Dispersive effects of weakly compressible and fast rotating inviscid fluids

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February 2018, 38(2): 749-789. doi: 10.3934/dcds.2018033

Dispersive effects of weakly compressible and fast rotating inviscid fluids

Van-Sang Ngo ^1,, and Stefano Scrobogna ^2,3,

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

* Corresponding author: Van-Sang Ngo

Received November 2016 Revised August 2017 Published February 2018

Fund Project: The research of the second author was partially supported by the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa accreditation SEV-2013-0323.

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.

Keywords: Compressible fluids, Strichartz estimates, symmetric hyperbolic systems.

Mathematics Subject Classification: 35A01, 35A02, 35Q31, 76N10, 76U05.

Citation: Van-Sang Ngo, Stefano Scrobogna. Dispersive effects of weakly compressible and fast rotating inviscid fluids. Discrete and Continuous Dynamical Systems, 2018, 38 (2) : 749-789. doi: 10.3934/dcds.2018033

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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