Hyperbolic Navier-Stokes equations I: Local well-posedness

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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

Abstract
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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 and Control Theory, 2012, 1 (1) : 195-215. doi: 10.3934/eect.2012.1.195

References:

[1]	R. A. Adams, "Sobolev Spaces,'' Pure Appl. Math., 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
[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, 16 pp.
[3]	B. Carbonaro and F. Rosso, Some remarks on a modified fluid dynamics equation, Rendiconti Del Circolo Matematico Di Palermo (2), 30 (1981), 111-122. doi: 10.1007/BF02845131.
[4]	M. Carrassi and A. Morro, A modified Navier-Stokes equation and its consequences on sound dispersion, II Nuovo Cimento B, 9 (1972).
[5]	T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Ration. Mech. Anal., 63 (1976), 273-294.
[6]	D. D. Joseph, "Fluid Dynamics of Viscoleastic Liquids,'' Appl. Math. Sciences, 84, Springer-Verlag, New York, 1990.
[7]	T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in "Spectral Theory and Differential Equations" (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jögens), Lecture Notes in Math., 448, Springer, Berlin, (1975), 25-70.
[8]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.
[9]	T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems,'' Lezioni Fermiane [Fermi Lectures], Scuola Normale Superiore, Pisa, Accademia Nazionale dei Lincei, Rome, 1985.
[10]	A. Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,'' Appl. Math. Sci., 53, Springer-Verlag, New York, 1984.
[11]	M. Paicu and G. Raugel, Une perturbation hyperbolique des équations de Navier-Stokes, 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.
[12]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
[13]	R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems,'' Aspects of Mathematics, E19, Friedr. Vieweg & Sohn, Braunschweig, 1992.
[14]	R. Racke, Thermoelasticity, in "Handbook of Differential Equations. Evolutionary Equations," Vol. V (eds. C. M. Dafermos and M. Pokorný), Elsevier/North-Holland, Amsterdam, (2009), 315-420.
[15]	A. Schöwe, "Langzeitasymptotik der Hyperbolischen Navier-Stokes Gleichung im $\mathbb R^3$,'' Diploma thesis, University of Konstanz, 2011.
[16]	R. Temam, "The Navier-Stokes Equations. Theory and Numerical Analysis,'' Revised edition, With an appendix by F. Thomasset, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.
[17]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'' North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.

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References:

[1]	R. A. Adams, "Sobolev Spaces,'' Pure Appl. Math., 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975.
[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, 16 pp.
[3]	B. Carbonaro and F. Rosso, Some remarks on a modified fluid dynamics equation, Rendiconti Del Circolo Matematico Di Palermo (2), 30 (1981), 111-122. doi: 10.1007/BF02845131.
[4]	M. Carrassi and A. Morro, A modified Navier-Stokes equation and its consequences on sound dispersion, II Nuovo Cimento B, 9 (1972).
[5]	T. J. R. Hughes, T. Kato and J. E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Ration. Mech. Anal., 63 (1976), 273-294.
[6]	D. D. Joseph, "Fluid Dynamics of Viscoleastic Liquids,'' Appl. Math. Sciences, 84, Springer-Verlag, New York, 1990.
[7]	T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations, in "Spectral Theory and Differential Equations" (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jögens), Lecture Notes in Math., 448, Springer, Berlin, (1975), 25-70.
[8]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems, Arch. Ration. Mech. Anal., 58 (1975), 181-205. doi: 10.1007/BF00280740.
[9]	T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems,'' Lezioni Fermiane [Fermi Lectures], Scuola Normale Superiore, Pisa, Accademia Nazionale dei Lincei, Rome, 1985.
[10]	A. Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,'' Appl. Math. Sci., 53, Springer-Verlag, New York, 1984.
[11]	M. Paicu and G. Raugel, Une perturbation hyperbolique des équations de Navier-Stokes, 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.
[12]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'' Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
[13]	R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems,'' Aspects of Mathematics, E19, Friedr. Vieweg & Sohn, Braunschweig, 1992.
[14]	R. Racke, Thermoelasticity, in "Handbook of Differential Equations. Evolutionary Equations," Vol. V (eds. C. M. Dafermos and M. Pokorný), Elsevier/North-Holland, Amsterdam, (2009), 315-420.
[15]	A. Schöwe, "Langzeitasymptotik der Hyperbolischen Navier-Stokes Gleichung im $\mathbb R^3$,'' Diploma thesis, University of Konstanz, 2011.
[16]	R. Temam, "The Navier-Stokes Equations. Theory and Numerical Analysis,'' Revised edition, With an appendix by F. Thomasset, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York, 1979.
[17]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'' North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.

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