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Recent Mathematical Results and Open Problems about Shallow Water Equations

  • Didier Bresch
  • Benoît Desjardins
  • Guy Métivier
Part of the Advances in Mathematical Fluid Mechanics book series (AMFM)

Abstract

The purpose of this work is to present recent mathematical results about the shallow water model. We will also mention related open problems of high mathematical interest.

Keywords

Viscous and inviscid flow shallow-water model lake equations quasi-geostrophic equations weak and strong solutions degenerate viscosities 

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References

  1. [1]
    S. Agmon, A. Douglis, L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations I, Comm. Pures and Appl. Math., 12, (1959), 623–727.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    S. Agmon, A. Douglis, L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations II, Comm. Pures and Appl. Math., 17, (1964), 35–92.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    C. Bernardi, O. Pironneau. On the shallow water equations at low Reynolds number. Commun. Partial Diff. Eqs. 16, 59–104 (1991).zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    A.L. Bertozzi, A.J. Majda. Vorticity and incompressible flows. Cambridge University Press, (2001).Google Scholar
  5. [5]
    P. Bolley, J. Camus, G. Métivier, Estimations de Schauder et régularité Hölderienne pour une classe de problèmes aux limites singuliers, Comm. Partial Diff. Equ. 11 (1986), 1135–1203.zbMATHCrossRefGoogle Scholar
  6. [6]
    F. Bouchut, A. Mangeney-Castelnau, B. Perthame, J.P. Vilotte. A new model of Saint-Venant and Savage-Hutter type for gravity driven shallow water flows. C.R. Acad. Sci. Paris, série I, 336(6):531–536, (2003).zbMATHMathSciNetGoogle Scholar
  7. [7]
    F. Bouchut, M. Westdickenberg. Gravity driven shallow water models for arbitrary topography. Comm. in Math. Sci., 2(3):359–389, (2004).zbMATHMathSciNetGoogle Scholar
  8. [8]
    D. Bresch, B. Desjardins. Numerical approximation of compressible fluid models with density dependent viscosity. In preparation (2005).Google Scholar
  9. [9]
    D. Bresch, B. Desjardins, J.-M. Ghidaglia. On bi-fluid compressible models. In preparation (2005).Google Scholar
  10. [10]
    D. Bresch, B. Desjardins. Existence globale de solutions pour les équations de Navier-Stokes compressibles complètes avec conduction thermique. C. R. Acad. Sci., Paris, Section mathématiques. Submitted (2005).Google Scholar
  11. [11]
    D. Bresch, B. Desjardins. Some diffusive capillary models of Korteweg type. C.R. Acad. Sciences, Paris, Section Mécanique. Vol. 332 no. 11 (2004), pp. 881–886.Google Scholar
  12. [12]
    D. Bresch, B. Desjardins. Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasi-geostrophic model. Commun. Math. Phys., 238,1–2, (2003), pp. 211–223.zbMATHMathSciNetGoogle Scholar
  13. [13]
    D. Bresch, M. Gisclon, C.K. Lin. An example of low Mach (Froude) number effects for compressible flows with nonconstant density (height) limit. M2AN, Vol. 39, No 3, pp. 477–486, (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    D. Bresch, A. Juengel, H.-L. Li, Z.P. Xin. Effective viscosity and dispersion (capillarity) approximations to hydrodynamics. Forthcoming paper, (2005).Google Scholar
  15. [15]
    D. Bresch, G. Métivier. Global existence and uniqueness for the lake equations with vanishing topography: elliptic estimates for degenerate equations. To appear in Nonlinearity, (2005).Google Scholar
  16. [16]
    D. Bresch, B. Desjardins, C.K. Lin. On some compressible fluid models: Korteweg, lubrication and shallow water systems. Comm. Partial Differential Equations, 28,3–4, (2003), p. 1009–1037.MathSciNetGoogle Scholar
  17. [17]
    A.T. Bui. Existence and uniqueness of a classical solution of an initial boundary value problem of the theory of shallow waters. SIAM J. Math. Anal. 12 (1981) 229–241.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    J.F. Chatelon, P. Orenga. Some smoothness and uniqueness for a shallow water problem. Adv. Diff. Eqs., 3,1 (1998), 155–176.zbMATHMathSciNetGoogle Scholar
  19. [19]
    C. Cheverry. Propagation of oscillations in Real Vanishing Viscosity Limit, Commun. Math. Phys. 247, 655–695 (2004).zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    D. Coutand, S. Shkoller. Unique solvability of the free-boundary Navier-Stokes equations with surface tension, Arch. Rat. Mech. Anal. (2005).Google Scholar
  21. [21]
    D. Coutand, S. Shkoller. Well-posedness of the free-surface incompressible Euler equations with or without surface tension, Submitted (2005).Google Scholar
  22. [22]
    E. Feireisl. Dynamics of viscous compressible fluids. Oxford Science Publication, Oxford, (2004).zbMATHGoogle Scholar
  23. [23]
    P.R. Gent. The energetically consistent shallow water equations. J. Atmos. Sci., 50, 1323–1325, (1993).CrossRefGoogle Scholar
  24. [24]
    J.F. Gerbeau, B. Perthame. Derivation of Viscous Saint-Venant System for Laminar Shallow Water; Numerical Validation, Discrete and Continuous Dynamical Systems, Ser. B, Vol. 1, Num. 1, 89–102, (2001).zbMATHMathSciNetGoogle Scholar
  25. [25]
    C. Goulaouic, N. Shimakura, Régularité Höldérienne de certains problèmes aux limites dégénérés, Ann. Scuola Norm. Sup. Pisa, 10, (1983), 79–108.MathSciNetGoogle Scholar
  26. [26]
    P. Gwiazda. An existence result for a model of granular material with non-constant density. Asymptotic Analysis, 30 Google Scholar
  27. [27]
    D. Hoff, D. Serre. The failure of continuous dependence on initial data for the Navier-Stokes equations of compressible flow. SIAM J. Appl. Math., 51(4):887–898, (1991).zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    A.V. Kazhikhov, Initial-boundary value problems for the Euler equations of an ideal incompressible fluid. Moscow Univ. Math. Bull. 46 (1991), no. 5, 10–14.zbMATHMathSciNetGoogle Scholar
  29. [29]
    A.V. Kazhikhov, A. Veigant. Global solutions of equations of potential fluids for small Reynolds number. Diff. Eqs., 30, (1994), 935–947.zbMATHGoogle Scholar
  30. [30]
    P.E. Kloeden, Global existence of classical solutions in the dissipative shallow water equations. SIAM J. Math. Anal. 16 (1985), 301–315.zbMATHCrossRefMathSciNetGoogle Scholar
  31. [31]
    D. Levermore, M. Oliver, E.S. Titi, Global well-posedness for models of shallow water in a basin with a varying bottom, Indiana Univ. Math. J. 45 (1996), 479–510.zbMATHCrossRefMathSciNetGoogle Scholar
  32. [32]
    D. Levermore, B. Sammartino. A shallow water model in a basin with varying bottom topography and eddy viscosity, Nonlinearity, Vol. 14, n. 6, 1493–1515 (2001).zbMATHCrossRefMathSciNetGoogle Scholar
  33. [33]
    J. Li, Z.P. Xin. Some Uniform Estimates and Blowup Behavior of Global Strong Solutions to the Stokes Approximation Equations for Two-Dimensional Compressible Flows. To appear in J. Diff. Eqs. (2005)Google Scholar
  34. [34]
    P.-L. Lions. Mathematical topics in fluid dynamics, Vol. 2, Compressible models. Oxford Science Publication, Oxford, (1998).Google Scholar
  35. [35]
    P.-L. Lions, B. Perthame, P. E. Souganidis, Existence of entropy solutions to isentropic gas dynamics System. Comm. Pure Appl. Math. 49 (1996), no. 6, 599–638.zbMATHCrossRefMathSciNetGoogle Scholar
  36. [36]
    A. Majda. Introduction to PDEs and waves for the atmosphere and ocean. Courant lecture notes in Mathematics, (2003).Google Scholar
  37. [37]
    F. Marche. Derivation of a new two-dimensional shallow water model with varying topography, bottom friction and capillary effects. Submitted (2005).Google Scholar
  38. [38]
    A. Mellet, A. Vasseur. On the isentropic compressible Navier-Stokes equation. Submitted (2005).Google Scholar
  39. [39]
    L. Min, A. Kazhikhov, S. Ukai. Global solutions to the Cauchy problem of the Stokes approximation equations for two-dimensional compressible flows. Comm. Partial Diff. Eqs., 23,5–6, (1998), 985–1006.zbMATHMathSciNetGoogle Scholar
  40. [40]
    M.L. Muoz-Ruiz, F.-J. Chatelon, P. Orenga. On a bi-layer shallow-water problem. Nonlinear Anal. Real World Appl. 4 (2003), no. 1, 139–171.CrossRefMathSciNetGoogle Scholar
  41. [41]
    A. Novotny, I. Straskraba. Introduction to the mathematical theory of compressible flow. Oxford lecture series in Mathematics and its applications, (2004).Google Scholar
  42. [42]
    M. Oliver, Justification of the shallow water limit for a rigid lid flow with bottom topography, Theoretical and Computational Fluid Dynamics 9 (1997), 311–324.zbMATHCrossRefGoogle Scholar
  43. [43]
    P. Orenga. Un théorème d’existence de solutions d’un problème de shallow water, Arch. Rational Mech. Anal. 130 (1995) 183–204.zbMATHCrossRefMathSciNetGoogle Scholar
  44. [44]
    L. Sundbye. Existence for the Cauchy Problem for the Viscous Shallow Water Equations. Rocky Mountain Journal of Mathematics, 1998, 28(3), 1135–1152.zbMATHMathSciNetCrossRefGoogle Scholar
  45. [45]
    L. Sundbye. Global existence for Dirichlet problem for the viscous shallow water equations. J. Math. Anal. Appl. 202 (1996), 236–258.zbMATHCrossRefMathSciNetGoogle Scholar
  46. [46]
    J.-P. Vila. Shallow water equations for laminar flows of newtonian fluids. Paper in preparation and private communication, (2005).Google Scholar
  47. [47]
    W. Wang, C.-J. Xu. The Cauchy problem for viscous shallow water equations Rev. Mat. Iberoamericana 21, no. 1 (2005), 1–24.MathSciNetGoogle Scholar
  48. [48]
    V.I. Yudovich. The flow of a perfect, incompressible liquid through a given region. Soviet Physics Dokl. 7 (1962) 789–791.zbMATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2006

Authors and Affiliations

  • Didier Bresch
    • 1
    • 2
  • Benoît Desjardins
    • 3
    • 4
  • Guy Métivier
    • 5
  1. 1.LMC-IMAG, (CNRS-INPG-UJF)Grenoble cedexFrance
  2. 2.Institute of Mathematical SciencesThe Chinese University of Hong-KongShatin, NT Hong-Kong
  3. 3.CEA/DIFBruyères le ChâtelFrance
  4. 4.E.N.S. Ulm, D.M.A.Paris cedex 05France
  5. 5.MAB, Université Bordeaux 1Talence cedexFrance

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