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
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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 ${L^{1}}$.
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G.-Q. Chen, J. W. Jerome, and Bo Zhang, Particle hydrodynamic moment models in biology and microelectronics: Singular relaxation limits, preprint, 1996
G.-Q. Chen, J. W. Jerome, and Bo Zhang, Existence and the singular relaxation limit for the inviscid hydrodynamic energy model, preprint, 1996
R. DiPerna, Convergence of approximate solutions of conservation laws, Arch. Rational Mech. Anal. 82, 27–70 (1983)
I. Ekeland and R. Temam, Analyse convexe et problèmes variationnels, Dunod, Gauthier-Villars, 1974
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T. Nishida, Global solutions for an initial boundary value problem of a quasilinear hyperbolic system, Japan Acad. 44, 642–646 (1968)
P. Marcati and R. Natalini, Weak Solutions to a hydrodynamic model for semiconductors: the Cauchy problem, Proc. Roy. Soc. Edinburgh, to appear
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)
P. A. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Wien-New York, 1990
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)
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
B. Zhang, Convergence of the Godunov scheme for a simplified one dimensional hydrodynamic model for semiconductor devices, preprint, Dept. Math., Purdue Univ., 1992
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