We study the low Mach number limit of the full Navier–Stokes–Fourier system in the case of low stratification with ill-prepared initial data for the problem stated on a moving domain with a prescribed motion of the boundary. Similarly as in the case of a fixed domain, we recover as a limit the Oberback–Boussinesq system; however, we identify one additional term in the temperature equation of the limit system which is related to the motion of the domain and which is not present in the case of a fixed domain. One of the main ingredients in the proof is the properties of the Helmholtz decomposition on moving domains and the dependence of eigenvalues and eigenspaces of the Neumann Laplace operator on time.
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T. Alazard. Low Mach number limit of the full Navier-Stokes equations. Arch. Ration. Mech. Anal., 180, 1–73, 2006.
R. A. Adams and J. J. F. Fournier. Sobolev spaces. Second edition. Pure and Applied Mathematics (Amsterdam), 140, Elsevier/Academic Press, Amsterdam, 2003.
V. I. Burenkov, P. D. Lamberti and M. Lanza de Cristoforis. Spectral stability of nonnegative self-adjoint operators. J. Math. Sci. (N. Y.), 149, 1417–1452, 2008.
R. Danchin. Zero Mach number limit for compressible flows with periodic boundary conditions. Amer. J. Math., 124, 1153–1219, 2002.
B. Desjardins and E. Grenier. Low Mach number limit of viscous compressible flows in the whole space. R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci., 455, 2271–2279, 1999.
B. Desjardins, E. Grenier, P.-L. Lions, and N. Masmoudi. Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl., 78, 461–471, 1999.
R. J. DiPerna and P.-L. Lions. Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math., 98, 511–547, 1989.
D. B. Ebin. The motion of slightly compressible fluids viewed as a motion with strong constraining force. Ann. Math., 105, 141–200, 1977.
R. Farwig, H. Kozono and H. Sohr. On the Helmholtz decomposition in general unbounded domains. Arch. Math. (Basel), 88, 239–248, 2007.
E. Feireisl, J. Neustupa, and J. Stebel. Convergence of a Brinkman-type penalization for compressible fluid flows. J. Differential Equations, 250, 596–606, 2011.
E. Feireisl. Dynamics of viscous compressible fluids. Oxford University Press, Oxford, 2004.
E. Feireisl. On the motion of a viscous, compressible, and heat conducting fluid. Indiana Univ. Math. J., 53, 1707–1740, 2004.
E. Feireisl and A. Novotný. Singular limits in thermodynamics of viscous fluids. Birkhäuser-Verlag, Basel, 2009.
E. Feireisl, O. Kreml, Š. Nečasová, J. Neustupa, and J. Stebel. Incompressible limits of fluids excited by moving boundary. SIAM J. Math. Anal. 46, 1456–1471, 2014.
E. Feireisl, O. Kreml, Š. Nečasová, J. Neustupa, and J. Stebel. Weak solutions to the barotropic Navier - Stokes system with slip boundary conditions in time dependent domains. J. Differential Equations, 254, 125–140, 2013.
E. Feireisl, O. Kreml, V. Mácha, Š. Nečasová. On the low Mach number limit of compressible flows in exterior moving domains. J. Evol. Equ., 16, 705–722, 2016.
E. Feireisl, A. Novotný, and H. Petzeltová. On the existence of globally defined weak solutions to the Navier-Stokes equations of compressible isentropic fluids. J. Math. Fluid Mech., 3, 358–392, 2001.
I. Gallagher. Résultats récents sur la limite incompressible. Astérisque, (299) Exp. No. 926, vii, 29–57, 2005. Séminaire Bourbaki. Vol. 2003/2004.
D. Hoff. Dynamics of singularity surfaces for compressible viscous flows in two space dimensions. Comm. Pure Appl. Math., 55, 1365–1407, 2002.
R. Klein, N. Botta, T. Schneider, C. D. Munz, S. Roller, A. Meister, L. Hoffmann and T. Sonar. Asymptotic adaptive methods for multi-scale problems in fluid mechanics J. Engrg. Math., 39, 261–343, 2001.
S. Klainerman and A. Majda. Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Comm. Pure Appl. Math., 34, 481–524, 1981.
O. Kreml, V. Mácha, Š. Nečasová, A. Wróblewska-Kamińska. Flow of heat conducting fluid in a time dependent domain. Z. Angew. Math. Phys, 69, Art. 119, 1–27, 2018.
O. Kreml, V. Mácha, Š. Nečasová, A. Wróblewska-Kamińska. Weak solutions to the full Navier-Stokes-Fourier system with slip boundary conditions in time dependent domain. J. Math. Pures Appl., 109, 67–92, 2018.
P.-L. Lions. Mathematical topics in fluid dynamics, Vol.2, Compressible models. Oxford Science Publication, Oxford, 1998.
S. Schochet. The mathematical theory of low Mach number flows. M2AN Math. Model Numer. anal., 39, 441–458, 2005.
J. Sokołowski and J.-P. Zolésio. Introduction to Shape Optimization. Shape Sensitivity Analysis. Springer Series in Computational Mathematics, 16, Springer-Verlag, Berlin, 1992.
R. Kh. Zeytounian, Asymptotic modeling of atmospheric flows. Springer-Verlag, Berlin, 1990.
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The works of O.K., V.M. and Š.N. were supported by project GA16-03230S and by RVO 67985840. Last version of the paper was supported by project GA19-04243S. Part of work was done during stay of O.K. at Imperial College London which was supported by the grant Iuventus Plus 0871/IP3/2016/74. The work of A.W.-K. is partially supported by a Newton Fellowship of the Royal Society and by the grant Iuventus Plus 0871/IP3/2016/74 of Ministry of Sciences and Higher Education RP. Her stay at Institute of Mathematics of Academy of Sciences, Prague, was supported by 7AMB16PL060.
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Kreml, O., Mácha, V., Nečasová, Š. et al. Low stratification of a heat-conducting fluid in a time-dependent domain. J. Evol. Equ. (2021). https://doi.org/10.1007/s00028-020-00653-3