Critical thresholds in Euler-Poisson equations

Indiana University Mathematics Journal

Volumn 50, Issue SUPPL., 2001, Pages 109-157

  Engelberg, Shlomo ^a   Liu, Hailiang ^a   Tadmor, Eitan ^b  

^a Jerusalem Coll Technol Machon L   (Israel)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Breakdown; Critical threshold; Euler Poisson equations; Global existence; Isotropic flow

Indexed keywords

EID: 0013184506     PISSN: 00222518     EISSN: None     Source Type: Journal    
DOI: 10.1512/iumj.2001.50.2177     Document Type: Article

References (27)

1. U.M. ASCHER, PETER A. MARKOWICH, P. PIETRA & C. SCHMEISER, A phase plane analysis of transonic solutions for the hydrodynamic semiconductor model, Math. Models Methods Appl. Sci. 1 (1991), 347-376.
2. U. BRAUER, A. RENDAL & O. REULA, The cosmic no-hair theorem and the non-linear stability of homogeneous Newtonian cosmological models, Class. Quantum Grav. 11 (1994), 2283-2296.
3. C. CERCIGNANI, The Boltzmann Equation and Its Applications, Springer-Verlag, New York, 1988.
4. E.A. CODDINGTON & N. LEVINSON, Theory of Ordinary Differential Equations, Robert E. Krieger Publishing Company, Malabar, Florida, 1987.
5. G.-Q. CHEN & D. WANG, Convergence of shock capturing scheme for the compressible Euler-Poisson equations, Comm. Math. Phys. 179 (1996), 333-364.
6. J. DOLBEAULT & G. REIN, Time-dependent rescalings and Lyapunov functionals for the Vlasov-Poisson and Euler-Poisson systems, and for related models of kinetic equations, fluid dynamics and quantum physics, Mathematical Models and methods in Applied Sciences 11 (2001), 407-432.
7. S. ENGELBERG, Formation of singularities in the Euler-Poisson equations, Physica D 98 (1996), 67-74.
8. P. GAMBLIN, Solution régulière à temps petit pour l'équation d'Euler-Poisson, Comm. Partial Differential Equations 18 (1993), 731-745.
9. I. GASSER, C-K LIN & P.A. MARKOWICH, A review of dispersive limits of (non)linear Schrödinger-type equations, preprint, 2000.
10. 3+1, Comm. math. Phys. 195 (1998), 249-265.
11. D. HOLM, S.F. JOHNSON & K.E. LONNGREN, Expansion of a cold ion cloud, Appl. Phys. Lett. 38 (1981), 519-521.
12. J.D. JACKSON, Classical Electrodynamics, 2nd ed., Wiley, New York, 1975.
13. F. JOHN, Partial Differential Equations, Fourth Edition, Springer-Verlag, New York 1982.
14. S. JUNCA & M. RASCLE, Relaxation of the isothermal Euler-Poisson system to the drifi-diffusion equations, Quart. Appl. Math. 58 (2000), 511-521.
15. T. LUO, R. NATALINI & Z. XIN, Large time behavior of the solutions to a hydrodynamic model for semiconductors, SIAM J. Appl. Math. 59 (1999), 810-830.
16. T. MAKINO, On a local existence theorem for the evolution of gaseous stars, In: Patterns and Waves (T. Nishida, M. Mimura & H. Fujii, eds.), North-Holland/Kinokuniya, 1986, pp. 459-479.
17. P.A. MARKOWICH, A non-isentropic Euler-Poisson Model for a Collisionless Plasma, Math. Methods Appl. Sci. 16 (1993), 409-442.
18. T. MAKINO & B. PERTHAME, Sur les solutions à symétrie sphérique de l'équation d'Euler-Poisson pour l'évolution d'étoiles gazeuses, Japan J. Appl. math. 7 (1990), 165-170.
19. P. MARCATI & R. NATALINI, Weak solutions to a hydrodynamic model for semiconductors and relaxation to the drift-diffusion equation, Arch. Rat. Mech. Anal. 129 (1995), 129-145.
20. P.A. MARKOWICH, C. RINGHOFER & C. SCHMEISER, Semiconductor Equations, Springer, Berlin, Heidelberg, New York, 1990.
21. T. MAKINO & S. UKAI, Sur l'éxistence des solutions locales de l'équation d'Euler-Poisson pour l'évolution d'étoiles gazeuses, J. Math. Kyoto Univ. 27 (1987), 387-399.
22. B. PERTHAME, Nonexistence of global solutions to the Euler-Poisson equations for repulsive forces, Japan J. Appl. Math. 7 (1990), 363-367.
23. P. ROSENAU, Evolution and breaking of ion-acoustic waves, Phys. Fluids 31 (1988), 1317-1319.
24. D. WANG, Global solutions and relaxation limits of Euler-Poisson equations, Z. Angew. Math. Phys., to appear.
25. _, Global solutions to the equations of viscous gas flows, Pro. Royal Soc. Edinburgh: Section A, to appear.
26. _, Global solutions to the Euler-Poisson equations of two-carrier types in one-dimension, Z. Angew. Math. Phys. 48 (1997), 680-693.
27. D. WANG & G.-Q. CHEN, Formation of singularities in compressible Euler-Poisson fluids with heat diffusion and damping relaxation, J. Diff. Eqs. 144 (1998), 44-65.