On a relaxation approximation of the incompressible Navier-Stokes equations

Proceedings of the American Mathematical Society

Volumn 132, Issue 4, 2004, Pages 1021-1028

  Brenier, Yann ^a   Natalini, Roberto ^b   Puel, Marjolaine ^c  

^a UNIVERSITÉ DE NICE SOPHIA ANTIPOLIS   (France)

^b CNR   (Italy)

^c SORBONNE UNIVERSITÉ   (France)

Author keywords

Hyperbolic singular perturbations; Incompressible Navier Stokes equations; Modulated energy method; Relaxation approximations

Indexed keywords

EID: 1642330013     PISSN: 00029939     EISSN: None     Source Type: Journal    
DOI: 10.1090/s0002-9939-03-07230-7     Document Type: Conference Paper

References (23)

1. D. Aregba-Driollet, R. Natalini, and S. Q. Tang, Diffusive kinetic explicit schemes for nonlinear degenerate parabolic systems, Math. Comp. 73 (2004), 3-34.
2. M. K. Banda, A. Klar, L. Pareschi, and M. Seaid, Lattice-Boltzmann type relaxation systems and high order relaxation schemes for the incompressible Navier-Stokes equations, preprint, 2001.
3. Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations 25 (2000), no. 3-4, 737-754. MR 2001c:76124
4. F. Bouchut, F. Guarguaglini, and R. Natalini, Diffusive BGK approximations for nonlinear multidimensional parabolic equations, Indiana Univ. Math. J. 49 (2000), 723-749. MR 2001k:35162
5. H. Brezis and T. Gallouet, Nonlinear Schrödinger evolution equations, Nonlinear Anal. 4 (1980), no. 4, 677-681. MR 81i:35139
6. C. Cattaneo, Sulla conduzione del calore, Atti Sem. Mat. Fis. Univ. Modena 3 (1949), 83-101. MR 11:362d
7. D. Donatelli and P. Marcati, Convergence of singular limits for multi-D semilinear hyperbolic systems to parabolic systems, Technical Report 12, Scuola Normale Superiore, Pisa, 2000, to appear in Trans. Amer. Math. Soc.
8. F. Bouchut, F. Golse, and M. Pulvirenti, Kinetic equations and asymptotic theory, Series in Appl. Math., Gauthiers-Villars, Paris, 2000.
9. E. Grenier, Boundary layers of 2D inviscid fluids from a Hamiltonian viewpoint, Math. Res. Lett. 6 (1999), 257-269. MR 2002b:76048
10. S. Klainerman, Global existence for nonlinear wave equations, Comm. Pure Appl. Math. 33 (1980), no. 1, 43-101. MR 81b:35050
11. T. Kurtz, Convergence of sequences of semigroups of nonlinear operators with an application to gas kinetics, Trans. Amer. Math. Soc. 186 (1973), 259-272. MR 49:1256
12. S. Jin and H. L. Liu, Diffusion limit of a hyperbolic system with relaxation, Methods Appl. Anal. 5 (1998), 317-334. MR 2000k:35176
13. S. Jin, L. Pareschi, and G. Toscani, Diffusive relaxation schemes for multiscale discrete-velocity kinetic equations, SIAM J. Numer. Anal. 35 (1998), 2405-2439. MR 99k:76100
14. S. Jin and Z. Xin, The relaxation schemes for systems of conservation laws in arbitrary space dimensions, Comm. Pure Appl. Math. 48 (1995), 235-276. MR 96c:65134
15. P. L. Lions and G. Toscani, Diffusive limit for finite velocity Boltzmann kinetic models, Rev. Mat. Iberoamericana 13 (1997), 473-513. MR 99g:76127
16. H. Liu and R. Natalini, Long-Time Diffusive Behavior of Solutions to a Hyperbolic Relaxation System, Asymptot. Anal. 25 (2001), no. 1, 21-38. MR 2001m:35208
17. P. Marcati and A. Milani, The one-dimensional Darcy's law as the limit of a compressible Euler flow, J. Differential Equations 84 (1990), 129-147, MR 91i:35156
18. P. Marcati, A. Milani, and P. Secchi, Singular convergence of weak solutions for a quasilinear nonhomogeneous hyperbolic system, Manuscripta Math. 60 (1988), 49-69. MR 89f:35127
19. P. Marcati and B. Rubino, Hyperbolic to Parabolic Relaxation Theory for Quasilinear First Order Systems, J. Differential Equations 162 (2000), 359-399. MR 2001d:35125
20. H. P. McKean, The central limit theorem for Carleman's equation, Israel J. Math. 21(1) (1975), 54-92.
21. J. C. Saut, Some remarks on the limit of viscoelastic fluids as the relaxation time tends to zero, Trends in applications of pure mathematics to mechanics (Bad Honnef, 1985), 364-369, Lecture Notes in Phys., 249, Springer, Berlin, 1986. MR 87i:76012
22. A. E. Tzavaras, Materials with internal variables and relaxation to conservation laws, Arch. Rational Mech. Anal. 146 (1999), 129-155. MR 2000i:74004
23. H. T. Yau, Relative entropy and hydrodynamics of Ginzburg-Landau models, Lett. Math. Phys. 22 (1991), 63-80. MR 93e:82035