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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 & 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. Google Scholar

[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. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

show all references

References:
[1]

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

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

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

[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.  Google Scholar

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