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Hyperbolic Navier-Stokes equations I: Local well-posedness
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 |
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). Google Scholar |
[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,'', 2nd 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. Google Scholar |
[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, (). Google Scholar |
[28] |
A. Schöwe, "Langzeitasymptotik der Hyperbolischen Navier-Stokes Gleichung im $\mathbb R^3$,'', Diploma thesis, (2011). Google Scholar |
[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).
|
show all references
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). Google Scholar |
[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,'', 2nd 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. Google Scholar |
[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, (). Google Scholar |
[28] |
A. Schöwe, "Langzeitasymptotik der Hyperbolischen Navier-Stokes Gleichung im $\mathbb R^3$,'', Diploma thesis, (2011). Google Scholar |
[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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