-
Previous Article
Hyperbolic Navier-Stokes equations II: Global existence of small solutions
- EECT Home
- This Issue
-
Next Article
On well-posedness of incompressible two-phase flows with phase transitions: The case of equal densities
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 |
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. |
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] |
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. |
[1] |
Daoyuan Fang, Ruizhao Zi. On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 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 and Continuous Dynamical Systems, 2013, 33 (11&12) : 5143-5151. doi: 10.3934/dcds.2013.33.5143 |
[3] |
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 |
[4] |
Quanrong Li, Shijin Ding. Global well-posedness of the Navier-Stokes equations with Navier-slip boundary conditions in a strip domain. Communications on Pure and Applied Analysis, 2021, 20 (10) : 3561-3581. doi: 10.3934/cpaa.2021121 |
[5] |
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 and Continuous Dynamical Systems, 2018, 38 (7) : 3459-3477. doi: 10.3934/dcds.2018148 |
[6] |
Xiaopeng Zhao, Yong Zhou. Well-posedness and decay of solutions to 3D generalized Navier-Stokes equations. Discrete and Continuous Dynamical Systems - B, 2021, 26 (2) : 795-813. doi: 10.3934/dcdsb.2020142 |
[7] |
Bin Han, Changhua Wei. Global well-posedness for inhomogeneous Navier-Stokes equations with logarithmical hyper-dissipation. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6921-6941. doi: 10.3934/dcds.2016101 |
[8] |
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 and Applied Analysis, 2002, 1 (1) : 35-50. doi: 10.3934/cpaa.2002.1.35 |
[9] |
Keyan Wang, Yao Xiao. Local well-posedness for Navier-Stokes equations with a class of ill-prepared initial data. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2987-3011. doi: 10.3934/dcds.2020158 |
[10] |
Weimin Peng, Yi Zhou. Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable. Discrete and Continuous Dynamical Systems, 2016, 36 (7) : 3845-3856. doi: 10.3934/dcds.2016.36.3845 |
[11] |
Yoshihiro Shibata. Local well-posedness of free surface problems for the Navier-Stokes equations in a general domain. Discrete and Continuous Dynamical Systems - S, 2016, 9 (1) : 315-342. doi: 10.3934/dcdss.2016.9.315 |
[12] |
Jingjing Zhang, Ting Zhang. Local well-posedness of perturbed Navier-Stokes system around Landau solutions. Electronic Research Archive, 2021, 29 (4) : 2719-2739. doi: 10.3934/era.2021010 |
[13] |
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 |
[14] |
Giovanna Bonfanti, Fabio Luterotti. A well-posedness result for irreversible phase transitions with a nonlinear heat flux law. Discrete and Continuous Dynamical Systems - S, 2013, 6 (2) : 331-351. doi: 10.3934/dcdss.2013.6.331 |
[15] |
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 and Continuous Dynamical Systems, 2014, 34 (2) : 437-459. doi: 10.3934/dcds.2014.34.437 |
[16] |
Gaocheng Yue, Chengkui Zhong. On the global well-posedness to the 3-D Navier-Stokes-Maxwell system. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5817-5835. doi: 10.3934/dcds.2016056 |
[17] |
Jishan Fan, Yueling Jia. Local well-posedness of the full compressible Navier-Stokes-Maxwell system with vacuum. Kinetic and Related Models, 2018, 11 (1) : 97-106. doi: 10.3934/krm.2018005 |
[18] |
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 |
[19] |
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 |
[20] |
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 |
2021 Impact Factor: 1.169
Tools
Metrics
Other articles
by authors
[Back to Top]