November  2014, 13(6): 2331-2350. doi: 10.3934/cpaa.2014.13.2331

Vanishing viscosity limit to rarefaction waves for the full compressible fluid models of Korteweg type

1. 

College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China

2. 

School of Mathematics and Center for Nonlinear Studies, Northwest University, Xi'an 710127, China

Received  October 2013 Revised  April 2014 Published  July 2014

We prove the solution of the full compressible fluid models of Korteweg type with centered rarefaction wave data of large strength exists globally in time. As the viscosity, heat-conductivity and capillary coefficients tend to zero, the global solution converges to the centered rarefaction wave solution of the corresponding Euler equations uniformly when the initial perturbation is small. Our analysis is based on the energy method.
Citation: Wenjun Wang, Lei Yao. Vanishing viscosity limit to rarefaction waves for the full compressible fluid models of Korteweg type. Communications on Pure and Applied Analysis, 2014, 13 (6) : 2331-2350. doi: 10.3934/cpaa.2014.13.2331
References:
[1]

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

[2]

Z. Z. Chen, Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 394 (2012), 438-448. doi: 10.1016/j.jmaa.2012.04.008.

[3]

Z. Z. Chen and Q. H. Xiao, Nonlinear stability of viscous contact wave for the one-dimensional compressible fluid models of Korteweg type, Math. Methods Appl. Sci., 36 (2013), 2265-2279. doi: 10.1002/mma.2750.

[4]

Z. Z. Chen and H. J. Zhao, Existence and nonlinear stability of stationary solutions to the full compressible Navier-Stokes-Korteweg system, J. Math. Pures Appl., 101 (2014), 330-371. doi: 10.1016/j.matpur.2013.06.005.

[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]

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

[7]

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.

[8]

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.

[9]

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.

[10]

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

[11]

F. M. Huang, M. J. Li and Y. Wang, Zero dissipation limit to rarefaction wave with vacuum for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 44 (2012), 1742-1759. doi: 10.1137/100814305.

[12]

S. Jiang, G. X. Ni and W. J. Sun, Vanishing viscosity limit to rarefaction waves for the Navier-Stokes Equations of one-dimensional compressible heat-conducting fluids, SIAM J. Math. Anal., 38 (2006), 368-384. doi: 10.1137/050626478.

[13]

S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 249-252.

[14]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité, Archives Néerlandaises de Sciences Exactes et Naturelles II}, 6 (1901), 1-24.

[15]

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

[16]

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.

[17]

A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13. doi: 10.1007/BF03167088.

[18]

Y. J. Meng and L. Ding, Convergence to the rarefaction waves for the 1D compressible fluid models of Korteweg type, preprint, 2013.

[19]

K. Nishihara, T. Yang and H. J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597. doi: 10.1137/S003614100342735X.

[20]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[21]

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.

[22]

Z. Xin, Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Comm. Pure Appl. Math., 46 (1993), 621-665. doi: 10.1002/cpa.3160460502.

show all references

References:
[1]

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

[2]

Z. Z. Chen, Asymptotic stability of strong rarefaction waves for the compressible fluid models of Korteweg type, J. Math. Anal. Appl., 394 (2012), 438-448. doi: 10.1016/j.jmaa.2012.04.008.

[3]

Z. Z. Chen and Q. H. Xiao, Nonlinear stability of viscous contact wave for the one-dimensional compressible fluid models of Korteweg type, Math. Methods Appl. Sci., 36 (2013), 2265-2279. doi: 10.1002/mma.2750.

[4]

Z. Z. Chen and H. J. Zhao, Existence and nonlinear stability of stationary solutions to the full compressible Navier-Stokes-Korteweg system, J. Math. Pures Appl., 101 (2014), 330-371. doi: 10.1016/j.matpur.2013.06.005.

[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]

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

[7]

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.

[8]

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.

[9]

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.

[10]

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

[11]

F. M. Huang, M. J. Li and Y. Wang, Zero dissipation limit to rarefaction wave with vacuum for one-dimensional compressible Navier-Stokes equations, SIAM J. Math. Anal., 44 (2012), 1742-1759. doi: 10.1137/100814305.

[12]

S. Jiang, G. X. Ni and W. J. Sun, Vanishing viscosity limit to rarefaction waves for the Navier-Stokes Equations of one-dimensional compressible heat-conducting fluids, SIAM J. Math. Anal., 38 (2006), 368-384. doi: 10.1137/050626478.

[13]

S. Kawashima, A. Matsumura and K. Nishihara, Asymptotic behavior of solutions for the equations of a viscous heat-conductive gas, Proc. Japan Acad. Ser. A Math. Sci., 62 (1986), 249-252.

[14]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité, Archives Néerlandaises de Sciences Exactes et Naturelles II}, 6 (1901), 1-24.

[15]

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

[16]

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.

[17]

A. Matsumura and K. Nishihara, Asymptotics toward the rarefaction waves of the solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 3 (1986), 1-13. doi: 10.1007/BF03167088.

[18]

Y. J. Meng and L. Ding, Convergence to the rarefaction waves for the 1D compressible fluid models of Korteweg type, preprint, 2013.

[19]

K. Nishihara, T. Yang and H. J. Zhao, Nonlinear stability of strong rarefaction waves for compressible Navier-Stokes equations, SIAM J. Math. Anal., 35 (2004), 1561-1597. doi: 10.1137/S003614100342735X.

[20]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd edition, Springer-Verlag, New York, 1994. doi: 10.1007/978-1-4612-0873-0.

[21]

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.

[22]

Z. Xin, Zero dissipation limit to rarefaction waves for the one-dimensional Navier-Stokes equations of compressible isentropic gases, Comm. Pure Appl. Math., 46 (1993), 621-665. doi: 10.1002/cpa.3160460502.

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