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

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

* 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.

Citation: Van-Sang Ngo, Stefano Scrobogna. Dispersive effects of weakly compressible and fast rotating inviscid fluids. Discrete & Continuous Dynamical Systems - A, 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. CheminB. DesjardinsI. 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. CheminB. DesjardinsI. 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. DesjardinsE. GrenierP.-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. FeireislI. GallagherD. 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. FeireislI. 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. KreissJ. 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. CheminB. DesjardinsI. 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. CheminB. DesjardinsI. 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. DesjardinsE. GrenierP.-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. FeireislI. GallagherD. 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. FeireislI. 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. KreissJ. 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.

[1]

Jin-Cheng Jiang, Chengbo Wang, Xin Yu. Generalized and weighted Strichartz estimates. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1723-1752. doi: 10.3934/cpaa.2012.11.1723

[2]

Robert Schippa. Generalized inhomogeneous Strichartz estimates. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3387-3410. doi: 10.3934/dcds.2017143

[3]

Younghun Hong, Changhun Yang. Uniform Strichartz estimates on the lattice. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3239-3264. doi: 10.3934/dcds.2019134

[4]

Gong Chen. Strichartz estimates for charge transfer models. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1201-1226. doi: 10.3934/dcds.2017050

[5]

Robert Schippa. Sharp Strichartz estimates in spherical coordinates. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2047-2051. doi: 10.3934/cpaa.2017100

[6]

Renato Manfrin. On the global solvability of symmetric hyperbolic systems of Kirchhoff type. Discrete & Continuous Dynamical Systems - A, 1997, 3 (1) : 91-106. doi: 10.3934/dcds.1997.3.91

[7]

Paolo Secchi. An alpha model for compressible fluids. Discrete & Continuous Dynamical Systems - S, 2010, 3 (2) : 351-359. doi: 10.3934/dcdss.2010.3.351

[8]

Roberto Guglielmi. Indirect stabilization of hyperbolic systems through resolvent estimates. Evolution Equations & Control Theory, 2017, 6 (1) : 59-75. doi: 10.3934/eect.2017004

[9]

Stefano Bianchini. Interaction estimates and Glimm functional for general hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2003, 9 (1) : 133-166. doi: 10.3934/dcds.2003.9.133

[10]

Chungen Liu. Minimal period estimates for brake orbits of nonlinear symmetric Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 337-355. doi: 10.3934/dcds.2010.27.337

[11]

Manil T. Mohan, Sivaguru S. Sritharan. New methods for local solvability of quasilinear symmetric hyperbolic systems. Evolution Equations & Control Theory, 2016, 5 (2) : 273-302. doi: 10.3934/eect.2016005

[12]

Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271

[13]

Michael Ruzhansky, Jens Wirth. Dispersive type estimates for fourier integrals and applications to hyperbolic systems. Conference Publications, 2011, 2011 (Special) : 1263-1270. doi: 10.3934/proc.2011.2011.1263

[14]

Chu-Hee Cho, Youngwoo Koh, Ihyeok Seo. On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 1905-1926. doi: 10.3934/dcds.2016.36.1905

[15]

Vladimir Georgiev, Atanas Stefanov, Mirko Tarulli. Smoothing-Strichartz estimates for the Schrodinger equation with small magnetic potential. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 771-786. doi: 10.3934/dcds.2007.17.771

[16]

Youngwoo Koh, Ihyeok Seo. Strichartz estimates for Schrödinger equations in weighted $L^2$ spaces and their applications. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4877-4906. doi: 10.3934/dcds.2017210

[17]

Younghun Hong. Strichartz estimates for $N$-body Schrödinger operators with small potential interactions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5355-5365. doi: 10.3934/dcds.2017233

[18]

Michael Goldberg. Strichartz estimates for Schrödinger operators with a non-smooth magnetic potential. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 109-118. doi: 10.3934/dcds.2011.31.109

[19]

Eugenio Aulisa, Lidia Bloshanskaya, Akif Ibragimov. Well productivity index for compressible fluids and gases. Evolution Equations & Control Theory, 2016, 5 (1) : 1-36. doi: 10.3934/eect.2016.5.1

[20]

Tohru Nakamura, Shinya Nishibata, Naoto Usami. Convergence rate of solutions towards the stationary solutions to symmetric hyperbolic-parabolic systems in half space. Kinetic & Related Models, 2018, 11 (4) : 757-793. doi: 10.3934/krm.2018031

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (34)
  • HTML views (55)
  • Cited by (0)

Other articles
by authors

[Back to Top]