October  2016, 21(8): 2745-2766. doi: 10.3934/dcdsb.2016071

On the compressible Navier-Stokes-Korteweg equations

1. 

Department of Mathematics, College of Sciences, Hohai University, Nanjing 210098

2. 

Jiangsu Provincial Key Laboratory for Numerical Simulation of Large Scale Complex Systems, School of Mathematical Science, Nanjing Normal University, Nanjing 210023

Received  October 2015 Revised  July 2016 Published  September 2016

In this paper, we consider compressible Navier-Stokes-Korteweg (N-S-K) equations with more general pressure laws, that is the pressure $P$ is non-monotone. We prove the stability of weak solutions in the periodic domain $\Omega=\mathbb{T}^{N}$, when $N = 2,3$. Utilizing an interesting Sobolev inequality to tackle the complicated Korteweg term, we obtain the global existence of weak solutions in one dimensional case. Moreover, when the initial data is compactly supported in the whole space $\mathbb{R}$, we prove the compressible N-S-K equations will blow-up in finite time.
Citation: Tong Tang, Hongjun Gao. On the compressible Navier-Stokes-Korteweg equations. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2745-2766. doi: 10.3934/dcdsb.2016071
References:
[1]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D Viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223. doi: 10.1007/s00220-003-0859-8.

[2]

D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl., 86 (2006), 362-368. doi: 10.1016/j.matpur.2006.06.005.

[3]

D. Bresch, B. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Commun. Partial Diff. Eqns., 28 (2003), 843-868. doi: 10.1081/PDE-120020499.

[4]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129. doi: 10.1007/s00229-006-0637-y.

[5]

R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 97-133. doi: 10.1016/S0294-1449(00)00056-1.

[6]

B. Ducomet, E. Feireisl, A. Petzeltová and I. Straškraba, Global in time weak solutions for compressible barotropic self-gravitating fluids, Discrete Contin. Dyn. Syst., 11 (2004), 113-130. doi: 10.3934/dcds.2004.11.113.

[7]

B. Ducomet, S. Nečasová and A. Vasseur, On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities, Z. Angew. Math. Phys., 61 (2010), 479-491. doi: 10.1007/s00033-009-0035-x.

[8]

B. Ducomet, S. Nečasová and A. Vasseur, On spherically symmetric motions of a viscous compressible barotropic and selfgravitating gas, J. Math. Fluid Mech., 13 (2011), 191-211. doi: 10.1007/s00021-009-0010-5.

[9]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907.

[10]

I. Gamba, A. Jüngel and A. Vasseur, Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations, J. Differential Equations, 247 (2009), 3117-3135. doi: 10.1016/j.jde.2009.09.001.

[11]

P. Germain and P. G. Lefloch, The finite energy method for compressible fluids-The Naiver-Stokes-Korteweg model, Comm. Pure Appl. Math., 69 (2016), 3-61. doi: 10.1002/cpa.21622.

[12]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Commun. Partial Diff. Eqns., 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.

[13]

B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249. doi: 10.1007/s00021-009-0013-2.

[14]

H. Hattori and D. Li, Global solutions of a high-dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069.

[15]

H. Hattori and D. Li, The existence of global solutions to a fluid dynamic model for materials for Korteweg type, J. Partial Diff. Eqns, 9 (1996), 323-342.

[16]

L. Hsiao and H. L. Li, Dissipation and dispersion approximation to hydrodynamical equations and asymptotic limit, J. Partial Diff. Eqns, 21 (2008), 59-76.

[17]

S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal., 12 (2005), 239-251. doi: 10.4310/MAA.2005.v12.n3.a2.

[18]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068.

[19]

M. Kotschote, Strong solutions for a compressible fluid of Korteweg type, Ann. Inst. H. Poincaré Anal. Non Lineaire, 25 (2008), 679-696. doi: 10.1016/j.anihpc.2007.03.005.

[20]

H. L. Li, J. Li and Z. P. Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 401-444. doi: 10.1007/s00220-008-0495-4.

[21]

P. L. Lions, Mathematical Topics in Fluid Dynamics 2, Compressible Models, Oxford Science Publication, Oxford, 1998.

[22]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.

[23]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Diff. Eqns, 32 (2007), 431-452. doi: 10.1080/03605300600857079.

[24]

Z. Tan, X. Zhang and H. Q. Wang, Asymptotic behavior of Navier-Stokes-Korteweg with friction in $R^3$, Discrete Contin. Dyn. Syst., 34 (2014), 2243-2259. doi: 10.3934/dcds.2014.34.2243.

[25]

T. Tang and J. Kuang, Blow-up of compressible Naiver-Stokes-Korteweg equations, Acta Applicanda Mathematicae, 130 (2014), 1-7. doi: 10.1007/s10440-013-9836-1.

[26]

T. Tang, Blow-up of smooth solutions to the compressible barotropic Navier-Stokes-Korteweg equations on bounded domains, Acta Applicanda Mathematicae, 136 (2015), 55-61. doi: 10.1007/s10440-014-9884-1.

[27]

Y. J. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271. doi: 10.1016/j.jmaa.2011.01.006.

[28]

A. Valli and W. M. Zajączkowski, Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys, 103 (1986), 259-296. doi: 10.1007/BF01206939.

[29]

Z. P. Xin, Blow-up of smooth solution to the compressible Naiver-Stokes equations with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[30]

T. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6.

[31]

Y. H. Zhang and Z. Tan, Blow-up of smooth solutions to the compressible fluid models of Korteweg type, Acta Math. Sin. (Engl. Ser.), 28 (2012), 645-652. doi: 10.1007/s10114-012-9042-5.

show all references

References:
[1]

D. Bresch and B. Desjardins, Existence of global weak solutions for a 2D Viscous shallow water equations and convergence to the quasi-geostrophic model, Comm. Math. Phys., 238 (2003), 211-223. doi: 10.1007/s00220-003-0859-8.

[2]

D. Bresch and B. Desjardins, On the construction of approximate solutions for the 2D viscous shallow water model and for compressible Navier-Stokes models, J. Math. Pures Appl., 86 (2006), 362-368. doi: 10.1016/j.matpur.2006.06.005.

[3]

D. Bresch, B. Desjardins and C. K. Lin, On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Commun. Partial Diff. Eqns., 28 (2003), 843-868. doi: 10.1081/PDE-120020499.

[4]

Y. Cho and H. Kim, On classical solutions of the compressible Navier-Stokes equations with nonnegative initial densities, Manuscripta Math., 120 (2006), 91-129. doi: 10.1007/s00229-006-0637-y.

[5]

R. Danchin and B. Desjardins, Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. H. Poincaré Anal. Non Linéaire, 18 (2001), 97-133. doi: 10.1016/S0294-1449(00)00056-1.

[6]

B. Ducomet, E. Feireisl, A. Petzeltová and I. Straškraba, Global in time weak solutions for compressible barotropic self-gravitating fluids, Discrete Contin. Dyn. Syst., 11 (2004), 113-130. doi: 10.3934/dcds.2004.11.113.

[7]

B. Ducomet, S. Nečasová and A. Vasseur, On global motions of a compressible barotropic and selfgravitating gas with density-dependent viscosities, Z. Angew. Math. Phys., 61 (2010), 479-491. doi: 10.1007/s00033-009-0035-x.

[8]

B. Ducomet, S. Nečasová and A. Vasseur, On spherically symmetric motions of a viscous compressible barotropic and selfgravitating gas, J. Math. Fluid Mech., 13 (2011), 191-211. doi: 10.1007/s00021-009-0010-5.

[9]

J. E. Dunn and J. Serrin, On the thermomechanics of interstitial working, Arch. Rational Mech. Anal., 88 (1985), 95-133. doi: 10.1007/BF00250907.

[10]

I. Gamba, A. Jüngel and A. Vasseur, Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations, J. Differential Equations, 247 (2009), 3117-3135. doi: 10.1016/j.jde.2009.09.001.

[11]

P. Germain and P. G. Lefloch, The finite energy method for compressible fluids-The Naiver-Stokes-Korteweg model, Comm. Pure Appl. Math., 69 (2016), 3-61. doi: 10.1002/cpa.21622.

[12]

Y. Guo and Y. J. Wang, Decay of dissipative equations and negative Sobolev spaces, Commun. Partial Diff. Eqns., 37 (2012), 2165-2208. doi: 10.1080/03605302.2012.696296.

[13]

B. Haspot, Existence of global weak solution for compressible fluid models of Korteweg type, J. Math. Fluid Mech., 13 (2011), 223-249. doi: 10.1007/s00021-009-0013-2.

[14]

H. Hattori and D. Li, Global solutions of a high-dimensional system for Korteweg materials, J. Math. Anal. Appl., 198 (1996), 84-97. doi: 10.1006/jmaa.1996.0069.

[15]

H. Hattori and D. Li, The existence of global solutions to a fluid dynamic model for materials for Korteweg type, J. Partial Diff. Eqns, 9 (1996), 323-342.

[16]

L. Hsiao and H. L. Li, Dissipation and dispersion approximation to hydrodynamical equations and asymptotic limit, J. Partial Diff. Eqns, 21 (2008), 59-76.

[17]

S. Jiang, Z. P. Xin and P. Zhang, Global weak solutions to 1D compressible isentropic Navier-Stokes equations with density-dependent viscosity, Methods Appl. Anal., 12 (2005), 239-251. doi: 10.4310/MAA.2005.v12.n3.a2.

[18]

A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068.

[19]

M. Kotschote, Strong solutions for a compressible fluid of Korteweg type, Ann. Inst. H. Poincaré Anal. Non Lineaire, 25 (2008), 679-696. doi: 10.1016/j.anihpc.2007.03.005.

[20]

H. L. Li, J. Li and Z. P. Xin, Vanishing of vacuum states and blow-up phenomena of the compressible Navier-Stokes equations, Comm. Math. Phys., 281 (2008), 401-444. doi: 10.1007/s00220-008-0495-4.

[21]

P. L. Lions, Mathematical Topics in Fluid Dynamics 2, Compressible Models, Oxford Science Publication, Oxford, 1998.

[22]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.

[23]

A. Mellet and A. Vasseur, On the barotropic compressible Navier-Stokes equations, Comm. Partial Diff. Eqns, 32 (2007), 431-452. doi: 10.1080/03605300600857079.

[24]

Z. Tan, X. Zhang and H. Q. Wang, Asymptotic behavior of Navier-Stokes-Korteweg with friction in $R^3$, Discrete Contin. Dyn. Syst., 34 (2014), 2243-2259. doi: 10.3934/dcds.2014.34.2243.

[25]

T. Tang and J. Kuang, Blow-up of compressible Naiver-Stokes-Korteweg equations, Acta Applicanda Mathematicae, 130 (2014), 1-7. doi: 10.1007/s10440-013-9836-1.

[26]

T. Tang, Blow-up of smooth solutions to the compressible barotropic Navier-Stokes-Korteweg equations on bounded domains, Acta Applicanda Mathematicae, 136 (2015), 55-61. doi: 10.1007/s10440-014-9884-1.

[27]

Y. J. Wang and Z. Tan, Optimal decay rates for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 379 (2011), 256-271. doi: 10.1016/j.jmaa.2011.01.006.

[28]

A. Valli and W. M. Zajączkowski, Navier-Stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case, Comm. Math. Phys, 103 (1986), 259-296. doi: 10.1007/BF01206939.

[29]

Z. P. Xin, Blow-up of smooth solution to the compressible Naiver-Stokes equations with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240. doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.

[30]

T. Yang and C. J. Zhu, Compressible Navier-Stokes equations with degenerate viscosity coefficient and vacuum, Comm. Math. Phys., 230 (2002), 329-363. doi: 10.1007/s00220-002-0703-6.

[31]

Y. H. Zhang and Z. Tan, Blow-up of smooth solutions to the compressible fluid models of Korteweg type, Acta Math. Sin. (Engl. Ser.), 28 (2012), 645-652. doi: 10.1007/s10114-012-9042-5.

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