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On a relaxation approximation of the incompressible Navier-Stokes equations


Authors: Yann Brenier, Roberto Natalini and Marjolaine Puel
Journal: Proc. Amer. Math. Soc. 132 (2004), 1021-1028
MSC (2000): Primary 35Q30; Secondary 76D05
DOI: https://doi.org/10.1090/S0002-9939-03-07230-7
Published electronically: November 14, 2003
MathSciNet review: 2045417
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Abstract: We consider a hyperbolic singular perturbation of the incompressible Navier Stokes equations in two space dimensions. The approximating system under consideration arises as a diffusive rescaled version of a standard relaxation approximation for the incompressible Euler equations. The aim of this work is to give a rigorous justification of its asymptotic limit toward the Navier Stokes equations using the modulated energy method.


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  • 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

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Additional Information

Yann Brenier
Affiliation: Laboratoire J. A. Dieudonné, U.M.R. C.N.R.S. No. 6621, Université de Nice Sophia-Antipolis, Parc Valrose, F–06108 Nice, France
Email: brenier@math.unice.fr

Roberto Natalini
Affiliation: Istituto per le Applicazioni del Calcolo “Mauro Picone”, Consiglio Nazionale delle Ricerche, Viale del Policlinico, 137, I-00161 Roma, Italy
Email: rnatalini@iac.rm.cnr.it

Marjolaine Puel
Affiliation: Université Pierre et Marie Curie, Laboratoire d’analyse numérique, Boite courrier 187, F–75252 Paris cedex 05, France
Email: mpuel@ceremade.dauphine.fr

DOI: https://doi.org/10.1090/S0002-9939-03-07230-7
Keywords: Incompressible Navier-Stokes equations, relaxation approximations, hyperbolic singular perturbations, modulated energy method
Received by editor(s): October 17, 2002
Published electronically: November 14, 2003
Additional Notes: Partially supported by European TMR projects NPPDE # ERB FMRX CT98 0201 and CNR Short Term Visiting program and European Union RTN HYKE Project: HPRN-CT-2002-00282
Communicated by: Suncica Canic
Article copyright: © Copyright 2003 American Mathematical Society

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