Inviscid quasi-neutral limit of a Navier-Stokes-Poisson-Korteweg system
Abstract
The combined quasi-neutral and inviscid limit of the Navier-Stokes-Poisson-Korteweg system with density-dependent viscosity and cold pressure in the torus T3 is studied. It is shown that, for the well-prepared initial data, the global weak solution of the Navier-Stokes-Poisson-Korteweg system converges strongly to the strong solution of the incompressible Euler equations when the Debye length and the viscosity coefficient go to zero simultaneously. Furthermore, the rate of convergence is also obtained.
Keyword : incompressible Euler equations, inviscid limit, Navier-Stokes-Poisson-Korteweg system, quasi-neutral limit
How to Cite
Wang, H., & Yang, J. (2018). Inviscid quasi-neutral limit of a Navier-Stokes-Poisson-Korteweg system. Mathematical Modelling and Analysis, 23(2), 205-216. https://doi.org/10.3846/mma.2018.013
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
[1] M. Bostan. The Vlasov-Maxwell system with strong initial magnetic field: guiding-center approximation. Multiscale Model. Simul, 6(3):1026–1058, 2007. https://doi.org/10.1137/070689383
[2] Y. Brenier. Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Commun. Part. Diff. Eqs., 25(3–4):737–754, 2000. https://doi.org/10.1080/03605300008821529
[3] D. Bresch and B. Desjardins. On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pures Appl., 87(1):57–90, 2007. https://doi.org/10.1016/j.matpur.2006.11.001
[4] D. Bresch, B. Desjardins and B. Ducomet. Quasi-neutral limit for a viscous capillary model of plasma. Ann. Inst. H. Poincaré Anal. Non Lianire, 22(1):1–9, 2005. https://doi.org/10.1016/j.anihpc.2004.02.001
[5] S. Cordier and E. Grenier. Quasineutral limit of an Euler-Poisson system arising from plasma physics. Communications in Partial Differential Equations, 25(5–6):1099–1113, 2000. https://doi.org/10.1080/03605300008821542
[6] D. Donatelli, E. Feireisl and A. Novotný. Scale analysis of a hydrodynamic model of plasma. M3AS. Math. Models Methods Appl. Sci., 25(2):371–394, 2015. https://doi.org/10.1142/S021820251550013X
[7] D. Donatelli and P. Marcati. Analysis of oscillations and defect measures for the quasineutral limit in plasma physics. Arch. Rat. Mech. Anal., 206(1):159–188, 2012. https://doi.org/10.1007/s00205-012-0531-6
[8] D. Donatelli and P. Marcati. The quasineutral limit for the Navier-Stokes-Fourier-Poisson system. Springer Proceedings in Mathematics & Statistics, 49:193–206, 2013. https://doi.org/10.1007/978-3-642-39007-4 9
[9] D. Donatelli and P. Marcati. Quasi-neutral limit, dispersion and oscillations for Korteweg-type fluids. SIAM J. Math. Anal., 47(3):2265–2282, 2015. https://doi.org/10.1137/140987651
[10] J.E. Dunn and J. Serrin. On the thermomechanics of interstitial working. Arch. Rat. Mech. Anal., 88(2):95–133, 1985. https://doi.org/10.1007/BF00250907
[11] I. Gasser, L. Hsiao, P. Markowich and S. Wang. Quasi-neutral limit of a non-linear drift diffusion model for semiconductor models. J. Math. Anal. Appl., 268(1):184–199, 2002. https://doi.org/10.1006/jmaa.2001.7813
[12] E. Grenier. Oscillations in quasineutral plasmas. Communications in Partial Differential Equations, 21(3–4):363–394, 1996. https://doi.org/10.1080/03605309608821189
[13] Q.C. Ju, F.C. Li and S. Wang. Convergence of the Navier-Stokes-Poisson system to the incompressible Navier-Stokes equations. J. Math. Phys., 49(7):073515, 2008. https://doi.org/10.1063/1.2956495
[14] A. Jüngel, C.K. Lin and K.C. Wu. An asymptotic limit of a Navier-Stokes system with capillary effects. Communications in Mathematical Physics, 329(2):725–744, 2014. https://doi.org/10.1007/s00220-014-1961-9
[15] T. Kato. Nonstationary flow of viscous and ideal fluids in R3. J. Funct. Anal, 9(3):296–305, 1972. https://doi.org/10.1016/0022-1236(72)90003-1
[16] 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 considé rables 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, 6:1–24, 1901.
[17] F.C. Li. Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics. J. Differential Equations, 246(9):3620–3641, 2009. https://doi.org/10.1016/j.jde.2009.01.027
[18] F.C. Li. Quasineutral limit of the viscous quantum hydrodynamic model for semiconductors. J. Math. Anal. Appl., 352(2):620–628, 2009. https://doi.org/10.1016/j.jmaa.2008.11.011
[19] H.L. Li and C.K. Lin. Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors. Comm. Math. Phys., 256(1):195–212, 2005. https://doi.org/10.1007/s00220-005-1316-7
[20] Y.P. Li and W. A Yong. Quasi-neutral limit in a 3D compressible Navier-Stokes-Poisson-Korteweg model. IMA Journal of Applied Mathematics, 80(3):712–727, 2015. https://doi.org/10.1093/imamat/hxu008
[21] P. Markowich, C.A. Ringhofer and C.A. Schmeiser. Semiconductor Equations. Springer-Verlag, 1955.
[22] N. Masmoudi. From Vlasov-Poisson system to the incompressible Euler system. Communications in Partial Differential Equations, 26(3):1913–1928, 2001. https://doi.org/10.1081/PDE-100107463
[23] Y.J. Peng and S. Wang. Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equation. SIAM J. Math. Anal., 40(2):540–565, 2008. https://doi.org/10.1137/070686056
[24] Y.J. Peng, Y.G.Wang and W.A. Yong. Quasi-neutral limit of the non-isentropic Euler-Poisson system. Proc. Roy. Soc. Edinburgh Sect. A, 136(5):1013–1026, 2006. https://doi.org/10.1017/S0308210500004856
[25] M. Slemrod and N. Sternberg. Quasi-neutral limit for Euler-Poisson system. J. Nonlinear Sci., 11(3):193–209, 2001. https://doi.org/10.1007/s00332-001-0004-9
[26] S. Wang. Quasineutral limit of Euler-Poisson system with and without viscosity. Communications in Partial Differential Equations, 29(3–4):419–456, 2005. https://doi.org/10.1081/PDE-120030403
[27] S. Wang and S. Jiang. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Comm. Partial. Differ. Equ., 31(4):571–591, 2006. https://doi.org/10.1080/03605300500361487
[28] S. Wang, Z.P. Xin and P. Markowich. Quasi-neutral limit of the drift-diffusion models for semiconductors: the case of general sign-changing doping profile. SIAM J. Math. Anal., 37(6):1854–1889, 2006. https://doi.org/10.1137/S0036141004440010
[29] J.W. Yang and Q.C. Ju. Convergence of the quantum Navier-Stokes-Poisson equations to the incompressible Euler equations for general initial data. Nonlinear Anal. Real World Appl., 23:148–159, 2015. https://doi.org/10.1016/j.nonrwa.2014.12.003
[30] J.W. Yang, Z.Y. Wang and F.X. Ding. Existence of global weak solutions for a 3D Navier-Stokes-Poisson-Korteweg equations. Applicable Analysis, 2017. https://doi.org/10.1080/00036811.2016.1276172
[2] Y. Brenier. Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Commun. Part. Diff. Eqs., 25(3–4):737–754, 2000. https://doi.org/10.1080/03605300008821529
[3] D. Bresch and B. Desjardins. On the existence of global weak solutions to the Navier-Stokes equations for viscous compressible and heat conducting fluids. J. Math. Pures Appl., 87(1):57–90, 2007. https://doi.org/10.1016/j.matpur.2006.11.001
[4] D. Bresch, B. Desjardins and B. Ducomet. Quasi-neutral limit for a viscous capillary model of plasma. Ann. Inst. H. Poincaré Anal. Non Lianire, 22(1):1–9, 2005. https://doi.org/10.1016/j.anihpc.2004.02.001
[5] S. Cordier and E. Grenier. Quasineutral limit of an Euler-Poisson system arising from plasma physics. Communications in Partial Differential Equations, 25(5–6):1099–1113, 2000. https://doi.org/10.1080/03605300008821542
[6] D. Donatelli, E. Feireisl and A. Novotný. Scale analysis of a hydrodynamic model of plasma. M3AS. Math. Models Methods Appl. Sci., 25(2):371–394, 2015. https://doi.org/10.1142/S021820251550013X
[7] D. Donatelli and P. Marcati. Analysis of oscillations and defect measures for the quasineutral limit in plasma physics. Arch. Rat. Mech. Anal., 206(1):159–188, 2012. https://doi.org/10.1007/s00205-012-0531-6
[8] D. Donatelli and P. Marcati. The quasineutral limit for the Navier-Stokes-Fourier-Poisson system. Springer Proceedings in Mathematics & Statistics, 49:193–206, 2013. https://doi.org/10.1007/978-3-642-39007-4 9
[9] D. Donatelli and P. Marcati. Quasi-neutral limit, dispersion and oscillations for Korteweg-type fluids. SIAM J. Math. Anal., 47(3):2265–2282, 2015. https://doi.org/10.1137/140987651
[10] J.E. Dunn and J. Serrin. On the thermomechanics of interstitial working. Arch. Rat. Mech. Anal., 88(2):95–133, 1985. https://doi.org/10.1007/BF00250907
[11] I. Gasser, L. Hsiao, P. Markowich and S. Wang. Quasi-neutral limit of a non-linear drift diffusion model for semiconductor models. J. Math. Anal. Appl., 268(1):184–199, 2002. https://doi.org/10.1006/jmaa.2001.7813
[12] E. Grenier. Oscillations in quasineutral plasmas. Communications in Partial Differential Equations, 21(3–4):363–394, 1996. https://doi.org/10.1080/03605309608821189
[13] Q.C. Ju, F.C. Li and S. Wang. Convergence of the Navier-Stokes-Poisson system to the incompressible Navier-Stokes equations. J. Math. Phys., 49(7):073515, 2008. https://doi.org/10.1063/1.2956495
[14] A. Jüngel, C.K. Lin and K.C. Wu. An asymptotic limit of a Navier-Stokes system with capillary effects. Communications in Mathematical Physics, 329(2):725–744, 2014. https://doi.org/10.1007/s00220-014-1961-9
[15] T. Kato. Nonstationary flow of viscous and ideal fluids in R3. J. Funct. Anal, 9(3):296–305, 1972. https://doi.org/10.1016/0022-1236(72)90003-1
[16] 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 considé rables 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, 6:1–24, 1901.
[17] F.C. Li. Quasineutral limit of the electro-diffusion model arising in electrohydrodynamics. J. Differential Equations, 246(9):3620–3641, 2009. https://doi.org/10.1016/j.jde.2009.01.027
[18] F.C. Li. Quasineutral limit of the viscous quantum hydrodynamic model for semiconductors. J. Math. Anal. Appl., 352(2):620–628, 2009. https://doi.org/10.1016/j.jmaa.2008.11.011
[19] H.L. Li and C.K. Lin. Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors. Comm. Math. Phys., 256(1):195–212, 2005. https://doi.org/10.1007/s00220-005-1316-7
[20] Y.P. Li and W. A Yong. Quasi-neutral limit in a 3D compressible Navier-Stokes-Poisson-Korteweg model. IMA Journal of Applied Mathematics, 80(3):712–727, 2015. https://doi.org/10.1093/imamat/hxu008
[21] P. Markowich, C.A. Ringhofer and C.A. Schmeiser. Semiconductor Equations. Springer-Verlag, 1955.
[22] N. Masmoudi. From Vlasov-Poisson system to the incompressible Euler system. Communications in Partial Differential Equations, 26(3):1913–1928, 2001. https://doi.org/10.1081/PDE-100107463
[23] Y.J. Peng and S. Wang. Rigorous derivation of incompressible e-MHD equations from compressible Euler-Maxwell equation. SIAM J. Math. Anal., 40(2):540–565, 2008. https://doi.org/10.1137/070686056
[24] Y.J. Peng, Y.G.Wang and W.A. Yong. Quasi-neutral limit of the non-isentropic Euler-Poisson system. Proc. Roy. Soc. Edinburgh Sect. A, 136(5):1013–1026, 2006. https://doi.org/10.1017/S0308210500004856
[25] M. Slemrod and N. Sternberg. Quasi-neutral limit for Euler-Poisson system. J. Nonlinear Sci., 11(3):193–209, 2001. https://doi.org/10.1007/s00332-001-0004-9
[26] S. Wang. Quasineutral limit of Euler-Poisson system with and without viscosity. Communications in Partial Differential Equations, 29(3–4):419–456, 2005. https://doi.org/10.1081/PDE-120030403
[27] S. Wang and S. Jiang. The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Comm. Partial. Differ. Equ., 31(4):571–591, 2006. https://doi.org/10.1080/03605300500361487
[28] S. Wang, Z.P. Xin and P. Markowich. Quasi-neutral limit of the drift-diffusion models for semiconductors: the case of general sign-changing doping profile. SIAM J. Math. Anal., 37(6):1854–1889, 2006. https://doi.org/10.1137/S0036141004440010
[29] J.W. Yang and Q.C. Ju. Convergence of the quantum Navier-Stokes-Poisson equations to the incompressible Euler equations for general initial data. Nonlinear Anal. Real World Appl., 23:148–159, 2015. https://doi.org/10.1016/j.nonrwa.2014.12.003
[30] J.W. Yang, Z.Y. Wang and F.X. Ding. Existence of global weak solutions for a 3D Navier-Stokes-Poisson-Korteweg equations. Applicable Analysis, 2017. https://doi.org/10.1080/00036811.2016.1276172