Local and global existence for the coupled Navier-Stokes-Poisson problem
Author:
Donatella Donatelli
Journal:
Quart. Appl. Math. 61 (2003), 345-361
MSC:
Primary 35Q30; Secondary 35D05, 76D03, 76X05
DOI:
https://doi.org/10.1090/qam/1976375
MathSciNet review:
MR1976375
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Abstract: In this paper we investigate the Cauchy Problem for coupled Navier-Stokes-Poisson equation. The global existence of weak solutions in Sobolev framework is proved by using some compactification properties deduced from the Poisson equation.
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A. M. Anile, An extended thermodynamic framework for the hydrodynamical modeling of semiconductors, Pitman Research Notes In Mathematics Series, 340, 1995, pp. 3–41.
A. M. Anile and S. Pennisi, Extended thermodynamics of the Blotekjaer hydrodynamical model for semiconductors. Continuum Mech. Therm., 4, 1992, pp. 187–193.
H. Brezis, Functional Analysis - Theory and Applications, Ligouri Editore.
R. Coifman, R. Rochberg, and G. Weiss, Ann. Math., 103, 1975, pp. 611-635.
C. Cercignani, The Boltzmann equation and its applications, Applied Mathematical Sciences, 67, Springer-Verlag, New York.
C. Cercignani, R. Illner, and M. Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, 106, 1994, Springer-Verlag, New York.
B. Desjardins, Weak solutions on the compressible isentropic Navier-Stokes equations. Appl. Math. Lett., 12, 1999, no. 7, 107–111.
L. C. Evans, Partial Differential Equations, volume 19 of Graduate Studies in Mathematics, American Mathematical Society.
V. I. Gerasimenko and D. Ya. Petrina, The Boltzmann-Grad limit theorem (Russian), Dokl. Akad. Nauk Ukrain. SSR Ser. A, 1989, no. 11, 12–16.
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, volume 224 of Grundlehren der matematischen Wissenschaften, Springer-Verlag, 1977.
C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Statist. Phys., 83, 1996, no. 5-6, 1021–1065.
C. D. Levermore, Entropy-based moment closures for kinetic equations, Proceedings of the International Conference on Latest Developments and Fundamental Advances in Radiative Transfer (Los Angeles, CA, 1996), Transport Theory Statist. Phys., 26, 1997, no. 4-5, 591–606.
P. L. Lions, Mathematical Topics in Fluid Mechanics, Vol. 1, Incompressible Models, Clarendon Press, Oxford Science Publications, Oxford, 1996
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P. Marcati and R. Natalini, Weak solutions to a hydrodynamic model for semiconductors: the Cauchy problem, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), no. 1, 115–131.
P. A. Markowich, The Steady-State Semiconductor Device Equations, Springer, New York, 1986.
P. A. Markowich, C. Ringhofer, and C. Schmeiser, Semiconductor Equations, 1990, Springer-Verlag, Wien, New York.
N. G. Meyers, An ${L^p}$ estimate for the gradient of solutions of the second order elliptic divergence equations, Annali della Scuola Normale Superiore, 1963, 189–205.
L. Nirenberg, Functional Analysis, Courant Institute of Math. Sciences, New York, 1984.
L. C. Piccinini, G. Stampacchia, and G. Vidossich, Ordinary Differential Equations in ${R^N}$ , Applied Mathematical Sciences 39, Springer-Verlag, New York, 1984.
R. Temam, Navier-Stokes Equations, North-Holland, Amsterdam, 1977.
S. Ukai, The Boltzmann-Grad limit and Cauchy-Kovalevskaya theorem, Recent topics in mathematics moving toward science and engineering. Japan J. Indust. Appl. Math., 18, 2001, no. 2, 383–392.
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