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 & 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. doi: 10.1081/PDE-120020499. Google Scholar

[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. doi: 10.1016/j.jmaa.2012.04.008. Google Scholar

[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. doi: 10.1002/mma.2750. Google Scholar

[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. doi: 10.1016/j.matpur.2013.06.005. Google Scholar

[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. doi: 10.1016/S0294-1449(00)00056-1. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[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. Google Scholar

[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. doi: 10.1137/100814305. Google Scholar

[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. doi: 10.1137/050626478. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. doi: 10.1016/j.jmaa.2011.11.006. Google Scholar

[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. doi: 10.1007/BF03167088. Google Scholar

[18]

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

[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. doi: 10.1137/S003614100342735X. Google Scholar

[20]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, 2nd edition, (1994). doi: 10.1007/978-1-4612-0873-0. Google Scholar

[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. doi: 10.1016/j.jmaa.2011.01.006. Google Scholar

[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. doi: 10.1002/cpa.3160460502. 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 systems,, Comm. Partial Differential Equations, 28 (2003), 843. doi: 10.1081/PDE-120020499. Google Scholar

[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. doi: 10.1016/j.jmaa.2012.04.008. Google Scholar

[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. doi: 10.1002/mma.2750. Google Scholar

[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. doi: 10.1016/j.matpur.2013.06.005. Google Scholar

[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. doi: 10.1016/S0294-1449(00)00056-1. Google Scholar

[6]

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

[7]

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

[8]

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

[9]

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

[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. Google Scholar

[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. doi: 10.1137/100814305. Google Scholar

[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. doi: 10.1137/050626478. Google Scholar

[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. Google Scholar

[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. Google Scholar

[15]

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

[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. doi: 10.1016/j.jmaa.2011.11.006. Google Scholar

[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. doi: 10.1007/BF03167088. Google Scholar

[18]

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

[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. doi: 10.1137/S003614100342735X. Google Scholar

[20]

J. Smoller, Shock Waves and Reaction-Diffusion Equations,, 2nd edition, (1994). doi: 10.1007/978-1-4612-0873-0. Google Scholar

[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. doi: 10.1016/j.jmaa.2011.01.006. Google Scholar

[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. doi: 10.1002/cpa.3160460502. Google Scholar

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