Abstract
We study the incompressible limit for the full Navier-Stokes-Fourier system on unbounded domains with boundaries, supplemented with the complete slip boundary condition for the velocity field. Using an abstract result of Tosio Kato we show that the energy of acoustic waves decays to zero on any compact subset of the physical space. This in turn implies strong convergence of the velocity field to its limit in the incompressible regime.
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References
Alazard T.: Low Mach number flows and combustion. SIAM J. Math. Anal. 38(4), 1186–1213 (2006) (electronic)
Alazard T.: Low Mach number limit of the full Navier-Stokes equations. Arch. Rat. Mech. Anal. 180, 1–73 (2006)
Bechtel S.E., Rooney F.J., Forest M.G.: Connection between stability, convexity of internal energy, and the second law for compressible Newtonian fuids. J. Appl. Mech. 72, 299–300 (2005)
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. Part. Diff. Eqs. 28(9–10), 1675–1683 (2003)
Burq N., Planchon F., Stalker J.G., Tahvildar-Zadeh A.S.: Strichartz estimates for the wave and Schrödinger equations with potentials of critical decay. Indiana Univ. Math. J. 53(6), 1665–1680 (2004)
Dermejian Y., Guillot J.-C.: Théorie spectrale de la propagation des ondes acoustiques dans un milieu stratifié perturbé. J. Diff. Eqs. 62, 357–409 (1986)
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. 455(1986), 2271–2279 (1999)
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. 78, 461–471 (1999)
Engquist B., Majda A.: Radiation boundary conditions for acoustic and elastic wave calculations. Comm. Pure Appl. Math. 32(3), 314–358 (1979)
Feireisl E., Novotný A.: Singular Limits in Thermodynamics of Viscous Fluids. Birkhäuser-Verlag, Basel (2009)
Feireisl E., Poul L.: On compactness of the velicity field in the incompressible limit of the full Navier-Stokes-Fourier system on large domains. Math. Meth. Appl. Sci. 32, 1269–1286 (2009)
Isozaki H.: Singular limits for the compressible Euler equation in an exterior domain. J. Reine Angew. Math. 381, 1–36 (1987)
Kato T.: Wave operators and similarity for some non-selfadjoint operators. Math. Ann. 162, 258–279 (1965/1966)
Klainerman S., Majda A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math. 34, 481–524 (1981)
Klein R.: Asymptotic analyses for atmospheric flows and the construction of asymptotically adaptive numerical methods. Z. Angw. Math. Mech. 80, 765–777 (2000)
Klein R.: Multiple spatial scales in engineering and atmospheric low Mach number flows. ESAIM: Math. Mod. Numer. Anal. 39, 537–559 (2005)
Klein R., Botta N., Schneider T., Munz C.D., Roller S., Meister A., Hoffmann L., Sonar T.: Asymptotic adaptive methods for multi-scale problems in fluid mechanics. J. Engrg. Math. 39, 261–343 (2001)
Leis, R.: Initial-boundary Value Problems in Mathematical Physics. Stuttgart: B. G. Teubner, 1986
Lighthill J.: On sound generated aerodynamically I. General theory. Proc. of the Royal Society of London A 211, 564–587 (1952)
Lighthill J.: Waves in Fluids. Cambridge University Press, Cambridge (1978)
Lions P.-L.: Mathematical Topics in Fluid Dynamics, Vol. 2, Compressible Models. Oxford Science Publication, Oxford (1998)
Lions P.-L., Masmoudi N.: Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl. 77, 585–627 (1998)
Lions P.-L., Masmoudi N.: Une approche locale de la limite incompressible. C.R. Acad. Sci. Paris Sér. I Math. 329(5), 387–392 (1999)
Masmoudi, N.: Asymptotic problems and compressible and incompressible limits. In: Advances in Mathematical Fluid Mechanics, edited by Málek, J., Nečas, J., Rokyta, M., Berlin: Springer-Verlag, 2000, pp. 119–158
Masmoudi, N.: Examples of singular limits in hydrodynamics. In: Handbook of Differential Equations, III, Dafermos, C., Feireisl, E., eds., Amsterdam: Elsevier, 2006
Masmoudi N.: Rigorous derivation of the anelastic approximation. J. Math. Pures Appl. 88, 230–240 (2007)
Metcalfe J.L.: Global Strichartz estimates for solutions to the wave equation exterior to a convex obstacle. Trans. Amer. Math. Soc. 356(12), 4839–4855 (2004) (electronic)
Morawetz C.S.: Decay for solutions of the exterior problem for the wave equation. Comm. Pure Appl. Math. 28, 229–264 (1975)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. New York: Academic Press [Harcourt Brace Jovanovich Publishers], 1978
Schochet S.: Fast singular limits of hyperbolic PDE’s. J. Diff. Eqs. 114, 476–512 (1994)
Schochet S.: The mathematical theory of low Mach number flows. M2AN Math. Model Numer. Anal. 39, 441–458 (2005)
Shimizu S.: The limiting absorption principle. Math. Meth. Appl. Sci. 19, 187–215 (1996)
Smith H.F., Sogge C.D.: Global Strichartz estimates for nontrapping perturbations of the Laplacian. Comm. Part. Diff. Eqs. 25(11–12), 2171–2183 (2000)
Vaigant V.A.: An example of the nonexistence with respect to time of the global solutions of Navier-Stokes equations for a compressible viscous barotropic fluid (in Russian). Dokl. Akad. Nauk 339(2), 155–156 (1994)
Vaĭnberg, B.R.: Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki. Moscow: Moskov. Gos. Univ., 1982
Walker H.F.: Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains. J. Diff. Eqs. 23(3), 459–471 (1977)
Wilcox, C.H.: Sound Propagation in Stratified Fluids. Appl. Math. Ser. 50. Berlin: Springer-Verlag, 1984
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Communicated by P. Constantin
The work of E.F. was supported by Grant 201/08/0315 of GA ČR as a part of the general research programme of the Academy of Sciences of the Czech Republic, Institutional Research Plan AV0Z10190503.
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Feireisl, E. Incompressible Limits and Propagation of Acoustic Waves in Large Domains with Boundaries. Commun. Math. Phys. 294, 73–95 (2010). https://doi.org/10.1007/s00220-009-0954-6
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DOI: https://doi.org/10.1007/s00220-009-0954-6