Skip to main content Accessibility help
×
Home
Hostname: page-component-5d6d958fb5-8cb25 Total loading time: 0.703 Render date: 2022-11-27T18:59:04.880Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

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

Published online by Cambridge University Press:  01 August 2022

Yeping Li
Affiliation:
School of Science, Nantong University, Nantong 226019, P. R. China (ypleemei@aliyun.com)
Jie Liao
Affiliation:
School of Mathematics, Shanghai University of Finance and Economics, Shanghai 200433, P. R. China (liaojie@mail.shufe.edu.cn)

Abstract

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 $t=0$.

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alazard, T.. Low Mach number limit of the full Navier–Stokes equations. Arch. Ration. Mech. Anal. 180 (2006), 173.CrossRefGoogle Scholar
Alazard, T.. A minicourse on the low Mach number limit. Discrete Contin. Dyn. Syst. Ser. S 1 (2008), 365404.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), 27452761.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), 44154427.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), 535557.CrossRefGoogle Scholar
Danchin, R.. Low Mach number limit for viscous compressible flows. Math. Model. Numer. Anal. 39 (2005), 459475.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), 22712279.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), 461471.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), 9851012.CrossRefGoogle Scholar
Donatelli, D. and Marcati, P.. A quasineutral type limit for the Navier–Stokes–Poisson system with large data. Nonlinearity 21 (2008), 135148.CrossRefGoogle Scholar
Duan, R.-J. and Liu, S.-Q.. Stability of rarefaction waves of the Navier–Stokes–Poisson system. J. Differ. Equ. 258 (2015), 24952530.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), 345361.Google Scholar
Ducomet, B.. A remark about global existence for the Navier–Stokes–Poisson system. Appl. Math. Lett. 12 (1999), 3137.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), 363394.CrossRefGoogle Scholar
Grenier, E.. Pseudo-differential energy estimates of singular perturbations. Commun. Pure Appl. Math. 50 (1997), 821865.3.0.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), 47914812.CrossRefGoogle Scholar
Jang, J. and Tice, L.. Instability theory of the Navier–Stokes–Poisson equations. Anal. Partial Differ. Equ. 6 (2013), 11211181.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), 203224.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), 10981116.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), 2134.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), 646663.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), 681713.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), 21112145.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), 94859501.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), 6596.CrossRefGoogle Scholar
Simon, J.. Nonhomogeneous viscous incompressible fluids: existence of velocity, density, and pressure. SIAM J. Math. Anal. 21 (1990), 10931117.CrossRefGoogle Scholar
Sitnko, A. and Malnev, V.. Plasma physics theory (London: Chapman & Hall, 1995).Google Scholar
Ukai, S.. The incompressible limit and the initial layer of the compressible Euler equation. J. Math. Kyoto Univ. 26 (1986), 323331.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), 571591.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), 16171636.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), 97117.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), 431447.CrossRefGoogle Scholar
Xu, J. and Zhang, T.. Zero-electron-mass limit of Euler–Poisson equations. Discrete Contin. Dyn. Syst. 33 (2013), 47434768.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), 195243.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), 305329.CrossRefGoogle Scholar

Save article to Kindle

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
Available formats
×

Save article to Dropbox

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.

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

Save article to Google Drive

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.

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

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *