No CrossRef data available.

Part of:
Partial differential equations
Equations of mathematical physics and other areas of application

Published online by Cambridge University Press:
**01 August 2022**

In this study, we consider the viscous compressible Navier–Stokes–Poisson equations, which consist of the balance laws for electron density and moment, and a Poisson equation for the electrostatic potential. The limit of vanishing electron mass of this system with both well/ill-prepared initial data on the whole space is rigorously justified within the framework of local smooth solution. We first make use of the symmetric hyperbolic–parabolic structure of the compressible Navier–Stokes–Poisson equation to obtain uniform estimate in the short time, by which we show uniform existence of local classical solution to the compressible Navier–Stokes–Poisson equation in $\mathbb {R}^d(d\geq 1)$. Further, with uniform estimate of time derivatives, we show the zero-electron-mass limit of the solutions for the compressible Navier–Stokes–Poisson equation with well-prepared initial data in $\mathbb {R}^d(d\geq 1)$ by using Aubin's lemma. A detailed spectral analysis on the linearized system is done so that we are able to prove the zero-electron-mass limit of the solutions with ill-prepared initial data in $\mathbb {R}^d(d\geq 3)$, where the convergence occurs away from the time $t=0$. Finally, note that the dissipation mechanism for the linearized compressible Navier–Stokes–Poisson system is different from that of the compressible Euler equations in Grenier (Commun. Partial Diff. Eqns.21 (1996), 363–394); Grenier (Commun. Pure Appl. Math.50 (1997), 821–865); Ukai (J. Math. Kyoto Univ.26 (1986), 323–331), or that of the compressible Euler–Poisson equations in Ali and Chen (Nonlinearity24 (2011), 2745–2761), since its eigenvalues are somehow similar to that of heat equation, and the fundamental solution contains a part behaving like the heat kernel, thus a big difficulty is the singularity of the heat kernel at $t=0$.

Primary:
35B25: Singular perturbations

- Type
- Research Article
- Information
- Copyright
- Copyright © The Author(s), 2022. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Alazard, T.. Low Mach number limit of the full Navier–Stokes equations. Arch. Ration. Mech. Anal. 180 (2006), 1–73.CrossRefGoogle Scholar

Alazard, T.. A minicourse on the low Mach number limit. Discrete Contin. Dyn. Syst. Ser. S 1 (2008), 365–404.Google Scholar

Ali, G. and Chen, L.. The zero-electron-mass limit in the Euler–Poisson system for both well- and ill-prepared initial data. Nonlinearity 24 (2011), 2745–2761.CrossRefGoogle Scholar

Ali, G., Chen, L., Jüngel, A. and Peng, Y.-J.. The zero-electron-mass limit in the hydrodynamic model for plasmas. Nonlinear Anal. 72 (2010), 4415–4427.CrossRefGoogle Scholar

Chae, D.. On the nonexistence of global weak solution to the Navier–Stokes–Poisson equations in $\mathbb {R}^N$. Commun. Partial Differ. Equ. 35 (2010), 535–557.CrossRefGoogle Scholar

Danchin, R.. Low Mach number limit for viscous compressible flows. Math. Model. Numer. Anal. 39 (2005), 459–475.CrossRefGoogle Scholar

Degond, P., Mathematical modelling of microelectronics semiconductor devices, In: *Some current topics on nonlinear conservation laws*, AMS/IPStud. Adv. Math 15, Providence, RI: Am. Math. Soc., 2000, 77–110.Google Scholar

Desjardins, B. and Grenier, E.. Low Mach number limit of viscous compressible flows in the whole space. Proc. R. Soc. Lond. 455 (1999), 2271–2279.CrossRefGoogle Scholar

Dasjardins, B., Grenier, E., Lions, P.-L. and Masmoudi, N.. In compressible limit for solutions of the isentropic Navier–Stokes equations with Dirichlet boundary conditions. J. Math. Pures Appl. 78 (1999), 461–471.CrossRefGoogle Scholar

Donatelli, D., Feireisl, E. and Novotný, A.. On the vanishing electron-mass limit in plasma hydrodynamics in unbounded media. J. Nonlinear Sci. 22 (2012), 985–1012.CrossRefGoogle Scholar

Donatelli, D. and Marcati, P.. A quasineutral type limit for the Navier–Stokes–Poisson system with large data. Nonlinearity 21 (2008), 135–148.CrossRefGoogle Scholar

Duan, R.-J. and Liu, S.-Q.. Stability of rarefaction waves of the Navier–Stokes–Poisson system. J. Differ. Equ. 258 (2015), 2495–2530.CrossRefGoogle Scholar

Duan, R.-J., Liu, S.-Q. and Zhang, Z., Ion-acoustic shock in a collisional plasmas, Preprint, 2019.Google Scholar

Ducomet, B.. Local and global existence for the coupled Navier–Stokes–Poisson problem. Quart. Appl. Math. 61 (2003), 345–361.Google Scholar

Ducomet, B.. A remark about global existence for the Navier–Stokes–Poisson system. Appl. Math. Lett. 12 (1999), 31–37.CrossRefGoogle Scholar

Feireisl, E. and Novotny, A.. Singular limits in thermodynamics of viscous fluids (Basel: Birkhauser, 2009).CrossRefGoogle Scholar

Grenier, E.. Oscillations in quasineutral plasmas. Commun. Partial Diff. Eqns. 21 (1996), 363–394.CrossRefGoogle Scholar

Grenier, E.. Pseudo-differential energy estimates of singular perturbations. Commun. Pure Appl. Math. 50 (1997), 821–865.3.0.CO;2-7>CrossRefGoogle Scholar

Hao, C.-C. and Li, H.-L.. Global existence for compressible Navier–Stokes–Poisson equations in three and higher dimensions. J. Differ. Equ. 246 (2009), 4791–4812.CrossRefGoogle Scholar

Jang, J. and Tice, L.. Instability theory of the Navier–Stokes–Poisson equations. Anal. Partial Differ. Equ. 6 (2013), 1121–1181.Google Scholar

Jüngel, A., *Quasi-hydrodynamic semiconductor equations*, Progress in Nonlinear Differential Equations (Birkhäuser, 2001).CrossRefGoogle Scholar

Ju, Q.-C., Li, F.-C. and Li, H.-L.. The quasineutral limit of compressible Navier–Stokes– Poisson system with heat conductivity and general initial date. J. Differ. Equ. 247 (2009), 203–224.CrossRefGoogle Scholar

Jiang, M.-N., Lai, S.-H., Yin, H.-Y. and Zhu, C.-J.. The stability of stationary solution for outflow problem on the Navier–Stokes–Poisson system. Acta Math. Sci. Ser. B 36 (2016), 1098–1116.CrossRefGoogle Scholar

Kawashima, S., *Systems of a hyperbolic–parabolic composite type, with applications to the equations of magnetohydrodynamics*, Thesis, Kyoto Univ., 1983.Google Scholar

Kong, H.-H. and Li, H.-L.. Free boundary value problem to 3D spherically symmetric compressible Navier–Stokes–Poisson equations. Z. Angew. Math. Phys. 68 (2017), 21–34.CrossRefGoogle Scholar

Li, Y.-P. and Liao, J.. Existence and zero-electron-mass limit of strong solutions to the stationary compressible Navier–Stokes–Poisson equation with large external force. Math. Methods Appl. Sci. 41 (2018), 646–663.CrossRefGoogle Scholar

Li, H.-L., Matsumura, A. and Zhang, G.-J.. Optimal decay rate of the compressible Navier–Stokes–Poisson system in $\mathbb {R}^3$. Arch. Ration. Mech. Anal. 196 (2010), 681–713.CrossRefGoogle Scholar

Li, Y.-P. and Zhu, P.-C.. Asymptotics towards a nonlinear wave for an out-flow problem of a model of viscous ions motion. Math. Model Methods Appl. Sci. 27 (2017), 2111–2145.CrossRefGoogle Scholar

Li, Y.-P., Zhou, G. and Liao, J.. Zero-electron-mass limit of the two-dimensional compressible Navier–Stokes–Poisson equations over bound domain. Math. Methods Appl. Sci. 41 (2018), 9485–9501.CrossRefGoogle Scholar

Masmoudi, N., *Examples of singular limits in hydrodynamics*, In: Handbook of Differential Equations: Evolutionary equations, Vol. III, 195–275 (Elsevier/North-Holland, Amsterdam, 2007).CrossRefGoogle Scholar

Majda, A.. Compressible fluid flow and systems of conservation laws in several space variables (New York: Springer, 1984).CrossRefGoogle Scholar

Schochet, S., *The mathematical theory of the incompressible limit in fluid dynamics*, in: Handbook of mathematical fluid dynamics, Vol. IV, 123–157 (Elsevier/North-Holland, Amsterdam, 2007).CrossRefGoogle Scholar

Simon, J.. Compact sets in the space $L^p(O,\, T; B)$. Annal. Matemat. Pura Appl. 146 (1986), 65–96.CrossRefGoogle Scholar

Simon, J.. Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21 (1990), 1093–1117.CrossRefGoogle Scholar

Ukai, S.. The incompressible limit and the initial layer of the compressible Euler equation. J. Math. Kyoto Univ. 26 (1986), 323–331.Google Scholar

Wang, S. and Jiang, S.. The convergence of the Navier–Stokes–Poisson system to the incompressible Euler equations. Commun. Partial Differ. Equ. 31 (2006), 571–591.CrossRefGoogle Scholar

Wang, W.-K. and Wu, Z.-G.. Pointwise estimates of solution for the Navier–Stokes–Poisson equations in multi-dimensions. J. Differ. Equ. 248 (2010), 1617–1636.CrossRefGoogle Scholar

Wang, L., Zhang, G.-J. and Zhang, K.-J.. Existence and stability of stationary solution to compressible Navier–Stokes–Poisson equations in half line. Nonlinear Anal. 145 (2016), 97–117.CrossRefGoogle Scholar

Xu, J. and Yong, W.-A.. Zero-electron-mass limit of hydrodynamic models for plasmas. Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), 431–447.CrossRefGoogle Scholar

Xu, J. and Zhang, T.. Zero-electron-mass limit of Euler–Poisson equations. Discrete Contin. Dyn. Syst. 33 (2013), 4743–4768.CrossRefGoogle Scholar

Zhang, T. and Fang, D.-Y.. Global behavior of spherically symmetric Navier–Stocks–Poisson system with degenerate viscosity coefficients. Arch. Ration. Mech. Anal. 191 (2009), 195–243.CrossRefGoogle Scholar

Zhang, Y.-H. and Tan, Z.. On the existence of solutions to the Navier–Stokes–Poisson equations of a two-dimensional compressible flow. Math. Meth. Appl. Sci. 30 (2007), 305–329.CrossRefGoogle Scholar

To save this article to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Zero-electron-mass limit of the compressible Navier–Stokes–Poisson equations with well/ill-prepared initial data

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

×
####
Submit a response