Abstract
We consider the barotropic Navier–Stokes system describing the motion of a compressible viscous fluid confined to a straight layer \({\Omega_\varepsilon = \omega\times (0, \varepsilon)}\) , where ω is a particular 2-D domain (a periodic cell, bounded domain or the whole 2-D space). We show that the weak solutions in the 3D domain converge to a (strong) solutions of the 2-D Navier–Stokes system on ω as \({\varepsilon \to 0}\) on the maximal life time of the strong solution.
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Adams R.A.: Sobolev spaces. Academic Press, New York (1975)
Bella, P., Feireisl, E., Novotny, A.: Dimensional reduction for compressible viscous fluids. Preprint IM-2013-21. http://www.math.cas.cz
Dafermos C.M.: The second law of thermodynamics and stability. Arch. Ration. Mech. Anal. 70, 167–179 (1979)
Dain S.: Generalized Korn’s inequality and conformal killing vectors. Calc. Var. Part. Differ. Equ. 25, 535–540 (2006)
Feireisl E.: Dynamics of viscous compressible fluids. Oxford University Press, Oxford (2004)
Feireisl E., Jin B., Novotný A.: Relative entropies, suitable weak solutions, and weak-strong uniqueness for the compressible Navier–Stokes system. J. Math. Fluid Mech. 14, 712–730 (2012)
Feireisl E., Novotny A.: Singular limits in thermodynamics of viscous fluids. Birkhauser, Basel (2009)
Feireisl E., Novotný A., Sun Y.: Suitable weak solutions to the Navier–Stokes equations of compressible viscous fluids. Indiana Univ. Math. J. 60, 611–631 (2011)
Germain P.: Weak-strong uniqueness for the isentropic compressible Navier–Stokes system. J. Math. Fluid Mech. 13(1), 137–146 (2011)
Iftimie D., Raugel G., Sell G.R.: Navier–Stokes equations in thin 3D domains with Navier boundary conditions. Indiana Univ. Math. J. 56, 1083–1156 (2007)
Jesslé, D., Jin, B.J., Novotny, A.: Navier–Stokes–Fourier system on unbounded domains: weak solutions, relative entropies, weak-strong uniqueness. SIAM J. Math. Anal. 45(3), 1003–1026 (2013)
Kazhikhov A.V.: Correctness “in the large” of mixed boundary value problems for a model of equations of a viscous gas (in Russian). Din. Sphlosn. Sredy 21, 18–47 (1975)
Lewicka M., Müller S.: The uniform Korn Poincaré inequality in thin domains. Ann. I. H. Poincaré 28, 443–469 (2011)
Lions P.-L.: Mathematical topics in fluid dynamics, vol. 1, Incompressible models. Oxford Science Publication, Oxford (1996)
Lions P.-L.: Mathematical topics in fluid dynamics, vol. 2, Compressible models. Oxford Science Publication, Oxford (1998)
Mellet A., Vasseur A.: Existence and uniqueness of global strong solutions for one-dimensional compressible Navier–Stokes equations. SIAM J. Math. Anal. 39(4), 1344–1365 (2007/08)
Novotny A., Straskraba I.: Introduction to the mathematical theory of compressible fluids. Oxford Science Publication, Oxford (2004)
Rajagopal, K. R.: A new development and interpretation of the Navier–Stokes fluid which reveals why the “Stokes assumption” is inapt. Int. J. Non-Linear Mech. 50, 141–151 (2013)
Raugel G., Sell G.R.: Navier–Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions. J. Am. Math. Soc. 6, 503–568 (1993)
Reshetnyak Yu. G.: Stability theorems in geometry and analysis, vol. 304 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht (1994). Translated from the 1982 Russian original by N. S. Dairbekov and V. N. Dyatlov, and revised by the author, Translation edited and with a foreword by S. S. Kutateladze.
Valli A., Zajaczkowski W.: Navier–Stokes equations for compressible fluids: global existence and qualitative properties of the solutions in the general case. Commun. Math. Phys. 103, 259–296 (1986)
Vodák R.: Asymptotic analysis of steady and nonsteady Navier–Stokes equations for barotropic compressible flow. Acta Appl. Math. 110, 991–1009 (2010)
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Communicated by E. Feireisl
A. Novotný work was supported by the MODTERCOM project within the APEX programme of the region Provence-Alpe-Côte d’Azur.
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Maltese, D., Novotný, A. Compressible Navier–Stokes Equations on Thin Domains. J. Math. Fluid Mech. 16, 571–594 (2014). https://doi.org/10.1007/s00021-014-0177-2
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DOI: https://doi.org/10.1007/s00021-014-0177-2