LOW MACH NUMBER LIMIT ON EXTERIOR DOMAINS

doi:10.1016/S0252-9602(12)60010-1

Abstract

Abstract:

This paper is concerned with the low Mach number limit for the compressible Navier-Stokes equations in an exterior domain. We present here an approach based on Strichartz estimate defined on a non trapping exterior domain and we will be able to show the compactness and strong convergence of the velocity vector field.

Key words: exterior domain, compressible and incompressible Navier-Stokes equation, low Mach number limit, hyperbolic equations, wave equations, rescaling, dispersive estimates, Strichartz estimates

CLC Number:

35L65

Donatella Donatelli, Pierangelo Marcati. LOW MACH NUMBER LIMIT ON EXTERIOR DOMAINS[J].Acta mathematica scientia,Series B, 2012, 32(1): 164-176.

Trendmd

References

[1] Adams R A. Sobolev Spaces. New York: Academic Press, 1975

[2] Burq N. Global Strichartz estimates for nontrapping geometries: about an article by H.F. Smith and C.D. Sogge: “Global Strichartz estimates for nontrapping perturbations of the Laplacian” [Comm Partial
Differential Equation, 2000, 25(11/12): 2171–2183; MR1789924 (2001j:35180)]. Comm Partial Differ Equ,
2003, 28(9/10): 1675–1683

[3] Chorin A J, Marsden J E. A Mathematical Introduction to Fluid Mechanics. Third ed. Texts in Applied
Mathematics, Vol 4. New York: Springer-Verlag, 1993

[4] Desjardins B, Grenier E. Low Mach number limit of viscous compressible flows in the whole space. R Soc
Lond Proc Ser A Math Phys Eng Sci, 1999, 455(1986): 2271–2279

[5] Desjardins B, Grenier E, Lions P -L, Masmoudi N. Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J Math Pures Appl, 1999, 78(5): 461–471

[6] Donatelli D, Marcati P. A quasineutral type limit for the Navier-Stokes-Poisson system with large data.
Nonlinearity, 2008, 21(1): 135–148

[7] Donatelli D,Marcati P.Leray weak solutions ofthe incompressiblenavierstokes system on exteriordomains
via the artificial compressibility method. Indiana Univ Math J, 2010, 59: 1831–1852

[8] Donatelli D, Feireisl E, Novotn′y A. On incompressible limits for the Navier-Stokes system on unbounded
domains under slip boundary conditions. Discrete Contin Dyn Syst Ser B, 2010, 13(4): 783–798

[9] Ducomet B, Feireisl E, Petzeltov′a H, Straˇskraba I. Existence globale pour un fluide barotrope autogravi-
tant. C R Acad Sci Paris S′er I Math, 2001, 332(7): 627–632

[10] DucometB,FeireislE,Petzeltov′a H,Straˇskraba I.Globalintimeweak solutionsforcompressiblebarotropic
self-gravitating fluids. Discrete Contin Dyn Syst, 2004, 11(1): 113–130

[11] Farwig R, Kozono H, Sohr H. On the Helmholtz decomposition in general unbounded domains. Arch Math
(Basel), 2007, 88(3): 239–248

[12] Lions P -L. Mathematical Topics in Fluid Dynamics, Incompressible Models. Claredon Press, Oxford
Science Publications, 1996

[13] Lions P -L, Masmoudi N. Incompressible limit for a viscous compressible fluid. J Math Pures Appl, 1998,
77(6) (9): 585–627

[14] Metcalfe J L. Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle.
Trans Amer Math Soc, 2004, 356(12): 4839–4855 (electronic)

[15] Michelson R C, Naqvi M A. Extraterrestrial flight (entomopter-based mars surveyor). Tech report, von Karman Institute, 2003

[16] Simon J. Compact sets in the space L (0,T;B). Ann Mat Pura Appl, 1987, 146(4): 65–96

[17] Smith H F, Sogge C D. On the critical semilinear wave equation outside convex obstacles. J Amer Math
Soc, 1995, 8(4): 879–916

[18] Smith H F, Sogge C D. Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm
Partial Di?erential Equations, 2000, 25(11/12): 2171–2183

[19] Strichartz R S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave
equations. Duke Math J, 1977, 44(3): 705–714