Hyperbolic Navier-Stokes equations I: Local well-posedness

Previous Article
On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities
EECT Home
This Issue
Next Article
Hyperbolic Navier-Stokes equations II: Global existence of small solutions

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

Hyperbolic Navier-Stokes equations I: Local well-posedness

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
Related Papers

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

References:

[1]	R. A. Adams, "Sobolev Spaces,'', Pure Appl. Math., 65 (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]	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. Google Scholar
[4]	M. Carrassi and A. Morro, A modified Navier-Stokes equation and its consequences on sound dispersion,, II Nuovo Cimento B, 9 (1972). Google Scholar
[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. Google Scholar
[6]	D. D. Joseph, "Fluid Dynamics of Viscoleastic Liquids,'', Appl. Math. Sciences, 84 (1990). Google Scholar
[7]	T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in, 448 (1975), 25. Google Scholar
[8]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, Arch. Ration. Mech. Anal., 58 (1975), 181. doi: 10.1007/BF00280740. Google Scholar
[9]	T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems,'', Lezioni Fermiane [Fermi Lectures], (1985). Google Scholar
[10]	A. Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,'', Appl. Math. Sci., 53 (1984). Google Scholar
[11]	M. Paicu and G. Raugel, Une perturbation hyperbolique des équations de Navier-Stokes,, in, 21 (2007), 65. Google Scholar
[12]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'', Applied Mathematical Sciences, 44 (1983). Google Scholar
[13]	R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems,'', Aspects of Mathematics, E19 (1992). Google Scholar
[14]	R. Racke, Thermoelasticity,, in, (2009), 315. Google Scholar
[15]	A. Schöwe, "Langzeitasymptotik der Hyperbolischen Navier-Stokes Gleichung im $\mathbb R^3$,'', Diploma thesis, (2011). Google Scholar
[16]	R. Temam, "The Navier-Stokes Equations. Theory and Numerical Analysis,'', Revised edition, 2 (1979). Google Scholar
[17]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'', North-Holland Mathematical Library, 18 (1978). Google Scholar

show all references

References:

[1]	R. A. Adams, "Sobolev Spaces,'', Pure Appl. Math., 65 (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]	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. Google Scholar
[4]	M. Carrassi and A. Morro, A modified Navier-Stokes equation and its consequences on sound dispersion,, II Nuovo Cimento B, 9 (1972). Google Scholar
[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. Google Scholar
[6]	D. D. Joseph, "Fluid Dynamics of Viscoleastic Liquids,'', Appl. Math. Sciences, 84 (1990). Google Scholar
[7]	T. Kato, Quasi-linear equations of evolution, with applications to partial differential equations,, in, 448 (1975), 25. Google Scholar
[8]	T. Kato, The Cauchy problem for quasi-linear symmetric hyperbolic systems,, Arch. Ration. Mech. Anal., 58 (1975), 181. doi: 10.1007/BF00280740. Google Scholar
[9]	T. Kato, "Abstract Differential Equations and Nonlinear Mixed Problems,'', Lezioni Fermiane [Fermi Lectures], (1985). Google Scholar
[10]	A. Majda, "Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables,'', Appl. Math. Sci., 53 (1984). Google Scholar
[11]	M. Paicu and G. Raugel, Une perturbation hyperbolique des équations de Navier-Stokes,, in, 21 (2007), 65. Google Scholar
[12]	A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,'', Applied Mathematical Sciences, 44 (1983). Google Scholar
[13]	R. Racke, "Lectures on Nonlinear Evolution Equations. Initial Value Problems,'', Aspects of Mathematics, E19 (1992). Google Scholar
[14]	R. Racke, Thermoelasticity,, in, (2009), 315. Google Scholar
[15]	A. Schöwe, "Langzeitasymptotik der Hyperbolischen Navier-Stokes Gleichung im $\mathbb R^3$,'', Diploma thesis, (2011). Google Scholar
[16]	R. Temam, "The Navier-Stokes Equations. Theory and Numerical Analysis,'', Revised edition, 2 (1979). Google Scholar
[17]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'', North-Holland Mathematical Library, 18 (1978). Google Scholar

[1]	Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3517-3541. doi: 10.3934/dcds.2013.33.3517
[2]	Matthias Hieber, Sylvie Monniaux. Well-posedness results for the Navier-Stokes equations in the rotational framework. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143
[3]	Maxim A. Olshanskii, Leo G. Rebholz, Abner J. Salgado. On well-posedness of a velocity-vorticity formulation of the stationary Navier-Stokes equations with no-slip boundary conditions. Discrete & Continuous Dynamical Systems - A, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148
[4]	Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101
[5]	Daniel Coutand, J. Peirce, Steve Shkoller. Global well-posedness of weak solutions for the Lagrangian averaged Navier-Stokes equations on bounded domains. Communications on Pure & Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35
[6]	Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete & Continuous Dynamical Systems - A, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845
[7]	Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315
[8]	Giovanna Bonfanti, Fabio Luterotti. A well-posedness result for irreversible phase transitions with a nonlinear heat flux law. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 331-351. doi: 10.3934/dcdss.2013.6.331
[9]	Chao Deng, Xiaohua Yao. Well-posedness and ill-posedness for the 3D generalized Navier-Stokes equations in $\dot{F}^{-\alpha,r}_{\frac{3}{\alpha-1}}$. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437
[10]	Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056
[11]	Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic & Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005
[12]	Tohru Nakamura, Shuichi Kawashima. Viscous shock profile and singular limit for hyperbolic systems with Cattaneo's law. Kinetic & Related Models, 2018, 11 (4) : 795-819. doi: 10.3934/krm.2018032
[13]	Kaitai Li, Yanren Hou. Fourier nonlinear Galerkin method for Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 1996, 2 (4) : 497-524. doi: 10.3934/dcds.1996.2.497
[14]	Alain Miranville. Asymptotic behavior of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1971-1987. doi: 10.3934/cpaa.2014.13.1971
[15]	Ahmad Makki, Alain Miranville, Georges Sadaka. On the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures and logarithmic potentials. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1341-1365. doi: 10.3934/dcdsb.2019019
[16]	Ben-Yu Guo, Yu-Jian Jiao. Mixed generalized Laguerre-Fourier spectral method for exterior problem of Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 315-345. doi: 10.3934/dcdsb.2009.11.315
[17]	Changjie Fang, Weimin Han. Well-posedness and optimal control of a hemivariational inequality for nonstationary Stokes fluid flow. Discrete & Continuous Dynamical Systems - A, 2016, 36 (10) : 5369-5386. doi: 10.3934/dcds.2016036
[18]	Pavel I. Plotnikov, Jan Sokolowski. Compressible Navier-Stokes equations. Conference Publications, 2009, 2009 (Special) : 602-611. doi: 10.3934/proc.2009.2009.602
[19]	Jan W. Cholewa, Tomasz Dlotko. Fractional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 2967-2988. doi: 10.3934/dcdsb.2017149
[20]	Vanessa Barros, Felipe Linares. A remark on the well-posedness of a degenerated Zakharov system. Communications on Pure & Applied Analysis, 2015, 14 (4) : 1259-1274. doi: 10.3934/cpaa.2015.14.1259

2018 Impact Factor: 1.048

American Institute of Mathematical Sciences