American Institute of Mathematical Sciences

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.
 [1] Jianwei Yang, Peng Cheng, Yudong Wang. Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity. Electronic Research Announcements, 2015, 22: 20-31. doi: 10.3934/era.2015.22.20 [2] Hong Cai, Zhong Tan, Qiuju Xu. Time periodic solutions to Navier-Stokes-Korteweg system with friction. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 611-629. doi: 10.3934/dcds.2016.36.611 [3] 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 [4] Wenjun Wang, Weike Wang. Decay rates of the compressible Navier-Stokes-Korteweg equations with potential forces. Discrete and Continuous Dynamical Systems, 2015, 35 (1) : 513-536. doi: 10.3934/dcds.2015.35.513 [5] Zhong Tan, Xu Zhang, Huaqiao Wang. Asymptotic behavior of Navier-Stokes-Korteweg with friction in $\mathbb{R}^{3}$. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2243-2259. doi: 10.3934/dcds.2014.34.2243 [6] Guangrong Wu, Ping Zhang. The zero diffusion limit of 2-D Navier-Stokes equations with $L^1$ initial vorticity. Discrete and Continuous Dynamical Systems, 1999, 5 (3) : 631-638. doi: 10.3934/dcds.1999.5.631 [7] Thomas Strömberg. A system of the Hamilton--Jacobi and the continuity equations in the vanishing viscosity limit. Communications on Pure and Applied Analysis, 2011, 10 (2) : 479-506. doi: 10.3934/cpaa.2011.10.479 [8] Hua Chen, Jian-Meng Li, Kelei Wang. On the vanishing viscosity limit of a chemotaxis model. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1963-1987. doi: 10.3934/dcds.2020101 [9] Renjun Duan, Xiongfeng Yang. Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Communications on Pure and Applied Analysis, 2013, 12 (2) : 985-1014. doi: 10.3934/cpaa.2013.12.985 [10] Huancheng Yao, Haiyan Yin, Changjiang Zhu. Stability of rarefaction wave for the compressible non-isentropic Navier-Stokes-Maxwell equations. Communications on Pure and Applied Analysis, 2021, 20 (3) : 1297-1317. doi: 10.3934/cpaa.2021021 [11] Giuseppe Maria Coclite, Nicola De Nitti, Mauro Garavello, Francesca Marcellini. Vanishing viscosity for a $2\times 2$ system modeling congested vehicular traffic. Networks and Heterogeneous Media, 2021, 16 (3) : 413-426. doi: 10.3934/nhm.2021011 [12] Alberto Bressan, Yilun Jiang. The vanishing viscosity limit for a system of H-J equations related to a debt management problem. Discrete and Continuous Dynamical Systems - S, 2018, 11 (5) : 793-824. doi: 10.3934/dcdss.2018050 [13] Cheng-Jie Liu, Feng Xie, Tong Yang. Uniform regularity and vanishing viscosity limit for the incompressible non-resistive MHD system with TMF. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2725-2750. doi: 10.3934/cpaa.2021073 [14] Grzegorz Karch, Maria E. Schonbek, Tomas P. Schonbek. Singularities of certain finite energy solutions to the Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2020, 40 (1) : 189-206. doi: 10.3934/dcds.2020008 [15] Sijia Zhong, Daoyuan Fang. $L^2$-concentration phenomenon for Zakharov system below energy norm II. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1117-1132. doi: 10.3934/cpaa.2009.8.1117 [16] J. Huang, Marius Paicu. Decay estimates of global solution to 2D incompressible Navier-Stokes equations with variable viscosity. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4647-4669. doi: 10.3934/dcds.2014.34.4647 [17] Sun-Ho Choi. Weighted energy method and long wave short wave decomposition on the linearized compressible Navier-Stokes equation. Networks and Heterogeneous Media, 2013, 8 (2) : 465-479. doi: 10.3934/nhm.2013.8.465 [18] Igor Kukavica. Interior gradient bounds for the 2D Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 873-882. doi: 10.3934/dcds.2001.7.873 [19] Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471 [20] Zhigang Wang. Vanishing viscosity limit of the rotating shallow water equations with far field vacuum. Discrete and Continuous Dynamical Systems, 2018, 38 (1) : 311-328. doi: 10.3934/dcds.2018015

2020 Impact Factor: 1.916

Metrics

• HTML views (0)
• Cited by (3)

• on AIMS