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May  2014, 34(5): 2243-2259. doi: 10.3934/dcds.2014.34.2243

Asymptotic behavior of Navier-Stokes-Korteweg with friction in $\mathbb{R}^{3}$

Zhong Tan 1, , Xu Zhang 2, and  Huaqiao Wang 2,

1. 

School of Mathematical Sciences, Xiamen University, Xiamen, Fujian 361005
2. 

School of Mathematical Sciences, Xiamen University, Fujian 361005, China, China

Received  March 2013 Revised  July 2013 Published  October 2013

We consider the compressible barotropic Navier-Stokes-Korteweg system with friction in this paper. The global solutions and optimal convergence rates are obtained by pure energy method provided the initial perturbation around a constant state is small enough. In particular, the decay rates of the higher-order spatial derivatives of the solution are obtained. Our proof is based on a family of scaled energy estimates and interpolations among them without linear decay analysis.
Keywords: energy method, Sobolev interpolation., optimal decay rates, Korteweg, Compressible Navier-Stokes equations, optimal decay rates.
Mathematics Subject Classification: Primary: 35Q30, 76N10; Secondary: 76D0.
Citation: Zhong Tan, Xu Zhang, Huaqiao Wang. Asymptotic behavior of Navier-Stokes-Korteweg with friction in $\mathbb{R}^{3}$. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2243-2259. doi: 10.3934/dcds.2014.34.2243
References:
[1]

D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water system, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: 10.1081/PDE-120020499.  Google Scholar

[2]

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[3]

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

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R. J. Duan, H. X. Liu, S. Ukai and T. Yang, Optimal $L^p-L^q$ convergence rate for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233. doi: 10.1016/j.jde.2007.03.008.  Google Scholar

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R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rate for the compressible Navier-Stokes equations with potential force, Math. Models Methods Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X.  Google Scholar

[8]

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.  Google Scholar

[9]

H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X.  Google Scholar

[10]

B. Haspot, Existence of Global Strong Solution for the Compressible Navier-Stokes System and the Korteweg System in Two-Dimension,, preprint, ().   Google Scholar

[11]

D. Hoff and K. Zumbrun, Multidimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003.  Google Scholar

[12]

D. Hoff and K. Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys., 48 (1997), 597-614. Google Scholar

[13]

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.  Google Scholar

[14]

D. J. Korteweg, Sur La Forme Que Prennent Les Équations Du Mouvement Des Fluides Si Lón Tient Compte Des Forces Capillaires Causées Par Des Variations De Densité Considérables Mais Continues Et Sur La Théorie De La Capillarité Dans Lhypothse Dúne Variation Continue De La Densité, Archives Néerlandaises de Sciences Exactes et Naturelles, (1901), 1-24. Google Scholar

[15]

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[16]

Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal., 177 (2005), 231-330. doi: 10.1007/s00205-005-0365-6.  Google Scholar

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

[18]

Y. P. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232. doi: 10.1016/j.jmaa.2011.11.006.  Google Scholar

[19]

A. Matsumura, An energy method for the equations of motion of compressible viscous and heat-conductive fluids in "MRC Technical Summary Report", University of Wisconsin-Madison, 1981, 1-16. Google Scholar

[20]

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

[21]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A, 55 (1979), 337-342. doi: 10.3792/pjaa.55.337.  Google Scholar

[22]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  Google Scholar

[23]

G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 339-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[24]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N. J. 1970 xiv+290 pp.  Google Scholar

[25]

Z. Tan, H. Q. Wang and J. K. Xu, Global existence and optimal $L^2$ decay rate for the strong solutions to the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 390 (2012), 181-187. doi: 10.1016/j.jmaa.2012.01.028.  Google Scholar

[26]

S. Ukai, T. Yang and H. J. Zhao, Convergence rate for the compressible Navier-Stokes equations with external force, J. Hyperbolic Differ. Equ., 3 (2006), 561-574. doi: 10.1142/S0219891606000902.  Google Scholar

[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.  Google Scholar

[28]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential equations, 253 (2012), 273-297. doi: 10.1016/j.jde.2012.03.006.  Google Scholar

show all references

References:
[1]

D. Bresch, B. Desjardins and C.-K. Lin, On some compressible fluid models: Korteweg, lubrication and shallow water system, Comm. Partial Differential Equations, 28 (2003), 843-868. doi: 10.1081/PDE-120020499.  Google Scholar

[2]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system, I. Interfacial free energy, J. Chem. Phys., 28 (1998), 258-267. Google Scholar

[3]

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

[4]

K. Deckelnick, $L^2$-decay for the compressible Navier-Stokes equations in unbounded domains, Comm. Partial Differential Equations, 18 (1993), 1445-1476. doi: 10.1080/03605309308820981.  Google Scholar

[5]

R. J. Duan, H. X. Liu, S. Ukai and T. Yang, Optimal $L^p-L^q$ convergence rate for the compressible Navier-Stokes equations with potential force, J. Differential Equations, 238 (2007), 220-233. doi: 10.1016/j.jde.2007.03.008.  Google Scholar

[6]

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

[7]

R. J. Duan, S. Ukai, T. Yang and H. J. Zhao, Optimal convergence rate for the compressible Navier-Stokes equations with potential force, Math. Models Methods Appl. Sci., 17 (2007), 737-758. doi: 10.1142/S021820250700208X.  Google Scholar

[8]

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.  Google Scholar

[9]

H. Hattori and D. Li, Solutions for two dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25 (1994), 85-98. doi: 10.1137/S003614109223413X.  Google Scholar

[10]

B. Haspot, Existence of Global Strong Solution for the Compressible Navier-Stokes System and the Korteweg System in Two-Dimension,, preprint, ().   Google Scholar

[11]

D. Hoff and K. Zumbrun, Multidimensional diffusion waves for the Navier-Stokes equations of compressible flow, Indiana Univ. Math. J., 44 (1995), 603-676. doi: 10.1512/iumj.1995.44.2003.  Google Scholar

[12]

D. Hoff and K. Zumbrun, Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves, Z. Angew. Math. Phys., 48 (1997), 597-614. Google Scholar

[13]

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.  Google Scholar

[14]

D. J. Korteweg, Sur La Forme Que Prennent Les Équations Du Mouvement Des Fluides Si Lón Tient Compte Des Forces Capillaires Causées Par Des Variations De Densité Considérables Mais Continues Et Sur La Théorie De La Capillarité Dans Lhypothse Dúne Variation Continue De La Densité, Archives Néerlandaises de Sciences Exactes et Naturelles, (1901), 1-24. Google Scholar

[15]

Y. Kagei and T. Kobayashi, On large time behavior of solutions to the compressible Navier-Stokes equations in the half space in $\mathbbR^3$, Arch. Ration. Mech. Anal., 165 (2002), 89-159. doi: 10.1007/s00205-002-0221-x.  Google Scholar

[16]

Y. Kagei and T. Kobayashi, Asymptotic behavior of solutions of the compressible Navier-Stokes equations on the half space, Arch. Ration. Mech. Anal., 177 (2005), 231-330. doi: 10.1007/s00205-005-0365-6.  Google Scholar

[17]

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

[18]

Y. P. Li, Global existence and optimal decay rate of the compressible Navier-Stokes-Korteweg equations with external force, J. Math. Anal. Appl., 388 (2012), 1218-1232. doi: 10.1016/j.jmaa.2011.11.006.  Google Scholar

[19]

A. Matsumura, An energy method for the equations of motion of compressible viscous and heat-conductive fluids in "MRC Technical Summary Report", University of Wisconsin-Madison, 1981, 1-16. Google Scholar

[20]

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

[21]

A. Matsumura and T. Nishida, The initial value problem for the equations of motion of compressible viscous and heat-conductive fluids, Proc. Japan Acad. Ser. A, 55 (1979), 337-342. doi: 10.3792/pjaa.55.337.  Google Scholar

[22]

L. Nirenberg, On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13 (1959), 115-162.  Google Scholar

[23]

G. Ponce, Global existence of small solution to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 339-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar

[24]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30 Princeton University Press, Princeton, N. J. 1970 xiv+290 pp.  Google Scholar

[25]

Z. Tan, H. Q. Wang and J. K. Xu, Global existence and optimal $L^2$ decay rate for the strong solutions to the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 390 (2012), 181-187. doi: 10.1016/j.jmaa.2012.01.028.  Google Scholar

[26]

S. Ukai, T. Yang and H. J. Zhao, Convergence rate for the compressible Navier-Stokes equations with external force, J. Hyperbolic Differ. Equ., 3 (2006), 561-574. doi: 10.1142/S0219891606000902.  Google Scholar

[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.  Google Scholar

[28]

Y. J. Wang, Decay of the Navier-Stokes-Poisson equations, J. Differential equations, 253 (2012), 273-297. doi: 10.1016/j.jde.2012.03.006.  Google Scholar

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