 0, Math. Z., 184, 141

    Article  MathSciNet  MATH  Google Scholar 

  • Rionero, S., and Galdi, G.P., 1979, The Weight Function Approach to Uniqueness of Viscous Flows in Unbounded Domains, Arch. Ratl Mech. Anal., 69, 37

    Article  MathSciNet  MATH  Google Scholar 

  • Scheffer, V., 1976a, Turbulence and Hausdorff Dimension, Springer Lecture Notes in Mathematics, 565, 174

    Article  MathSciNet  Google Scholar 

  • Scheffer, V., 1976b, Partial Regularity of Solutions to the Navier-Stokes Equations, Pacific J. Math., 66, 535

    MathSciNet  MATH  Google Scholar 

  • Scheffer, V., 1977, Hausdorff Measure and the Navier-Stokes Equations, Commun Math. Phys., 55, 97

    Article  MathSciNet  MATH  Google Scholar 

  • Scheffer, V., 1978, The Navier-Stokes Equations in Space Dimension Four, Commun Math. Phys., 61, 41

    Article  MathSciNet  MATH  Google Scholar 

  • Scheffer, V., 1980, The Navier-Stokes Equations on a Bounded Domain, Commun Math. Phys., 73, 1

    Article  MathSciNet  MATH  Google Scholar 

  • Scheffer, V., 1982, Boundary Regularity for the Navier-Stokes Equations in Half-Space, Commun Math. Phys., 80, 275

    Article  MathSciNet  Google Scholar 

  • Scheffer, V., 1985, A Solution to the Navier-Stokes Equations with an Internal Singularity, Commun Math. Phys., 101, 47

    Article  MathSciNet  MATH  Google Scholar 

  • Serrin, J., 1962, On the Interior Regularity of Weak Solutions of the Navier-Stokes Equations, Arch. Ratl Mech. Anal., 9, 187

    Article  MathSciNet  MATH  Google Scholar 

  • Serrin, J., 1963, The Initial Value Problem for the Navier-Stokes Equations, Nonlinear Problems., R.E. Langer Ed., Madison: University of Wisconsin Press 9, 69

    Google Scholar 

  • Shinbrot, M., 1973, Lectures on Fluid Mechanics, Gordon and Breach, New York

    MATH  Google Scholar 

  • Shinbrot, M., 1974, The Energy Equation for the Navier-Stokes System, SIAM J. Math. Anal., 5, 948

    Article  MathSciNet  Google Scholar 

  • Simon, J., 1999, On the existence of the pressure for solutions of the variational Navier-Stokes equations, J. Math. Fluid Mech., in press

    Google Scholar 

  • Simon, L., 1983, Lectures on Geometric Measure Theory, University of Canberra, Australia

    MATH  Google Scholar 

  • Sohr, H., 1983, Zur Regularitätstheorie der Instationären Gleichungen von Navier-Stokes, Math. Z., 184, 359

    Article  MathSciNet  MATH  Google Scholar 

  • Sohr, H., 1984, Erweiterung eines Satzes von Serrin über die Gleichungen von Navier-Stokes, J. Reine Angew. Math., 352, 81

    MathSciNet  MATH  Google Scholar 

  • Sohr, H. and von Wahl, W., 1984, On the Singular Set and the Uniqueness of Weak Solutions of the Navier-Stokes Equations, Manuscripta Math., 49, 27

    Article  MathSciNet  MATH  Google Scholar 

  • Sohr, H. and von Wahl, W., 1985, A New Proof of Leray’s Structure Theorem and the Smoothness of Weak Solutions Navier-Stokes Equations for large ∣x∣, Bayreuth. Math. Schr., 20, 153

    Google Scholar 

  • Sohr, H. and von Wahl, W., 1986, On the Regularity of the Pressure of Weak Solutions of Navier-Stokes Equations, Arch. Math., 46, 428

    Article  MATH  Google Scholar 

  • Sohr, H., von Wahl, W. and Wiegner, W., 1986, Zur Asymptotik der Gleichungen von Navier-Stokes, Nachr. Akad. Wiss. Göttingen., 146, 1

    Google Scholar 

  • Solonnikov, V.A., 1964, Estimates of Solutions of Nonstationary Linearized Systems of Navier-Stokes equations, Trudy Mat. Inst. Steklov, 70, 213; English Transl.: A.M.S. Transl., 75, 1968, 1

    MathSciNet  MATH  Google Scholar 

  • Stein, E.M., 1970, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton

    MATH  Google Scholar 

  • Stokes, G.H., 1845, On the Theories of the Internal Friction of Fluids in Motion Trans. Cambridge Phil. Soc., 8, 287

    Google Scholar 

  • Struwe, M., 1988, On Partial Regularity Results for the Navier-Stokes Equations, Commun. Pure Appl. Math., 41, 437

    Article  MathSciNet  MATH  Google Scholar 

  • Takahashi, S., 1990, On Interior Regularity Criteria for Weak Solutions of the Navier-Stokes Equations, Manuscripta Math., 69, 237

    Article  MathSciNet  MATH  Google Scholar 

  • Takahashi, S., 1992, On a Regularity Criterion up to the Boundary for Weak Solutions of the Navier-Stokes Equations, Comm. Partial Diff. Eq., 17, 261; Corrigendum

    Article  MATH  Google Scholar 

  • Takahashi, S., 1992, On a Regularity Criterion up to the Boundary for Weak Solutions of the Navier-Stokes Equations, Comm. Partial Diff. Eq., 19, 1994, 1015

    Google Scholar 

  • Tanaka, A., 1987, Régularité par Rapport au Temps des Solutions Faibles de l’Équation de Navier-Stokes, J. Math. Kyoto Univ., 27, 217

    MathSciNet  MATH  Google Scholar 

  • Taniuchi Y., 1997, On generalized energy equality of the Navier-Stokes equations, Manuscripta Math, 94 365

    Article  MathSciNet  MATH  Google Scholar 

  • Temam, R., 1977, Navier-Stokes Equations, North-Holland

    Google Scholar 

  • Temam, R., 1980, Behavior at t = 0 of the Solutions of Semi-Linear Evolution Equations, MRC Technical Summary Report 2162, Madison: University of Wisconsin

    Google Scholar 

  • Titchmarsh, M.A., 1964, The Theory of Functions, Oxford University Press

    Google Scholar 

  • Tsai, T.P., 1998, On Leray’s Self-Similar Solutions of the Navier-Stokes Equations Satisfying Local Energy Estimates, Arch. Ratl Mech. Anal., 143, 29 Corrigendum

    Article  MATH  Google Scholar 

  • Tsai, T.P., 1998, On Leray’s Self-Similar Solutions of the Navier-Stokes Equations Satisfying Local Energy Estimates, Arch. Ratl Mech. Anal., 147, 1999, 363

    Google Scholar 

  • von Wahl, W., 1983, Über das Behlten für t → 0 der Lösungen Nichtlinearer Parabolischer Gleichungen, insbesondere der Gleichungen von Navier-Stokes, Sonderforschungsbereich, Universität Bonn

    Google Scholar 

  • von Wahl, W., 1985, The Equations of Navier-Stokes and Abstract Parabolic Equations, Vieweg Verlag

    Google Scholar 

  • von Wahl, W., 1986, Regularity of Weak Solutions of the Navier-Stokes Equations, Proc. Symp. Pure Appl. Math., 45, 497

    Google Scholar 

  • Wu, L., 1991, Partial Regularity Problems for Navier-Stokes Equations in Three-Dimensional Space, Acta Math. Sinica, 34, 541

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, University of Pittsburgh, USA

    Giovanni P. Galdi

Authors
  1. Giovanni P. Galdi
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Engineering Department of Mechanical Engineering, University of Pittsburgh, 3700 O’Hara Street, Pittsburgh, PA, 15261, USA

    Giovanni P. Galdi

  2. Department of Mathematics, University of British Columbia, Vancouver, BC, Canada, V6T 1Y4

    John G. Heywood

  3. Institut für Angewandte Mathematik, Universität Heidelberg, Im Neuenheimer Feld 293/294, Germany

    Rolf Rannacher

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Basel AG

About this chapter

Cite this chapter

Galdi, G.P. (2000). An Introduction to the Navier-Stokes Initial-Boundary Value Problem. In: Galdi, G.P., Heywood, J.G., Rannacher, R. (eds) Fundamental Directions in Mathematical Fluid Mechanics. Advances in Mathematical Fluid Mechanics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8424-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8424-2_1

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9561-3

  • Online ISBN: 978-3-0348-8424-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation