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On uniqueness of measure-valued solutions to Liouville's equation of Hamiltonian PDEs
Dispersive effects of weakly compressible and fast rotating inviscid fluids
1. | Université de Rouen, Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, 76801 Saint-Etienne du Rouvray, France |
2. | IMB, Université de Bordeaux, 351, cours de la Libération, 33405 Talence, France |
3. | Basque Center for Applied Mathematics, Mazarredo, 14, E48009 Bilbao, Basque Country, Spain |
We consider a system describing the motion of an isentropic, inviscid, weakly compressible, fast rotating fluid in the whole space $\mathbb{R}^3$, with initial data belonging to $ H^s \left( \mathbb{R}^3 \right), s>5/2 $. We prove that the system admits a unique local strong solution in $ L^\infty \left( [0,T]; H^s\left( \mathbb{R}^3 \right) \right) $, where $ T $ is independent of the Rossby and Mach numbers. Moreover, using Strichartz-type estimates, we prove the longtime existence of the solution, i.e. its lifespan is of the order of $\varepsilon^{-\alpha}, \alpha >0$, without any smallness assumption on the initial data (the initial data can even go to infinity in some sense), provided that the rotation is fast enough.
References:
[1] |
H. Bahouri, J. -Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
G. K. Batchelor,
An Introduction to Fluid Dynamics Cambridge University Press, Cambridge, 1999. |
[3] |
J.-M. Bony,
Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Annales de l'École Normale Supérieure, 14 (1981), 209-246.
doi: 10.24033/asens.1404. |
[4] |
M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de NavierStokes, in Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, Exp. No. Ⅷ,
École Polytechnique, Palaiseau, (1994), 12pp.
doi: 10.1108/09533239410052824. |
[5] |
J. -Y. Chemin, Fluides parfaits incompressibles,
Astérisque 230 (1995), 177pp. |
[6] |
J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier,
Fluids with anisotropic viscosity, Special issue for R. Temam's 60th birthday, M2AN. Mathematical Modelling and Numerical Analysis, 34 (2000), 315-335.
doi: 10.1051/m2an:2000143. |
[7] |
J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier,
Anisotropy and dispersion in rotating fluids, Nonlinear Partial Differential Equations and their application, Collége de France Seminar, Studies in Mathematics and its Applications, 31 (2002), 171-191.
doi: 10.1016/S0168-2024(02)80010-8. |
[8] |
J. -Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier,
Mathematical Geophysics: An Introduction to Rotating Fluids and to the Navier-Stokes Equations Oxford University Press, 2006. |
[9] |
J.-Y. Chemin and N. Lerner,
Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, Journal of Differential Equations, 121 (1992), 314-328.
doi: 10.1006/jdeq.1995.1131. |
[10] |
R. Danchin,
Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.
doi: 10.1007/s002220000078. |
[11] |
R. Danchin,
Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.
doi: 10.1081/PDE-100106132. |
[12] |
R. Danchin,
Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup., 35 (2002), 27-75.
doi: 10.1016/S0012-9593(01)01085-0. |
[13] |
B. Desjardins and E. Grenier,
Low Mach number limit of viscous compressible flows in the whole space, Proceedings: Mathematical, Physical and Engineering Sciences, 455 (1999), 2271-2279.
doi: 10.1098/rspa.1999.0403. |
[14] |
B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi,
Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl., 78 (1999), 461-471.
doi: 10.1016/S0021-7824(99)00032-X. |
[15] |
A. Dutrifoy,
Examples of dispersive effects in non-viscous rotating fluids, Journal de Mathématiques Pures et Appliquées, 84 (2005), 331-356.
doi: 10.1016/j.matpur.2004.09.007. |
[16] |
A. Dutrifoy and T. Hmidi,
The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data, Comm. Pure Appl. Math., 57 (2004), 1159-1177.
doi: 10.1002/cpa.20026. |
[17] |
F. Fanelli,
Highly rotating viscous compressible fluids in presence of capillarity effects, Mathematische Annalen, 366 (2016), 981-1033.
doi: 10.1007/s00208-015-1358-x. |
[18] |
F. Fanelli,
A singular limit problem for rotating capillary fluids with variable rotation axis, Journal of Mathematical Fluid Mechanics, 18 (2016), 625-658.
doi: 10.1007/s00021-016-0256-7. |
[19] |
E. Feireisl, I. Gallagher, D. Gerard-Varet and A. Novotný,
Multi-scale analysis of compressible viscous and rotating fluids, Comm. Math. Phys., 314 (2012), 641-670.
doi: 10.1007/s00220-012-1533-9. |
[20] |
E. Feireisl, I. Gallagher and A. Novotný,
A singular limit for compressible rotating fluids, SIAM J. Math. Anal., 44 (2012), 192-205.
doi: 10.1137/100808010. |
[21] |
E. Feireisl and H. Petzeltová,
Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96.
doi: 10.1007/s002050050181. |
[22] |
E. Feireisl and H. Petzeltová,
On compactness of solutions to the Navier-Stokes equations of compressible flow, J. Differential Equations, 163 (2000), 57-75.
doi: 10.1006/jdeq.1999.3720. |
[23] |
H. Fujita and T. Kato,
On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[24] |
I. Gallagher and L. Saint-Raymond,
Weak convergence results for inhomogeneous rotating fluid equations, Journal d'Analyse Mathématique, 99 (2006), 1-34.
doi: 10.1007/BF02789441. |
[25] |
D. Hoff,
The zero-Mach limit of compressible flows, Comm. Math. Phys., 192 (1998), 543-554.
doi: 10.1007/s002200050308. |
[26] |
N. Itaya,
The existence and uniqueness of the solution of the equations describing compressible viscous fluid flow, Proc. Japan Acad., 46 (1970), 379-382.
doi: 10.3792/pja/1195520358. |
[27] |
S. Klainerman and A. Majda,
Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.
doi: 10.1002/cpa.3160340405. |
[28] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[29] |
H.-O. Kreiss, J. Lorenz and M. J. Naughton,
Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations, Adv. in Appl. Math., 12 (1991), 187-214.
doi: 10.1016/0196-8858(91)90012-8. |
[30] |
L. D. Landau and E. M. Lifschitz,
Lehrbuch Der Theoretischen Physik Band Ⅵ fifth ed., Akademie-Verlag, Berlin, 1991, Hydrodynamik. |
[31] |
C. -K. Lin,
On the Incompressible Limit of the Compressible Navier-Stokes Equations Ph. D. Thesis of The University of Arizona, 1992. |
[32] |
P. -L. Lions,
Mathematical Topics in Fluid Mechanics. Vol. 1 Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. |
[33] |
P. -L. Lions,
Mathematical Topics in Fluid Mechanics. Vol. 2 Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998. |
[34] |
P.-L. Lions and N. Masmoudi,
Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.
doi: 10.1016/S0021-7824(98)80139-6. |
[35] |
G. Métivier and S. Schochet,
The incompressible limit of the non-isentropic Euler equations, Archive for Rational Mechanics and Analysis, 158 (2001), 61-90.
doi: 10.1007/PL00004241. |
[36] |
J. Nash,
Le probléme de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France, 90 (1962), 487-497.
|
[37] |
V.-S. Ngo,
Rotating Fluids with small viscosity, International Mathematics Research Notices
IMRN, (2009), 1860-1890.
doi: 10.1093/imrn/rnp004. |
[38] |
J. Pedlosky,
Geophysical Fluid Dynamics Springer-Verlag, 1987. |
[39] |
M. Reed and B. Simon,
Methods of Modern Mathematical Physics. Ⅲ Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. |
show all references
References:
[1] |
H. Bahouri, J. -Y. Chemin and R. Danchin,
Fourier Analysis and Nonlinear Partial Differential Equations Grundlehren der Mathematischen Wissenschaften, vol. 343, Springer, Heidelberg, 2011.
doi: 10.1007/978-3-642-16830-7. |
[2] |
G. K. Batchelor,
An Introduction to Fluid Dynamics Cambridge University Press, Cambridge, 1999. |
[3] |
J.-M. Bony,
Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Annales de l'École Normale Supérieure, 14 (1981), 209-246.
doi: 10.24033/asens.1404. |
[4] |
M. Cannone, Y. Meyer and F. Planchon, Solutions auto-similaires des équations de NavierStokes, in Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, Exp. No. Ⅷ,
École Polytechnique, Palaiseau, (1994), 12pp.
doi: 10.1108/09533239410052824. |
[5] |
J. -Y. Chemin, Fluides parfaits incompressibles,
Astérisque 230 (1995), 177pp. |
[6] |
J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier,
Fluids with anisotropic viscosity, Special issue for R. Temam's 60th birthday, M2AN. Mathematical Modelling and Numerical Analysis, 34 (2000), 315-335.
doi: 10.1051/m2an:2000143. |
[7] |
J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier,
Anisotropy and dispersion in rotating fluids, Nonlinear Partial Differential Equations and their application, Collége de France Seminar, Studies in Mathematics and its Applications, 31 (2002), 171-191.
doi: 10.1016/S0168-2024(02)80010-8. |
[8] |
J. -Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier,
Mathematical Geophysics: An Introduction to Rotating Fluids and to the Navier-Stokes Equations Oxford University Press, 2006. |
[9] |
J.-Y. Chemin and N. Lerner,
Flot de champs de vecteurs non Lipschitziens et équations de Navier-Stokes, Journal of Differential Equations, 121 (1992), 314-328.
doi: 10.1006/jdeq.1995.1131. |
[10] |
R. Danchin,
Global existence in critical spaces for compressible Navier-Stokes equations, Invent. Math., 141 (2000), 579-614.
doi: 10.1007/s002220000078. |
[11] |
R. Danchin,
Local theory in critical spaces for compressible viscous and heat-conductive gases, Comm. Partial Differential Equations, 26 (2001), 1183-1233.
doi: 10.1081/PDE-100106132. |
[12] |
R. Danchin,
Zero Mach number limit in critical spaces for compressible Navier-Stokes equations, Ann. Sci. École Norm. Sup., 35 (2002), 27-75.
doi: 10.1016/S0012-9593(01)01085-0. |
[13] |
B. Desjardins and E. Grenier,
Low Mach number limit of viscous compressible flows in the whole space, Proceedings: Mathematical, Physical and Engineering Sciences, 455 (1999), 2271-2279.
doi: 10.1098/rspa.1999.0403. |
[14] |
B. Desjardins, E. Grenier, P.-L. Lions and N. Masmoudi,
Incompressible limit for solutions of the isentropic Navier-Stokes equations with Dirichlet boundary conditions, J. Math. Pures Appl., 78 (1999), 461-471.
doi: 10.1016/S0021-7824(99)00032-X. |
[15] |
A. Dutrifoy,
Examples of dispersive effects in non-viscous rotating fluids, Journal de Mathématiques Pures et Appliquées, 84 (2005), 331-356.
doi: 10.1016/j.matpur.2004.09.007. |
[16] |
A. Dutrifoy and T. Hmidi,
The incompressible limit of solutions of the two-dimensional compressible Euler system with degenerating initial data, Comm. Pure Appl. Math., 57 (2004), 1159-1177.
doi: 10.1002/cpa.20026. |
[17] |
F. Fanelli,
Highly rotating viscous compressible fluids in presence of capillarity effects, Mathematische Annalen, 366 (2016), 981-1033.
doi: 10.1007/s00208-015-1358-x. |
[18] |
F. Fanelli,
A singular limit problem for rotating capillary fluids with variable rotation axis, Journal of Mathematical Fluid Mechanics, 18 (2016), 625-658.
doi: 10.1007/s00021-016-0256-7. |
[19] |
E. Feireisl, I. Gallagher, D. Gerard-Varet and A. Novotný,
Multi-scale analysis of compressible viscous and rotating fluids, Comm. Math. Phys., 314 (2012), 641-670.
doi: 10.1007/s00220-012-1533-9. |
[20] |
E. Feireisl, I. Gallagher and A. Novotný,
A singular limit for compressible rotating fluids, SIAM J. Math. Anal., 44 (2012), 192-205.
doi: 10.1137/100808010. |
[21] |
E. Feireisl and H. Petzeltová,
Large-time behaviour of solutions to the Navier-Stokes equations of compressible flow, Arch. Ration. Mech. Anal., 150 (1999), 77-96.
doi: 10.1007/s002050050181. |
[22] |
E. Feireisl and H. Petzeltová,
On compactness of solutions to the Navier-Stokes equations of compressible flow, J. Differential Equations, 163 (2000), 57-75.
doi: 10.1006/jdeq.1999.3720. |
[23] |
H. Fujita and T. Kato,
On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., 16 (1964), 269-315.
doi: 10.1007/BF00276188. |
[24] |
I. Gallagher and L. Saint-Raymond,
Weak convergence results for inhomogeneous rotating fluid equations, Journal d'Analyse Mathématique, 99 (2006), 1-34.
doi: 10.1007/BF02789441. |
[25] |
D. Hoff,
The zero-Mach limit of compressible flows, Comm. Math. Phys., 192 (1998), 543-554.
doi: 10.1007/s002200050308. |
[26] |
N. Itaya,
The existence and uniqueness of the solution of the equations describing compressible viscous fluid flow, Proc. Japan Acad., 46 (1970), 379-382.
doi: 10.3792/pja/1195520358. |
[27] |
S. Klainerman and A. Majda,
Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids, Comm. Pure Appl. Math., 34 (1981), 481-524.
doi: 10.1002/cpa.3160340405. |
[28] |
H. Koch and D. Tataru,
Well-posedness for the Navier-Stokes equations, Adv. Math., 157 (2001), 22-35.
doi: 10.1006/aima.2000.1937. |
[29] |
H.-O. Kreiss, J. Lorenz and M. J. Naughton,
Convergence of the solutions of the compressible to the solutions of the incompressible Navier-Stokes equations, Adv. in Appl. Math., 12 (1991), 187-214.
doi: 10.1016/0196-8858(91)90012-8. |
[30] |
L. D. Landau and E. M. Lifschitz,
Lehrbuch Der Theoretischen Physik Band Ⅵ fifth ed., Akademie-Verlag, Berlin, 1991, Hydrodynamik. |
[31] |
C. -K. Lin,
On the Incompressible Limit of the Compressible Navier-Stokes Equations Ph. D. Thesis of The University of Arizona, 1992. |
[32] |
P. -L. Lions,
Mathematical Topics in Fluid Mechanics. Vol. 1 Oxford Lecture Series in Mathematics and its Applications, vol. 3, The Clarendon Press, Oxford University Press, New York, 1996. |
[33] |
P. -L. Lions,
Mathematical Topics in Fluid Mechanics. Vol. 2 Oxford Lecture Series in Mathematics and its Applications, vol. 10, The Clarendon Press, Oxford University Press, New York, 1998. |
[34] |
P.-L. Lions and N. Masmoudi,
Incompressible limit for a viscous compressible fluid, J. Math. Pures Appl., 77 (1998), 585-627.
doi: 10.1016/S0021-7824(98)80139-6. |
[35] |
G. Métivier and S. Schochet,
The incompressible limit of the non-isentropic Euler equations, Archive for Rational Mechanics and Analysis, 158 (2001), 61-90.
doi: 10.1007/PL00004241. |
[36] |
J. Nash,
Le probléme de Cauchy pour les équations différentielles d'un fluide général, Bull. Soc. Math. France, 90 (1962), 487-497.
|
[37] |
V.-S. Ngo,
Rotating Fluids with small viscosity, International Mathematics Research Notices
IMRN, (2009), 1860-1890.
doi: 10.1093/imrn/rnp004. |
[38] |
J. Pedlosky,
Geophysical Fluid Dynamics Springer-Verlag, 1987. |
[39] |
M. Reed and B. Simon,
Methods of Modern Mathematical Physics. Ⅲ Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. |
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