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June  2012, 1(1): 195-215. doi: 10.3934/eect.2012.1.195

Hyperbolic Navier-Stokes equations I: Local well-posedness

Reinhard Racke 1, and  Jürgen Saal 2,

1. 

Department of Mathematics, University of Konstanz, 78457 Konstanz, Germany
2. 

Center of Smart Interfaces, Technische Universität Darmstadt, Petersenstraße 32, 64287 Darmstadt, Germany

Received  September 2011 Revised  December 2011 Published  March 2012

We replace a Fourier type law by a Cattaneo type law in the derivation of the fundamental equations of fluid mechanics. This leads to hyperbolicly perturbed quasilinear Navier-Stokes equations. For this problem the standard approach by means of quasilinear symmetric hyperbolic systems seems to fail by the fact that finite propagation speed might not be expected. Therefore a somewhat different approach via viscosity solutions is developed in order to prove higher regularity energy estimates for the linearized system. Surprisingly, this method yields stronger results than previous methods, by the fact that we can relax the regularity assumptions on the coefficients to a minimum. This leads to a short and elegant proof of a local-in-time existence result for the corresponding first order quasilinear system, hence also for the original hyperbolicly perturbed Navier-Stokes equations.
Keywords: Navier-Stokes, Cattaneo law, Oldroyd., Fourier law, well-posedness.
Mathematics Subject Classification: Primary: 35L72, 35Q30, 76D0.
Citation: Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations I: Local well-posedness. Evolution Equations & Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195
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show all references

References:
[1]

Pure Appl. Math., 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar

[2]

G. M. de Araújo, S. B. de Menezes and A. O. Marinho, Existence of solutions for an Oldroyd model of viscoelastic fluids,, Electronic J. Differential Equations, 2009 ().   Google Scholar

[3]

Rendiconti Del Circolo Matematico Di Palermo (2), 30 (1981), 111-122. doi: 10.1007/BF02845131.  Google Scholar

[4]

II Nuovo Cimento B, 9 (1972). Google Scholar

[5]

Arch. Ration. Mech. Anal., 63 (1976), 273-294.  Google Scholar

[6]

Appl. Math. Sciences, 84, Springer-Verlag, New York, 1990.  Google Scholar

[7]

in "Spectral Theory and Differential Equations" (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jögens), Lecture Notes in Math., 448, Springer, Berlin, (1975), 25-70.  Google Scholar

[8]

Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.  Google Scholar

[9]

Lezioni Fermiane [Fermi Lectures], Scuola Normale Superiore, Pisa, Accademia Nazionale dei Lincei, Rome, 1985.  Google Scholar

[10]

Appl. Math. Sci., 53, Springer-Verlag, New York, 1984.  Google Scholar

[11]

in "ESAIM: Proceedings," Vol. 21 (2007) [Journées d'Analyse Fonctionnelle et Numérique en l'honneur de Michel Crouzeix], ESAIM Proc., 21, EDP Sci., Les Ulis, (2007), 65-87.  Google Scholar

[12]

Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.  Google Scholar

[13]

Aspects of Mathematics, E19, Friedr. Vieweg & Sohn, Braunschweig, 1992.  Google Scholar

[14]

in "Handbook of Differential Equations. Evolutionary Equations," Vol. V (eds. C. M. Dafermos and M. Pokorný), Elsevier/North-Holland, Amsterdam, (2009), 315-420.  Google Scholar

[15]

Diploma thesis, University of Konstanz, 2011. Google Scholar

[16]

Revised edition, With an appendix by F. Thomasset, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.  Google Scholar

[17]

North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

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