Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Relaxation of the isothermal Euler-Poisson system to the drift-diffusion equations

Authors: S. Junca and M. Rascle
Journal: Quart. Appl. Math. 58 (2000), 511-521
MSC: Primary 35Q05; Secondary 35L65, 35L70, 76X05, 82D37
DOI: https://doi.org/10.1090/qam/1770652
MathSciNet review: MR1770652
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the one-dimensional Euler-Poisson system in the isothermal case, with a friction coefficient $ {\varepsilon ^{ - 1}}$. When $ \varepsilon \to {0_ + }$, we show that the sequence of entropy-admissible weak solutions constructed in [PRV] converges to the solution to the drift-diffusion equations. We use the scaling introduced in [MN2], who proved a quite similar result in the isentropic case, using the theory of compensated compactness. On the one hand, this theory cannot be used in our case; on the other hand, exploiting the linear pressure law, we can give here a much simpler proof by only using the entropy inequality and de la Vallée-Poussin criterion of weak compactness in $ {L^{1}}$.

References [Enhancements On Off] (What's this?)

  • [B] H. Brezis, Analyse fonctionnelle, Masson, Paris, 1983 MR 697382
  • [DCL] X. Ding, G.-Q. Chen, and P. Luo, Convergence of the fractional step Lax-Friedrichs scheme: a Godounov scheme for isentropic gas dynamics, Comm. Math. Phys. 121, 63-84 (1989) MR 985615
  • [CJZ1] G.-Q. Chen, J. W. Jerome, and Bo Zhang, Particle hydrodynamic moment models in biology and microelectronics: Singular relaxation limits, preprint, 1996 MR 1489784
  • [CJZ2] G.-Q. Chen, J. W. Jerome, and Bo Zhang, Existence and the singular relaxation limit for the inviscid hydrodynamic energy model, preprint, 1996 MR 1677393
  • [DiP] R. DiPerna, Convergence of approximate solutions of conservation laws, Arch. Rational Mech. Anal. 82, 27-70 (1983) MR 684413
  • [ET] I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels, Dunod, Gauthier-Villars, 1974 MR 0463993
  • [G] J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18, 698-715 (1965) MR 0194770
  • [J] S. Junca, Optique géométrique non linéaire, chocs forts, relaxation, Thesis, (III), Univ. Nice-Sophia Antipolis, 1995
  • [Ni] T. Nishida, Global solutions for an initial boundary value problem of a quasilinear hyperbolic system, Japan Acad. 44, 642-646 (1968) MR 0236526
  • [MN1] P. Marcati and R. Natalini, Weak Solutions to a hydrodynamic model for semiconductors: the Cauchy problem, Proc. Roy. Soc. Edinburgh, to appear MR 1318626
  • [MN2] P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rational Mech. Anal. 129, 129-145 (1995) MR 1328473
  • [MRS] P. A. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Wien-New York, 1990 MR 1063852
  • [PRV] F. Poupaud, M. Rascle, and J.-P. Vila, Global solutions to the isothermal Euler-Poisson system with arbitrarily large data, J. Differential Equations 123, 93-121 (1995) MR 1359913
  • [T] L. Tartar, Compensated compactness and applications to partial differential equations, Research notes in mathematics, nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, ed. R. J. Knops, Pitman Press, Boston, MA, 1979, pp. 136-212 MR 584398
  • [Z] B. Zhang, Convergence of the Godunov scheme for a simplified one dimensional hydrodynamic model for semiconductor devices, preprint, Dept. Math., Purdue Univ., 1992 MR 1244856

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC: 35Q05, 35L65, 35L70, 76X05, 82D37

Retrieve articles in all journals with MSC: 35Q05, 35L65, 35L70, 76X05, 82D37

Additional Information

DOI: https://doi.org/10.1090/qam/1770652
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society