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

Previous Article
EECT Home
This Issue
Next Article
Hyperbolic Navier-Stokes equations I: Local well-posedness

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

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

Reinhard Racke ^1, and Jürgen Saal ^2,

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
Related Papers

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 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,'' 2^nd 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,'' 2^nd 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.

[1]	Yat Tin Chow, Ali Pakzad. On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021270
[2]	Tongtong Liang. The stability with the general decay rate of the solution for stochastic functional Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022127
[3]	Yuan Xu, Fujun Zhou, Weihua Gong. Global Well-posedness and Optimal Decay Rate of the Quasi-static Incompressible Navier–Stokes–Fourier–Maxwell–Poisson System. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1537-1565. doi: 10.3934/cpaa.2022028
[4]	J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647
[5]	Xin Zhong. Global strong solution and exponential decay for nonhomogeneous Navier-Stokes and magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3563-3578. doi: 10.3934/dcdsb.2020246
[6]	Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637
[7]	Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete and Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009
[8]	Tohru Nakamura, Shuichi Kawashima. Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law. Kinetic and Related Models, 2018, 11 (4) : 795-819. doi: 10.3934/krm.2018032
[9]	Pedro Roberto de Lima, Hugo D. Fernández Sare. General condition for exponential stability of thermoelastic Bresse systems with Cattaneo's law. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3575-3596. doi: 10.3934/cpaa.2020156
[10]	Yinsong Bai, Lin He, Huijiang Zhao. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2441-2474. doi: 10.3934/cpaa.2021049
[11]	Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497
[12]	Shuai Liu, Yuzhu Wang. Optimal time-decay rate of global classical solutions to the generalized compressible Oldroyd-B model. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021041
[13]	Yongfu Wang. Global strong solution to the two dimensional nonhomogeneous incompressible heat conducting Navier-Stokes flows with vacuum. Discrete and Continuous Dynamical Systems - B, 2020, 25 (11) : 4317-4333. doi: 10.3934/dcdsb.2020099
[14]	Lihuai Du, Ting Zhang. Local and global strong solution to the stochastic 3-D incompressible anisotropic Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2018, 38 (9) : 4745-4765. doi: 10.3934/dcds.2018209
[15]	Zhenhua Guo, Zilai Li. Global existence of weak solution to the free boundary problem for compressible Navier-Stokes. Kinetic and Related Models, 2016, 9 (1) : 75-103. doi: 10.3934/krm.2016.9.75
[16]	Qi S. Zhang. An example of large global smooth solution of 3 dimensional Navier-Stokes equations without pressure. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5521-5523. doi: 10.3934/dcds.2013.33.5521
[17]	Ruizhao Zi. Global solution in critical spaces to the compressible Oldroyd-B model with non-small coupling parameter. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6437-6470. doi: 10.3934/dcds.2017279
[18]	Anthony Suen. Existence and a blow-up criterion of solution to the 3D compressible Navier-Stokes-Poisson equations with finite energy. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1775-1798. doi: 10.3934/dcds.2020093
[19]	Paolo Maremonti. A note on the Navier-Stokes IBVP with small data in $L^n$. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 255-267. doi: 10.3934/dcdss.2016.9.255
[20]	Zhilei Liang. Convergence rate of solutions to the contact discontinuity for the compressible Navier-Stokes equations. Communications on Pure and Applied Analysis, 2013, 12 (5) : 1907-1926. doi: 10.3934/cpaa.2013.12.1907

2021 Impact Factor: 1.169

Tools

Metrics

Other articles
by authors

on AIMS
Reinhard Racke Jürgen Saal
on Google Scholar
Reinhard Racke Jürgen Saal

[Back to Top]