Hyperbolic Navier-Stokes equations II: Global existence of small solutions

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June 2012, 1(1): 217-234. doi: 10.3934/eect.2012.1.217

Hyperbolic Navier-Stokes equations II: Global existence of small solutions

1.	Department of Mathematics, University of Konstanz, 78457 Konstanz
2.	Center of Smart Interfaces, Technische Universität Darmstadt, Petersenstraße 32, 64287 Darmstadt

Received September 2011 Revised December 2011 Published March 2012

Abstract
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We consider a hyperbolicly perturbed Navier-Stokes initial value problem in ${\mathbb R}^n$, $n=2,3$, arising from using a Cattaneo type relation instead of a Fourier type one in the constitutive equations. The resulting system is an essentially hyperbolic one with quasilinear nonlinearities. The global existence of smooth solutions for small data is proved, and relations to the classical Navier-Stokes systems are discussed.

Keywords: decay rate, Oldroyd, Navier-Stokes, Cattaneo law, Fourier law, global small solution, blow-up..

Mathematics Subject Classification: Primary: 35L72, 35Q30, 76D0.

Citation: Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations & Control Theory, 2012, 1 (1) : 217-234. doi: 10.3934/eect.2012.1.217

References:

[1]	R. A. Adams, "Sobolev Spaces,", Pure Appl. Math., 65 (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 ().
[3]	A. Babin, A. Mahalov and B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity,, Indiana Univ. Math. J., 50 (2001), 1.
[4]	M. Cannone, "Ondelettes, Paraproduits et Navier-Stokes,'', With a preface by Yves Meyer, (1995).
[5]	B. Carbonaro and F. Rosso, Some remarks on a modified fluid dynamics equation,, Rendiconti Del Circolo Matematico Di Palermo (2), 30 (1981), 111. doi: 10.1007/BF02845131.
[6]	M. Carrassi and A. Morro, A modified Navier-Stokes equation and its consequences on sound dispersion,, II Nuovo Cimento B, 9 (1972).
[7]	P. Constantin and C. Foias, "Navier-Stokes Equations,'', Chicago Lectures in Mathematics, (1988).
[8]	M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics,, Appl. Math. Letters, 22 (2009), 1374. doi: 10.1016/j.aml.2009.03.010.
[9]	H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law,, Arch. Rational Mech. Anal., 194 (2009), 221. doi: 10.1007/s00205-009-0220-2.
[10]	H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u + u^{1+\alpha}$,, J. Fac. Sci. Univ. Tokyo, 13 (1966), 109.
[11]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,'', 2^nd edition, (2011).
[12]	Y. Giga, K. Inui, A. Mahalov and J. Saal, Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data,, Indiana Univ. Math. J., 57 (2008), 2775.
[13]	E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen,, Math. Nachrichten, 4 (1951), 213. doi: 10.1002/mana.3210040121.
[14]	D. D. Joseph, "Fluid Dynamics of Viscoleastic Liquids,'', Appl. Math. Sciences, 84 (1990).
[15]	O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'', Second English edition, (1969).
[16]	J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, Acta Math., 63 (1934), 193. doi: 10.1007/BF02547354.
[17]	A. Mahalov and B. Nicolaenko, Global solubility of the three-dimensional Navier-Stokes equations with uniformly large vorticity,, Russ. Math. Surveys, 58 (2003), 287. doi: 10.1070/RM2003v058n02ABEH000611.
[18]	A. Mahalov, E. S. Titi and S. Leibovich, Invariant helical subspaces for the Navier-Stokes equations,, Arch. Rational Mech. Anal., 112 (1990), 193. doi: 10.1007/BF00381234.
[19]	A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. RIMS, 12 (): 169. doi: 10.2977/prims/1195190962.
[20]	M. Paicu and G. Raugel, Une perturbation hyperbolique des équations de Navier-Stokes,, in, 21 (2007), 65.
[21]	G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations,, Nonlinear Analysis, 9 (1985), 399. doi: 10.1016/0362-546X(85)90001-X.
[22]	G. Ponce and R. Racke, Global existence of small solutions to the initial value problem for nonlinear thermoelasticity,, J. Differential Equations, 87 (1990), 70.
[23]	G. Ponce, R. Racke, T. C. Sideris and E. S. Titi, Global stability of large solutions to the 3d Navier-Stokes equations,, Commun. Math. Phys., 159 (1994), 329.
[24]	R. Quintanilla and R. Racke, Addendum to: Qualitative aspects of solutions in resonators,, Arch. Mech., 63 (2011), 429.
[25]	R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems,'', Aspects of Mathematics, E19 (1992).
[26]	R. Racke, Thermoelasticity,, in, (2009), 315.
[27]	R. Racke and J. Saal, Hyperbolic Navier-Stokes equations I: Local well-posedness,, Evolution Equations and Control Theory, ().
[28]	A. Schöwe, "Langzeitasymptotik der Hyperbolischen Navier-Stokes Gleichung im $\mathbb R^3$,'', Diploma thesis, (2011).
[29]	T. C. Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations,, Arch. Rational Mech. Anal., 86 (1984), 369. doi: 10.1007/BF00280033.
[30]	R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'', Revised edition, 2 (1979).
[31]	M .R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space,, J. Appl. Math. Mech., 32 (1968), 52. doi: 10.1016/0021-8928(68)90147-0.
[32]	W. von Wahl, "The Equations of Navier-Stokes and Abstract Parabolic Equations,'', Aspects of Mathematics, E8 (1985).

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References:

[1]	R. A. Adams, "Sobolev Spaces,", Pure Appl. Math., 65 (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 ().
[3]	A. Babin, A. Mahalov and B. Nicolaenko, 3D Navier-Stokes and Euler equations with initial data characterized by uniformly large vorticity,, Indiana Univ. Math. J., 50 (2001), 1.
[4]	M. Cannone, "Ondelettes, Paraproduits et Navier-Stokes,'', With a preface by Yves Meyer, (1995).
[5]	B. Carbonaro and F. Rosso, Some remarks on a modified fluid dynamics equation,, Rendiconti Del Circolo Matematico Di Palermo (2), 30 (1981), 111. doi: 10.1007/BF02845131.
[6]	M. Carrassi and A. Morro, A modified Navier-Stokes equation and its consequences on sound dispersion,, II Nuovo Cimento B, 9 (1972).
[7]	P. Constantin and C. Foias, "Navier-Stokes Equations,'', Chicago Lectures in Mathematics, (1988).
[8]	M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics,, Appl. Math. Letters, 22 (2009), 1374. doi: 10.1016/j.aml.2009.03.010.
[9]	H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law,, Arch. Rational Mech. Anal., 194 (2009), 221. doi: 10.1007/s00205-009-0220-2.
[10]	H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t = \Delta u + u^{1+\alpha}$,, J. Fac. Sci. Univ. Tokyo, 13 (1966), 109.
[11]	G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,'', 2^nd edition, (2011).
[12]	Y. Giga, K. Inui, A. Mahalov and J. Saal, Uniform global solvability of the rotating Navier-Stokes equations for nondecaying initial data,, Indiana Univ. Math. J., 57 (2008), 2775.
[13]	E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen,, Math. Nachrichten, 4 (1951), 213. doi: 10.1002/mana.3210040121.
[14]	D. D. Joseph, "Fluid Dynamics of Viscoleastic Liquids,'', Appl. Math. Sciences, 84 (1990).
[15]	O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'', Second English edition, (1969).
[16]	J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace,, Acta Math., 63 (1934), 193. doi: 10.1007/BF02547354.
[17]	A. Mahalov and B. Nicolaenko, Global solubility of the three-dimensional Navier-Stokes equations with uniformly large vorticity,, Russ. Math. Surveys, 58 (2003), 287. doi: 10.1070/RM2003v058n02ABEH000611.
[18]	A. Mahalov, E. S. Titi and S. Leibovich, Invariant helical subspaces for the Navier-Stokes equations,, Arch. Rational Mech. Anal., 112 (1990), 193. doi: 10.1007/BF00381234.
[19]	A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations,, Publ. RIMS, 12 (): 169. doi: 10.2977/prims/1195190962.
[20]	M. Paicu and G. Raugel, Une perturbation hyperbolique des équations de Navier-Stokes,, in, 21 (2007), 65.
[21]	G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations,, Nonlinear Analysis, 9 (1985), 399. doi: 10.1016/0362-546X(85)90001-X.
[22]	G. Ponce and R. Racke, Global existence of small solutions to the initial value problem for nonlinear thermoelasticity,, J. Differential Equations, 87 (1990), 70.
[23]	G. Ponce, R. Racke, T. C. Sideris and E. S. Titi, Global stability of large solutions to the 3d Navier-Stokes equations,, Commun. Math. Phys., 159 (1994), 329.
[24]	R. Quintanilla and R. Racke, Addendum to: Qualitative aspects of solutions in resonators,, Arch. Mech., 63 (2011), 429.
[25]	R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems,'', Aspects of Mathematics, E19 (1992).
[26]	R. Racke, Thermoelasticity,, in, (2009), 315.
[27]	R. Racke and J. Saal, Hyperbolic Navier-Stokes equations I: Local well-posedness,, Evolution Equations and Control Theory, ().
[28]	A. Schöwe, "Langzeitasymptotik der Hyperbolischen Navier-Stokes Gleichung im $\mathbb R^3$,'', Diploma thesis, (2011).
[29]	T. C. Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations,, Arch. Rational Mech. Anal., 86 (1984), 369. doi: 10.1007/BF00280033.
[30]	R. Temam, "Navier-Stokes Equations. Theory and Numerical Analysis,'', Revised edition, 2 (1979).
[31]	M .R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space,, J. Appl. Math. Mech., 32 (1968), 52. doi: 10.1016/0021-8928(68)90147-0.
[32]	W. von Wahl, "The Equations of Navier-Stokes and Abstract Parabolic Equations,'', Aspects of Mathematics, E8 (1985).

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