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

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.
Citation: Reinhard Racke, Jürgen Saal. Hyperbolic Navier-Stokes equations II: Global existence of small solutions. Evolution Equations and 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, 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]

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), 1-35.

[4]

M. Cannone, "Ondelettes, Paraproduits et Navier-Stokes,'' With a preface by Yves Meyer, Diderot Editeur, Paris, 1995.

[5]

B. Carbonaro and F. Rosso, Some remarks on a modified fluid dynamics equation, Rendiconti Del Circolo Matematico Di Palermo (2), 30 (1981), 111-122. 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, The University of Chicago Press, Chicago, IL, 1988.

[8]

M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Letters, 22 (2009), 1374-1379. 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), 221-251. 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, Sect I, 13 (1966), 109-124.

[11]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,'' 2nd edition, Springer Monographs in Mathematics, Springer, New York, 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), 2775-2791.

[13]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachrichten, 4 (1951), 213-231. doi: 10.1002/mana.3210040121.

[14]

D. D. Joseph, "Fluid Dynamics of Viscoleastic Liquids,'' Appl. Math. Sciences, 84, Springer-Verlag, New York, 1990.

[15]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Second English edition, revised and enlarged, Mathematics and its Applications, Vol. 2 , Gordon and Breachm, Science Publishers, New York-London-Paris, 1969.

[16]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. 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), 287-318. 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), 193-222. doi: 10.1007/BF00381234.

[19]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS, 12 (1976/77), 169-189. doi: 10.2977/prims/1195190962.

[20]

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.

[21]

G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Analysis, 9 (1985), 399-418. 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), 70-83.

[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), 329-341.

[24]

R. Quintanilla and R. Racke, Addendum to: Qualitative aspects of solutions in resonators, Arch. Mech., 63 (2011), 429-433.

[25]

R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems,'' Aspects of Mathematics, E19, Friedr. Vieweg & Sohn, Braunschweig, 1992.

[26]

R. Racke, Thermoelasticity, in "Handbook of Differential Equations: Evolutionary Equations," Vol. V (eds. C. M. Dafermos and M. Pokorný), Elsevier/North-Holland, Amesterdam, (2009), 315-420.

[27]

R. Racke and J. Saal, Hyperbolic Navier-Stokes equations I: Local well-posedness, Evolution Equations and Control Theory, to appear.

[28]

A. Schöwe, "Langzeitasymptotik der Hyperbolischen Navier-Stokes Gleichung im $\mathbb R^3$,'' Diploma thesis, University of Konstanz, 2011.

[29]

T. C. Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 86 (1984), 369-381. doi: 10.1007/BF00280033.

[30]

R. Temam, "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.

[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), 52-61. 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, Friedr. Vieweg & Sohn, Braunschweig, 1985.

show all references

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]

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), 1-35.

[4]

M. Cannone, "Ondelettes, Paraproduits et Navier-Stokes,'' With a preface by Yves Meyer, Diderot Editeur, Paris, 1995.

[5]

B. Carbonaro and F. Rosso, Some remarks on a modified fluid dynamics equation, Rendiconti Del Circolo Matematico Di Palermo (2), 30 (1981), 111-122. 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, The University of Chicago Press, Chicago, IL, 1988.

[8]

M. Dreher, R. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Appl. Math. Letters, 22 (2009), 1374-1379. 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), 221-251. 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, Sect I, 13 (1966), 109-124.

[11]

G. P. Galdi, "An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Steady-State Problems,'' 2nd edition, Springer Monographs in Mathematics, Springer, New York, 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), 2775-2791.

[13]

E. Hopf, Über die Anfangswertaufgabe für die hydrodynamischen Grundgleichungen, Math. Nachrichten, 4 (1951), 213-231. doi: 10.1002/mana.3210040121.

[14]

D. D. Joseph, "Fluid Dynamics of Viscoleastic Liquids,'' Appl. Math. Sciences, 84, Springer-Verlag, New York, 1990.

[15]

O. A. Ladyzhenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'' Second English edition, revised and enlarged, Mathematics and its Applications, Vol. 2 , Gordon and Breachm, Science Publishers, New York-London-Paris, 1969.

[16]

J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math., 63 (1934), 193-248. 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), 287-318. 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), 193-222. doi: 10.1007/BF00381234.

[19]

A. Matsumura, On the asymptotic behavior of solutions of semi-linear wave equations, Publ. RIMS, 12 (1976/77), 169-189. doi: 10.2977/prims/1195190962.

[20]

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.

[21]

G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Analysis, 9 (1985), 399-418. 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), 70-83.

[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), 329-341.

[24]

R. Quintanilla and R. Racke, Addendum to: Qualitative aspects of solutions in resonators, Arch. Mech., 63 (2011), 429-433.

[25]

R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems,'' Aspects of Mathematics, E19, Friedr. Vieweg & Sohn, Braunschweig, 1992.

[26]

R. Racke, Thermoelasticity, in "Handbook of Differential Equations: Evolutionary Equations," Vol. V (eds. C. M. Dafermos and M. Pokorný), Elsevier/North-Holland, Amesterdam, (2009), 315-420.

[27]

R. Racke and J. Saal, Hyperbolic Navier-Stokes equations I: Local well-posedness, Evolution Equations and Control Theory, to appear.

[28]

A. Schöwe, "Langzeitasymptotik der Hyperbolischen Navier-Stokes Gleichung im $\mathbb R^3$,'' Diploma thesis, University of Konstanz, 2011.

[29]

T. C. Sideris, Formation of singularities in solutions to nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 86 (1984), 369-381. doi: 10.1007/BF00280033.

[30]

R. Temam, "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.

[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), 52-61. 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, Friedr. Vieweg & Sohn, Braunschweig, 1985.

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