Asymptotic limit of the Navier-Stokes-Poisson-Korteweg system in the half-space. (English) Zbl 1504.35399
Summary: In this paper, we consider the quasi-neutral limit, zero-viscosity limit and vanishing capillarity limit for the compressible Navier-Stokes-Poisson system of Korteweg type in the half-space. The system is supplemented with the Neumann, Navier-slip and Dirichlet boundary conditions for density, velocity and electric potential, respectively. The stability of the approximation solutions involving the boundary layer is established by a conormal energy estimate, and then the convergence of solution of the Navier-Stokes-Poisson-Korteweg system to that of the compressible Euler equation is obtained with convergence rate.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q31 | Euler equations |
| 76X05 | Ionized gas flow in electromagnetic fields; plasmic flow |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
| 76N20 | Boundary-layer theory for compressible fluids and gas dynamics |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 82D10 | Statistical mechanics of plasmas |
Keywords:
Navier-Stokes-Poisson-Korteweg system; boundary layers; asymptotic limit; uniform energy estimatesReferences:
| [1] | Anderson, D. M.; Mcfadden, G. B.; Wheeler, A. A., Diffuse-interface methods in fluid mechanics, Annu. Rev. Fluid Mech., 30, 139-165 (1998) · Zbl 1398.76051 |
| [2] | Antonelli, P.; Spirito, S., On the compactness of weak solutions to the Navier-Stokes-Korteweg equations for capillary fluids, Nonlinear Anal., 187, 110-124 (2019) · Zbl 1428.35342 |
| [3] | Audiard, C.; Haspot, B., Global well-posedness of the Euler Korteweg system for small irrotational data, Commun. Math. Phys., 351, 201-247 (2017) · Zbl 1369.35047 |
| [4] | Bian, D.; Yao, L.; Zhu, C., Vanishing capillarity limit of the compressible fluid models of Korteweg type to the Navier-Stokes equations, SIAM J. Math. Anal., 46, 2, 1633-1650 (2014) · Zbl 1304.35531 |
| [5] | Bresch, D.; Desjardins, B.; Ducomet, B., Quasi-neutral limit for a viscous capillary model of plasma, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 22, 1, 1-9 (2005) · Zbl 1062.35061 |
| [6] | Bresch, D.; Desjardins, B.; Lin, C. K., On some compressible fluid models: Korteweg, lubrication, and shallow water systems, Commun. Partial Differ. Equ., 28, 843-868 (2003) · Zbl 1106.76436 |
| [7] | Bresch, D.; Gisclon, M.; Lacroix-Violet, I., On Navier-Stokes-Korteweg and Euler-Korteweg systems: application to quantum fluids models, Arch. Ration. Mech. Anal., 233, 975-1025 (2019) · Zbl 1433.35217 |
| [8] | Charve, F.; Haspot, B., Existence of a global strong solution and vanishing capilliary-viscosity limit in one dimensional for the Korteweg system, SIAM J. Math. Anal., 45, 2, 469-494 (2013) · Zbl 1294.35091 |
| [9] | Danchin, R.; Desjardins, B., Existence of solutions for compressible fluid models of Korteweg type, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 18, 1, 97-133 (2001) · Zbl 1010.76075 |
| [10] | Donatelli, D.; Marcati, P., Quasi-neutral limit, dispersion, and oscillations for Korteweg-type fluids, SIAM J. Math. Anal., 47, 3, 2265-2282 (2015) · Zbl 1320.35277 |
| [11] | Dunn, J. E.; Serrin, J., On the thermodynamics of interstitial working, Arch. Ration. Mech. Anal., 88, 2, 95-133 (1985) · Zbl 0582.73004 |
| [12] | Gérard-Varet, D.; Han-Kwan, D.; Rousset, F., Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries, Indiana Univ. Math. J., 62, 2, 359-402 (2013) · Zbl 1417.35119 |
| [13] | Gérard-Varet, D.; Han-Kwan, D.; Rousset, F., Quasineutral limit of the Euler-Poisson system for ions in a domain with boundaries II, J. Éc. Polytech. Math., 1, 343-386 (2014) · Zbl 1417.35120 |
| [14] | Hattori, H.; Li, D., Global solutions of a high dimensional system for Korteweg materials, J. Math. Anal. Appl., 198, 1, 84-97 (1996) · Zbl 0858.35124 |
| [15] | Hattori, H.; Li, D., Solutions for two-dimensional system for materials of Korteweg type, SIAM J. Math. Anal., 25, 1, 85-98 (1994) · Zbl 0817.35076 |
| [16] | Hoff, D., Local solutions of a compressible flow problem with Navier boundary conditions in general three-dimensional domains, SIAM J. Math. Anal., 44, 2, 633-650 (2012) · Zbl 1246.35149 |
| [17] | Q. Ju, T. Luo, X. Xu, Singular limits for the Navier-Stokes-Poisson equations of viscous plasma with strong density boundary layer, submitted for publication. |
| [18] | Ju, Q.; Xu, X., Quasi-neutral and zero-viscosity limits of Navier-Stokes-Poisson equations in the half-space, J. Differ. Equ., 264, 2, 867-896 (2018) · Zbl 1378.35242 |
| [19] | Jüngel, A.; Lin, C.; Wu, K., An asymptotic limit of a Navier-Stokes system with capillary effects, Commun. Math. Phys., 329, 725-744 (2014) · Zbl 1297.35169 |
| [20] | Korteweg, D. J., Sur la forme que prennent les équations des mouvements des fluides si l’on tient compte des forces capillaires par des variations de densité, Arch. Neerl. Sci. Exactes Nat. Ser. II, 6, 1-24 (1901) · JFM 32.0756.02 |
| [21] | Li, Y.; Peng, S.; Wang, S., Some limit analysis of a three dimensional viscous compressible capillary model for plasma, Math. Methods Appl. Sci., 41, 14, 5535-5551 (2018) · Zbl 1404.35361 |
| [22] | Li, Y.; Yong, W., Quasi-neutral limit in a 3D compressible Navier-Stokes-Poisson-Korteweg model, IMA J. Appl. Math., 80, 3, 712-727 (2015) · Zbl 1331.35282 |
| [23] | Majda, A., Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables (1984), Springer-Verlag: Springer-Verlag New York · Zbl 0537.76001 |
| [24] | Masmoudi, N.; Rousset, F., Uniform regularity and vanishing viscosity limit for the free surface Navier-Stokes equations, Arch. Ration. Mech. Anal., 223, 1, 301-417 (2017) · Zbl 1359.35133 |
| [25] | Murata, M.; Shibata, Y., The global well-posedness for the compressible fluid model of Korteweg type, SIAM J. Math. Anal., 52, 6, 6313-6337 (2020) · Zbl 1455.35202 |
| [26] | Paddick, M., The strong inviscid limit of the isentropic compressible Navier-Stokes equations with Navier boundary conditions, Discrete Contin. Dyn. Syst., 36, 5, 2673-2709 (2016) · Zbl 1332.35267 |
| [27] | Schochet, S., The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit, Commun. Math. Phys., 104, 1, 49-75 (1986) · Zbl 0612.76082 |
| [28] | Wang, Y.; Williams, M., The inviscid limit and stability of characteristic boundary layers for the compressible Navier-Stokes equations with Navier-friction boundary conditions, Ann. Inst. Fourier (Grenoble), 62, 6, 2257-2314 (2012) · Zbl 1334.76128 |
| [29] | Wang, Y.; Xin, Z.; Yong, Y., Uniform regularity and vanishing viscosity limit for the compressible Navier-Stokes with general Navier-slip boundary conditions in 3-dimensional domains, SIAM J. Math. Anal., 47, 6, 4123-4191 (2015) · Zbl 1332.35303 |
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