A Fluid Problem with Navier-Slip Boundary Conditions

  • Adriana Valentina Busuioc
  • T. S. Ratiu
Conference paper
Part of the Advances in Mechanics and Mathematics book series (AMMA, volume 6)

Abstract

We study the equations governing the motion of second grade fluids in a bounded domain of ℝ d , d = 2, 3, with Navier slip boundary conditions with and without viscosity (averaged Euler equations). The main results concern the global existence and uniqueness of H 3 solutions in dimension two and, for dimension three, local existence of H 3 solutions for arbitrary initial data and global existence for small initial data and positive viscosity. The last part discusses Liapunov stability conditions for stationary solutions for the averaged Euler equations similar to the Rayleigh-Arnold stability result for the classical Euler equations. This paper presents only the main ideas of the proofs that can be found in [4].

Keywords

Global Existence Potential Vorticity Fluid Problem Small Initial Data Grade Fluid 
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References

  1. [1]
    S. Agmon, A. Douglis and L. Nirenberg. Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math., XVII, 1964, pp. 35–92.MathSciNetCrossRefGoogle Scholar
  2. [2]
    V.I. Arnold. Conditions for nonlinear stability of the stationary plane curvilinear flows of an ideal fluid. Dokl. Akad. Nauk 162, n° 5, 1965, 773–777.Google Scholar
  3. [3]
    V.I. Arnold. An a priori estimate in the theory of hydrodynamic stability. [English translation] Amer. Math. Soc. Transi. 19, 1969, 267–269.Google Scholar
  4. [4]
    A.V. Busuioc and T.S. Ratiu The second grade fluid and averaged Euler equations with Navier-slip boundary conditions,submitted.Google Scholar
  5. [5]
    R. Camassa and D.D. Holm. An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71, n° 11, 1993, pp. 1661–1664.MathSciNetADSMATHCrossRefGoogle Scholar
  6. [6]
    S. Chen, C. Foias, D.D. Holm, E. Olson, E.S. Titi, S. Wynne. The Camassa-Holm equations as a closure model for turbulent channel and pipe flow. Phys. rev. Lett., 81, 1998, pp. 5338–5341.MathSciNetADSMATHCrossRefGoogle Scholar
  7. [7]
    D. Cioranescu and V. Girault. Weak and classical solutions of a family of second grade fluids. Internat. J. Non-Linear Mech., 32, n 2, 1997, pp. 317–335.MathSciNetADSMATHCrossRefGoogle Scholar
  8. [8]
    D. Cioranescu and E. H. Ouazar. Existence and uniqueness for fluids of second grade. Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. VI (Paris, 1982/1983), pp. 178–197. Boston, MA, Pitman, 1984.Google Scholar
  9. [9]
    G. Derks and T.S. Ratiu Attracting curves on families of stationary solutions. Proc. R. Soc. Lond. A, 454, 1998, pp. 1407–1444.MathSciNetADSMATHCrossRefGoogle Scholar
  10. [10]
    J. E. Dunn and R. L. Fosdick. Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade. Arch. Rational Mech. Anal., 56, 1974, pp. 191–252.MathSciNetADSMATHCrossRefGoogle Scholar
  11. [11]
    J. E. Dunn and K. R. Rajagopal. Fluids of differential type: critical review and thermodynamic analysis. Internat. J. Engrg. Sci., 33, 1995, pp. 689–729.MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    C. Foias and R. Temam. Remarques sur les équations de NavierStokes stationnaires et les phénomènes successifs de bifurcation. Ann. Sc. Norm. Pisa, V, 1978, pp. 29–63.Google Scholar
  13. [13]
    C. Foias, D.D. Holm, E.S.Titi. The three dimensional viscous Camassa-Holm Equations and their relation to the Navier-Stokes equation and turbulence theory. To appear in J. Dyn. Diff. Eq.Google Scholar
  14. [14]
    R. L. Fosdick and K. R. Rajagopal, Anomalous features in the model of “second order fluids”, Arch. Rational Mech. Anal. 70, n 2, 1979, pp. 145–152.MathSciNetADSMATHCrossRefGoogle Scholar
  15. [15]
    G. P. Galdi and A. Sequeira. Further existence results for classical solutions of the equations of a second-grade fluid. Arch. Rational Mech. Anal., 128, n° 4, 1994, pp. 297–312.MathSciNetADSMATHCrossRefGoogle Scholar
  16. [16]
    A.A. Himonas and G. Misiolek. The Cauchy Problem for an integrable shallow-water Equation. Differential and Integral Equations, Vol. 14, No. 7, 2001, pp. 821–831.MathSciNetMATHGoogle Scholar
  17. [17]
    D.D. Holm, J.E. Marsden, and T.S. Ratiu. The Euler-Poincaré equations and semidirect products withxl applications to continuum theories. Adv. Math., 137, n° 1, 1998, pp. 1–81.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    D.D. Holm, J.E. Marsden, and T.S. Ratiu. Euler-Poincaré models of ideal fluids with nonlinear dispersion. Phys. Rev. Lett., 349, 1998, pp. 4173–4177.ADSCrossRefGoogle Scholar
  19. [19]
    D.D. Holm, J.E. Marsden, T.S. Ratiu, and A.D. Weinstein. Nonlinear stability of fluid and plasma equilibria. Phys. Rep. 123, 1985, 1–116.MathSciNetADSMATHCrossRefGoogle Scholar
  20. [20]
    T. Kato. On classical solutions of the two-dimensional non—stationary Euler equations. Arch. Rat. Mech. Anal. 24 n° 3, 1967, 302–324.Google Scholar
  21. [21]
    J.E. Marsden, T.S. Ratiu, and S. Shkoller. The geometry and analysis of the averaged Euler equations and a new diffeomorphism group. Geom. and Func. Anal. 10, 2000, pp. 582–599.MathSciNetMATHCrossRefGoogle Scholar
  22. [22]
    G. Misiolek. A shallow water equation as a geodesic flow on the Bott-Virasoro group. J. Geom. Phys. 24, n° 3, 1998, pp. 203–208.MathSciNetADSMATHCrossRefGoogle Scholar
  23. [23]
    R. S. Rivlin and J. L. Ericksen. Stress-deformation relations for isotropic materials. J. Rational Mech. Anal., 4, 1955, pp. 323–425.MathSciNetMATHGoogle Scholar
  24. [24]
    V. E. Scadilov and V. A. Solonnikov. On a boundary value problem for a stationary system of Navier-Stokes equations Proc. Steklov Inst. Math., 125, 1973, pp. 186–199.MathSciNetMATHGoogle Scholar
  25. [25]
    V. A. Solonnikov. General boundary value problems for DouglisNirenberg elliptic systems. Proc. Steklov Inst. Math. 92, 1966, pp. 269–339.Google Scholar
  26. [26]
    M. Vishik. Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type. Ann. Sci. Ecole Norm. Sup. (4) 32, 1999, no. 6, pp. 769–812.MathSciNetMATHGoogle Scholar
  27. [27]
    W. Wolibner. Un théorème sur l’existence du mouvement plan d’un fluid parfait homogène, incompressible, pendant un temps infiniment longue. Math. Zeitschrift 37, 1933, 698–726.MathSciNetCrossRefGoogle Scholar
  28. [28]
    V.I. Yudovich. Non—stationary flow of an ideal incompressible liquid. Zh. Vych. Mat. 3 n° 6, 1963, 1032–1066.MATHGoogle Scholar
  29. [29]
    V.I. Yudovich. Uniqueness theorem for the basic nonstationary problem in dynamics of ideal incompressible fluid. Math. Res. Lett. 2, 1995, 27–38.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2004

Authors and Affiliations

  • Adriana Valentina Busuioc
    • 1
  • T. S. Ratiu
    • 1
  1. 1.Section de mathématiques, Institut BernoulliÉcole Polytechnique Fédérale de LausanneLausanneSwitzerland

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