The second grade fluid and averaged Euler equations with Navier-slip boundary conditions

Nonlinearity

Volumn 16, Issue 3, 2003, Pages 1119-1149

  Busuioc, Adriana Valentina ^a   Ratiu, Tudor S ^a  

^a EPFL   (Switzerland)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0242276990     PISSN: 09517715     EISSN: None     Source Type: Journal    
DOI: 10.1088/0951-7715/16/3/318     Document Type: Article

References (46)

1. Agmon S, Douglis A and Nirenberg L 1964 Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II Commun. Pure Appl. Math. XVII 35-92
2. Amrouche C, Bernardi C, Dauge M and Girault V 1998 Vector potentials in three-dimensional non-smooth domains Math. Meth. Appl Sci. 21 823-64
3. Arnold V I 1965 Conditions for nonlinear stability of the stationary plane curvilinear flows of an ideal fluid Dokl. Akad. Nauk 162 773-7
4. Arnold V I 1969 An a priori estimate in the theory of hydrodynamic stability (Eng. transl.) Am. Math. Soc. Transl. 19 267-9
5. Busuioc A V 1999 On second grade fluids with vanishing viscosity C.R. Acad. Sci., Paris Ser. I Math. 328 1241-6
6. Busuioc A V 2002 Sur les équations α Navier-Stokes dans un ouvert borné C.R. Acad. Sci., Paris Ser. I Math. 334 823-6
7. Camassa R and Holm D D 1993 An integrable shallow water equation with peaked solitons Phys. Rev. Lett. 71 1661-4
8. Chen S, Foias C, Holm D D, Olson E, Titi E S and Wynne S 1998 The Camassa-Holm equations as a closure model for turbulent channel and pipe flow Phys. Rev. Lett. 81 5338-41
9. Cioranescu D and Girault V 1997 Weak and classical solutions of a family of second grade fluids Int. J. Non-Linear Mech. 32 317-35
10. Cioranescu D and Ouazar E H 1984 Existence and uniqueness for fluids of second grade Nonlinear Partial Differential Equations and Their Applications (Collège de France Seminar, Paris, 1982/1983) vol 4 (Boston, MA: Pitman) pp 178-97
11. Clopeau T, Mikelič A and Robert R 1998 On the vanishing viscosity limit for the 2D imcompressible Navier-Stokes equations with the friction type boundary conditions Nonlinearity 11 1625-36
12. Coron J-M 1995/96 On the controllability of the 2D incompressible Navier-Stokes equations with the Navier-slip boundary conditions ESAIM Control Optim. Calc. Var. 1 35-75
13. Coutand D, Pierce J and Shkoller S 2002 Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains Commun. Pure Appl. Anal. 1 35-50
14. Derks G and Ratiu T S 1998 Attracting curves on families of stationary solutions Proc. R. Soc. A 454 1407-44
15. Dunn J E and Fosdick R L 1974 Thermodynamics, stability, and boundedness of fluids of complexity 2 and fluids of second grade Arch. Ration. Mech. Anal. 56 191-252
16. Dunn J E and Rajagopal K R 1995 Fluids of differential type: critical review and thermodynamic analysis Int. J. Eng. Sci. 33 689-729
17. Ebin D and Marsden J 1970 Groups of diffeomorphisms and the motion of an incompressible fluid Ann. Math. 92 102-63
18. Foias C and Temam R 1978 Remarques sur les équations de Navier-Stokes stationnaires et les phénomènes successifs de bifurcation Ann. Sci. Norm. Pisa V 29-63
19. Foias C, Holm D D and Titi E S The three dimensional viscous Camassa-Holm equations and their relation to the Navier-Stokes equation and turbulence theory J. Dyn. Diff. Eqns 14 1-35
20. Fosdick R L and Rajagopal K R 1979 Anomalous features in the model of 'second order fluids' Arch. Ration. Mech. Anal. 70 145-52
21. Galdi G P and Sequeira 1994 A Further existence results for classical solutions of the equations of a second-grade fluid Arch. Ration. Mech. Anal. 128 297-312
22. Himonas A A and Misiolek G 2001 The Cauchy problem for an integrable shallow-water equation Diff. Integral Eqns 14 621-31
23. Iftimie D 2002 Remarques sur la limite α → 0 pour les fluides de grade 2 C. R. Acad. Sci., Paris Sér. I Math. 334 83-6
24. Iftimie D 2002 Remarques sur la limite α → 0 pour les fluides de grade 2 Studies in Mathematics and its Applications, Nonlinear Partial Differential Equations and their Applications (Collège de France Seminar) vol XIV (Amsterdam: North-Holland) p 31
25. Holm D D, Marsden J E and Ratiu T S 1998 The Euler-Poincaré equations and semidirect products with applications to continuum theories Adv. Math. 137 1-81
26. Holm D D, Marsden J E and Ratiu T S 1998 Euler-Poincaré models of ideal fluids with nonlinear dispersion Phys. Rev. Lett. 349 4173-7
27. Holm D D, Marsden J E, Ratiu T S and Weinstein A D 1985 Nonlinear stability of fluid and plasma equilibria Phys. Rep. 123 1-116
28. Johnson R S 2002 Camassa-Holm, Korteweg-de-Vries and related models for water waves J. Fluid Mech. 455 63-82
29. Joseph D D 1990 Fluid Dynamics of Viscoelastic Liquids (New York: Springer)
30. Kato T 1967 On classical solutions of the two-dimensional non-stationary Euler equations Arch. Rat. Mech. Anal. 24 302-24
31. Kouranbaeva S 1999 The Camassa-Holm equation as a geodesic flow on the diffeomorphism group J. Math. Phys. 40 857-68
32. Marsden J E, Ratiu T S and Shkoller S 2000 The geometry and analysis of the averaged Euler equations and a new diffeomorphism group Geom. Funct. Anal. 10 582-99
33. Marsden J E and Shkoller S 2001 Global well-posedness for the LANS-α equations on bounded domains Phil. Trans. R. Soc. A 359 1449-68
34. Misiolek G 1998 A shallow water equation as a geodesic flow on the Bott-Virasoro group J. Geom. Phys. 24 203-8
35. Rivlin R S and Ericksen J L 1955 Stress-deformation relations for isotropie materials J. Ration. Mech. Anal. 4 323-425
36. Ščadilov V E and Solonnikov V A 1973 On a boundary value problem for a stationary system of Navier-Stokes equations Proc. Steklov Inst. Math. 125 186-99
37. Schaefer H H and Wolff M P 1999 Topological Vector Spaces 2nd edn (New York: Springer)
38. 1 metric and mean hydrodynamics J. Funct. Anal. 160 337-65
39. Shkoller S 2000 Analysis on groups of diffeomorphisms of manifolds with boundary and the averaged motion of a fluid J. Diff. Geom. 55 145-91
40. Shkoller S 2001 Smooth global Lagrangian flow for the 2D Euler and second-grade fluid equations Appl. Math. Lett. 14 539-43
41. Solonnikov V A 1966 General boundary value problems for Douglis-Nirenberg elliptic systems Proc. Steklov Inst. Math. 92 269-339
42. Vishik M 1999 Incompressible flows of an ideal fluid with vorticity in borderline spaces of Besov type Ann. Sci. Ecole Norm. Sup. 32 769-812
43. Wolibner W 1933 Un théorème sur 1'existence du mouvement plan d'un fluide parfait homogène, incompressible, pendant un temps infiniment long Math. Zeitschrift 37 698-726
44. Xin Z and Zhang P 2000 On the weak solutions to a shallow water equations Commun. Pure Appl. Math. 53 1411-33
45. Yudovich V I 1963 Non-stationary flow of an ideal incompressible liquid Zh. Vych. Mat. 3 1032-66
46. Yudovich V I 1995 Uniqueness theorem for the basic nonstationary problem in dynamics of ideal incompressible fluid Math. Res. Lett. 2 27-38