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January  2017, 16(1): 345-368. doi: 10.3934/cpaa.2017017

Global well posedness for the ghost effect system

Bilal Al Taki 1,2,,

1. 

LAMA, UMR 5127 CNRS, Université de Savoie Mont, Blanc 73376 Le Bourget du lac cedex, France
2. 

Laboratory of Mathematics-EDST, Lebanese University, Beirut, Lebanon

E-mail address: bilal.al-taki@univ-smb.fr

Received  August 2016 Revised  September 2016 Published  November 2016

The aim of this paper is to discuss the issue of global existence of weak solutions of the so called ghost effect system which has been derived recently in [C. D. LEVERMORE, W. SUN, K. TRIVISA, SIAM J. Math. Anal. 2012]. We extend the local existence of solutions proved in [C.D. LEVERMORE, W. SUN, K. TRIVISA, Indiana Univ. J., 2011] to a global existence result. The key tool in this paper is a new functional inequality inspired of what proposed in [A. JÜNGEL, D. MATTHES, SIAM J. Math. Anal., 2008]. Such an inequality being adapted in [D. BRESCH, A. VASSEUR, C. YU, 2016] to be useful for compressible Navier-Stokes equations with degenerate viscosities. Our strategy to prove the global existence of solution builds upon the framework developed in [D. BRESCH, V. GIOVANGILI, E. ZATORSKA, J. Math. Pures Appl., 2015] for low Mach number system.

Keywords: Ghost effect system, Navier-Stokes equations, low Mach number, Bohm potential.
Mathematics Subject Classification: 35G25, 35Q30, 35Q40, 35D30, 82D05, 82D37.
Citation: Bilal Al Taki. Global well posedness for the ghost effect system. Communications on Pure and Applied Analysis, 2017, 16 (1) : 345-368. doi: 10.3934/cpaa.2017017
References:
[1]

P. Antonelli and S. Spirito, A global existence result for a zero mach number system, arXiv: 1605.03510, 2016.

[2]

D. Bresch, F. Couderc, P. Noble and J. -P. Vila, New extended formulations of euler-korteweg equations based on a generalization of the quantum bohm identity, arXiv: 1503.08678, 2015.

[3]

D. Bresch, B. Desjardins and E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: Part ii existence of global κ-entropy solutions to the compressible navier-stokes systems with degenerate viscosities, J. Math. Pures Appl., 4 (2015), 801-836. doi: 10.1016/j.matpur.2015.05.004.

[4]

D. Bresch, E. H. Essoufi and M. Sy, Effect of density dependent viscosities on multiphasic incompressible fluid models, Journal of Math. Fluid Mech., 3 (2007), 377-397. doi: 10.1007/s00021-005-0204-4.

[5]

D. Bresch, V. Giovangigli and E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: Part i well posedness for zero mach number systems, Journal de Math. Pures. Appl., 4 (2015), 762-800. doi: 10.1016/j.matpur.2015.05.003.

[6]

D. Bresch, A. Vasseur and C. Yu, Global existence of compressible Navier-Stokes equation with degenerates viscosities, In preparation, (2016).

[7]

R. Danchin and X. Liao, On the well-posedness of the full low Mach number limit system in general critical Besov spaces, Commun. Contemp. Math., 3 (2012), 125-0022. doi: 10.1142/S0219199712500228.

[8]

J. Dong, A note on barotropic compressible quantum Navier-Stokes equations, Nonlinear Anal., 4 (2010), 854-856. doi: 10.1016/j.na.2010.03.047.

[9]

M. Gisclon and I. Lacroix-Violet, About the barotropic compressible quantum Navier-Stokes equations, Nonlinear Anal. , (2015), 106-121. doi: 10.1016/j.na.2015.07.006.

[10]

F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlinear Anal. Real World Appl., 3 (2011), 1733-1735. doi: 10.1016/j.nonrwa.2010.11.005.

[11]

A. Jüngel, Global weak solutions to compressible navier-stokes equations for quantum fluids, SIAM J. Math. Anal., 3 (2010), 1025-1045. doi: 10.1137/090776068.

[12]

A. Jüngel and D. Matthes, The derrida-lebowitz-speer-spohn equation: existence, nonuniqueness, and decay rates of the solutions, SIAM J. Math. Anal., 6 (2008), 1996-2015. doi: 10.1137/060676878.

[13]

C. D. Levermore, W. Sun and K. Trivisa, A low mach number limit of a dispersive navierstokes system, SIAM J. Math. Anal., 3 (2012), 1760-1807. doi: 10.1137/100818765.

[14]

C. D. Levermore, W. R. Sun and K. Trivisa, Local well-posedness of a ghost effect system, Indiana Univ. Math. J., 2 (2009), 517-576.

[15]

X. Liao, A global existence result for a zero Mach number system, J. Math. Fluid Mech., 1 (2014), 77-103. doi: 10.1007/s00021-013-0152-3.

[16]

P. -L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1996.

[17]

P. -L Lions, Mathematical Topics in Fluid Mechanics. Vol. 2 Compressible Models, Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998.

[18]

J. -C. Maxwell, On stresses in rarified gases arising from inequalities of temperature, Philos Trans R Soc Lond B Biol Sci, (1879), 231-256.

[19]

J. Simon, On the existence of the pressure for solutions of the variational Navier-Stokes equations, J. Math. Fluid Mech., 3 (1999), 225-234. doi: 10.1007/s000210050010.

[20]

Y. Sone, Flows induced by temperature fields in a rarefied gas and their ghost effect on the behavior of a gas in the continuum limit, Annu. Rev. Fluid Mech., 32 (2000), 779-811. doi: 10.1146/annurev.fluid.32.1.779.

[21]

Y. Sone, Kinetic Theory and Fluid Dynamics, Birkhäuser Boston, Inc. , Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.

[22]

C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. Ⅰ (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.

show all references

References:
[1]

P. Antonelli and S. Spirito, A global existence result for a zero mach number system, arXiv: 1605.03510, 2016.

[2]

D. Bresch, F. Couderc, P. Noble and J. -P. Vila, New extended formulations of euler-korteweg equations based on a generalization of the quantum bohm identity, arXiv: 1503.08678, 2015.

[3]

D. Bresch, B. Desjardins and E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: Part ii existence of global κ-entropy solutions to the compressible navier-stokes systems with degenerate viscosities, J. Math. Pures Appl., 4 (2015), 801-836. doi: 10.1016/j.matpur.2015.05.004.

[4]

D. Bresch, E. H. Essoufi and M. Sy, Effect of density dependent viscosities on multiphasic incompressible fluid models, Journal of Math. Fluid Mech., 3 (2007), 377-397. doi: 10.1007/s00021-005-0204-4.

[5]

D. Bresch, V. Giovangigli and E. Zatorska, Two-velocity hydrodynamics in fluid mechanics: Part i well posedness for zero mach number systems, Journal de Math. Pures. Appl., 4 (2015), 762-800. doi: 10.1016/j.matpur.2015.05.003.

[6]

D. Bresch, A. Vasseur and C. Yu, Global existence of compressible Navier-Stokes equation with degenerates viscosities, In preparation, (2016).

[7]

R. Danchin and X. Liao, On the well-posedness of the full low Mach number limit system in general critical Besov spaces, Commun. Contemp. Math., 3 (2012), 125-0022. doi: 10.1142/S0219199712500228.

[8]

J. Dong, A note on barotropic compressible quantum Navier-Stokes equations, Nonlinear Anal., 4 (2010), 854-856. doi: 10.1016/j.na.2010.03.047.

[9]

M. Gisclon and I. Lacroix-Violet, About the barotropic compressible quantum Navier-Stokes equations, Nonlinear Anal. , (2015), 106-121. doi: 10.1016/j.na.2015.07.006.

[10]

F. Jiang, A remark on weak solutions to the barotropic compressible quantum Navier-Stokes equations, Nonlinear Anal. Real World Appl., 3 (2011), 1733-1735. doi: 10.1016/j.nonrwa.2010.11.005.

[11]

A. Jüngel, Global weak solutions to compressible navier-stokes equations for quantum fluids, SIAM J. Math. Anal., 3 (2010), 1025-1045. doi: 10.1137/090776068.

[12]

A. Jüngel and D. Matthes, The derrida-lebowitz-speer-spohn equation: existence, nonuniqueness, and decay rates of the solutions, SIAM J. Math. Anal., 6 (2008), 1996-2015. doi: 10.1137/060676878.

[13]

C. D. Levermore, W. Sun and K. Trivisa, A low mach number limit of a dispersive navierstokes system, SIAM J. Math. Anal., 3 (2012), 1760-1807. doi: 10.1137/100818765.

[14]

C. D. Levermore, W. R. Sun and K. Trivisa, Local well-posedness of a ghost effect system, Indiana Univ. Math. J., 2 (2009), 517-576.

[15]

X. Liao, A global existence result for a zero Mach number system, J. Math. Fluid Mech., 1 (2014), 77-103. doi: 10.1007/s00021-013-0152-3.

[16]

P. -L. Lions, Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1996.

[17]

P. -L Lions, Mathematical Topics in Fluid Mechanics. Vol. 2 Compressible Models, Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York, 1998.

[18]

J. -C. Maxwell, On stresses in rarified gases arising from inequalities of temperature, Philos Trans R Soc Lond B Biol Sci, (1879), 231-256.

[19]

J. Simon, On the existence of the pressure for solutions of the variational Navier-Stokes equations, J. Math. Fluid Mech., 3 (1999), 225-234. doi: 10.1007/s000210050010.

[20]

Y. Sone, Flows induced by temperature fields in a rarefied gas and their ghost effect on the behavior of a gas in the continuum limit, Annu. Rev. Fluid Mech., 32 (2000), 779-811. doi: 10.1146/annurev.fluid.32.1.779.

[21]

Y. Sone, Kinetic Theory and Fluid Dynamics, Birkhäuser Boston, Inc. , Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.

[22]

C. Villani, A review of mathematical topics in collisional kinetic theory, Handbook of mathematical fluid dynamics, Vol. Ⅰ (2002), 71-305. doi: 10.1016/S1874-5792(02)80004-0.

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