Domain sensitivity in singular limits of compressible viscous fluids
Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 12, 16 p.

In this note, we discuss several recently developed methods for studying stability of a singular limit process with respect to the shape of the underlying physical space. As a model example, we consider a compressible viscous barotropic fluid occupying a spatial domain ΩR 3 . In what follows, we describe two rather different problems: (i) the choice of effective boundary conditions; (ii) the fluid flow in the low Mach number regime. In the remaining part of the paper, we analyze these two issues simultaneously comparing the impact of different scales on the form of the resulting effective equations as well as the boundary conditions. Such a “synthesis” of several mathematical techniques may be useful in analyzing much broader class of multiscale problems.

DOI : 10.5802/slsedp.9
Feireisl, Eduard 1

1 Institute of Mathematics of the Academy of Sciences of the Czech Republic Žitná 25 115 67 Praha 1 Czech Republic
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Feireisl, Eduard. Domain sensitivity in singular limits of compressible viscous fluids. Séminaire Laurent Schwartz — EDP et applications (2011-2012), Exposé no. 12, 16 p. doi : 10.5802/slsedp.9. http://archive.numdam.org/articles/10.5802/slsedp.9/

[1] A. A. Amirat, D. Bresch, J. Lemoine, and J. Simon. Effect of rugosity on a flow governed by stationary Navier-Stokes equations. Quart. Appl. Math., 59:768–785, 2001. | MR | Zbl

[2] J. M. Arrieta and D. Krejčiřík. Geometric versus spectral convergence for the Neumann Laplacian under exterior perturbations of the domain. In Integral methods in science and engineering. Vol. 1, pages 9–19. Birkhäuser Boston Inc., Boston, MA, 2010. | MR | Zbl

[3] D. Bucur, E. Feireisl, and Nečasová. Boundary behavior of viscous fluids: Influence of wall roughness and friction-driven boundary conditions. Arch. Rational Mech. Anal., 197:117–138, 2010. | MR

[4] D. Bucur, E. Feireisl, Š. Nečasová, and J. Wolf. On the asymptotic limit of the Navier–Stokes system on domains with rough boundaries. J. Differential Equations, 244:2890–2908, 2008. | MR | Zbl

[5] M. Bulíček, J. Málek, and K.R. Rajagopal. Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear- rate dependent viscosity. Indiana Univ. Math. J., 56:51–86, 2007. | MR | Zbl

[6] J. Casado-Díaz, E. Fernández-Cara, and J. Simon. Why viscous fluids adhere to rugose walls: A mathematical explanation. J. Differential Equations, 189:526–537, 2003. | MR | Zbl

[7] Luna-Laynez M. Suárez-Grau F.J. Casado-Díaz, J. Asymptotic behavior of a viscous fluid with slip boundary conditions on a slightly rough wall. Math. Models Meth. Appl. Sci., 20:121–156, 2010. | MR | Zbl

[8] F. Coron. Derivation of slip boundary conditions for the Navier-Stokes system from the Boltzmann equation. J. Statistical Phys., 54:829–857, 1989. | MR | Zbl

[9] H.L. Cycon, R.G. Froese, W. Kirsch, and B. Simon. Schrödinger operators: with applications to quantum mechanics and global geometry. Texts and monographs in physics, Springer-Verlag, Berlin,Heidelberg, 1987. | MR | Zbl

[10] R. Danchin. Zero Mach number limit for compressible flows with periodic boundary conditions. Amer. J. Math., 124:1153–1219, 2002. | MR | Zbl

[11] P. D’Ancona and R. Racke. Evolution equations in non-flat waveguides. 2010. arXiv:1010.0817.

[12] D. M. Eidus. Limiting amplitude principle (in Russian). Usp. Mat. Nauk, 24(3):91–156, 1969. | MR

[13] R. Farwig, H. Kozono, and H. Sohr. An L q -approach to Stokes and Navier-Stokes equations in general domains. Acta Math., 195:21–53, 2005. | MR | Zbl

[14] E. Feireisl. Local decay of acoustic waves in the low mach number limits on general unbounded domains under slip boundary conditions. Commun. Partial Differential Equations, 2010. Submitted. | MR | Zbl

[15] E. Feireisl, T. Karper, O. Kreml, and J. Stebel. Stability with respect to domain of the low Mach number limit of compressible viscous fluids. 2011. Preprint.

[16] 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. | MR | Zbl

[17] I. Gallagher. Résultats récents sur la limite incompressible. Astérisque, (299):Exp. No. 926, vii, 29–57, 2005. Séminaire Bourbaki. Vol. 2003/2004. | Numdam | MR

[18] A. Henrot and M. Pierre. Variation et optimisation de formes, volume 48 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Berlin, 2005. Une analyse géométrique. [A geometric analysis]. | MR | Zbl

[19] W. Jaeger and A. Mikelić. On the roughness-induced effective boundary conditions for an incompressible viscous flow. J. Differential Equations, 170:96–122, 2001. | MR | Zbl

[20] T. Kato. Remarks on the zero viscosity limit for nonstationary Navier–Stokes flows with boundary. In Seminar on PDE’s, S.S. Chern (ed.), Springer, New York, 1984. | MR | Zbl

[21] Y. Last. Quantum dynamics and decomposition of singular continuous spectra. J. Funct. Anal., 142:406–445, 1996. | MR | Zbl

[22] R. Leis. Initial-boundary value problems in mathematical physics. B.G. Teubner, Stuttgart, 1986. | MR | Zbl

[23] J. Lighthill. On sound generated aerodynamically I. General theory. Proc. of the Royal Society of London, A 211:564–587, 1952. | MR | Zbl

[24] J. Lighthill. On sound generated aerodynamically II. General theory. Proc. of the Royal Society of London, A 222:1–32, 1954. | MR | Zbl

[25] C.M. Linton, P.-L., and P. McIver. Embedded trapped modes in water waves and acoustics. Wave Motion, 45:16–29, 2007. | MR | Zbl

[26] P.-L. Lions. Mathematical topics in fluid dynamics, Vol.2, Compressible models. Oxford Science Publication, Oxford, 1998. | MR

[27] P.-L. Lions and N. Masmoudi. Incompressible limit for a viscous compressible fluid. J. Math. Pures Appl., 77:585–627, 1998. | MR | Zbl

[28] N. Masmoudi. Examples of singular limits in hydrodynamics. In Handbook of Differential Equations, III, C. Dafermos, E. Feireisl Eds., Elsevier, Amsterdam, 2006. | MR | Zbl

[29] B. Mohammadi, O. Pironneau, and F. Valentin. Rough boundaries and wall laws. Int. J. Numer. Meth. Fluids, 27:169–177, 1998. | MR | Zbl

[30] J.-C. Nédélec. Acoustic and electromagnetic equations. Springer-Verlag, Heidelberg, 2001. | MR | Zbl

[31] N. V. Priezjev and S.M. Troian. Influence of periodic wall roughness on the slip behaviour at liquid/solid interfaces: molecular versus continuum predictions. J. Fluid Mech., 554:25–46, 2006. | Zbl

[32] S. Richardson. On the no-slip boundary condition. J. Fluid Mech., 59:707–719, 1973. | Zbl

[33] S. Schochet. The mathematical theory of low Mach number flows. M2ANMath. Model Numer. anal., 39:441–458, 2005. | Numdam | MR | Zbl

[34] B. R. Vaĭnberg. Asimptoticheskie metody v uravneniyakh matematicheskoi fiziki. Moskov. Gos. Univ., Moscow, 1982. | MR

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