A geometric level-set formulation of a plasma-sheath interface

Archive for Rational Mechanics and Analysis

Volumn 178, Issue 1, 2005, Pages 81-123

  Feldman, Mikhail ^a   Ha, Seung Yeal ^b   Slemrod, Marshall ^a  

^a UNIVERSITY OF WISCONSIN MADISON   (United States)

^b Seoul National University   (South Korea)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 24944539889     PISSN: 00039527     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00205-005-0368-3     Document Type: Article

References (36)

1. ANDREWS, J.G., VAREY, R.H.: Sheath growth in a low pressure plasma. Physics of Fluids 14, 339-343 (1971)
2. ANGENENT, S.B., GURTIN, M.E.: Anisotropic motion of a phase interface. Well-posedness of the initial value problem and qualitative properties of the interface. J. Reine Angew. Math. 446, 1-47 (1994)
3. 2 well-posedness of planar Div-Curl systems. Arch. Rational Mech. Anal. 160, 91-134 (2001)
4. BEN ABDALLAH, N.: Convergence of the Child-Langmuir asymptotics of the Boltzmann equation of semiconductors. SIAM J. Math. Anal. 27, 92-109 (1996)
5. BEN ABDALLAH, N.: The Child-Langmuir regime for electron transport in a plasma including a background of positive ions. Math. Models Methods Appl. Sci. 4, 409-438 (1994)
6. BEN ABDALLAH, N., DEGOND, P., MARKOWICH, P.: The quantum Child-Langmuir problem. Nonlinear Anal. 31, 629-648 (998)
7. BEN ABDALLAH, N., DEGOND, P.: The Child-Langmuir law for the Boltzmann equation of semiconductor. SIAM J. Math. Anal. 26, 364-398 (1995)
8. CANIC, S., KEYFITZ, B.L., LIEBERMAN, G.M.: A proof of existence of perturbed steady transonic shocks via a free boundary problem. Comm. Pure Appl. Math 53, 484-511 (2003)
9. CHEN, G.-Q., FELDMAN, M.: Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type. J. Amer. Math. Soc. 16, 461-494 (2003)
10. CIPOLLA, J.W., SILEVITCH, M.B.: On the temporal development of a plasma sheath. Plasma Physics 25, 373-389 (1981)
11. CONRAD, J.R., RADTKE, J.R., DODD, R.A., WORZAKA, F.J., TRAN, N.C.: Plasma source ion-implantation technique for surface modification of materials. J. Appl. Phys. 62, 4591-4596 (1987)
12. COURANT, R., HILBERT, D.: Methods of mathematical physics, vol II, (Partial differential equations), A Wiley-Interscience Publication, New York: 1989
13. DEGOND, P., JAFFARD, S., POUPAUD, F., RAVIART, P.A.: The Child-Langmuir asymptotics of the Vlasov-Poisson equation for cylindrical or spherically symmetric diodes. I. (Statement of the problem and basic estimates). Math. Methods Appl. Sci. 19, 287-312 (1996)
14. DEGOND, P., PARZANI, C., VIGNAL, M.-H.: Un modèle d'expansion de plasma dans le vide. Comptes Rendus Academie Sciences (Paris), Series 1 335, 399-404 (2002)
15. DEGOND, P., RAVIART, P.A.: On a penalization of the Child-Langmuir emission condition for the one-dimensional Vlasov-Poisson equation. Asymptotic Analysis 6, 1-27 (1992)
16. DEGOND, P., RAVIART, P.A.: An asymptotic analysis of the one-dimensional Vlasov-Poisson system: the Child-Langmuir law. Asymptotic Analysis 4, 187-214 (1991)
17. DOUGLIS, A.: Some existence theorems for hyperbolic systems of partial differential equations in two independent variables. Comm. Pure and Appl. Math. 5, 119-154 (1952)
18. EVANS, C.E.: Partial differential equations. Graduate Studies in Mathematics 19, AMS, 1998
19. FRANKLIN, R.N., OCKENDON, J.R.: Asymptotic matching of plasma and sheath in an active low pressure discharge. Journal of Plasma Physics 4, 3521-3528 (1970)
20. FRIEDRICHS, K.O.: Nonlinear hyperbolic differential equations for functions of two independent variables. Amer. J. Math. 70, 555-589 (1948)
21. GIERLING, J., RIEMANN, K.-U.: Comparison of a consistent theory of radio frequency sheaths with step models. Journal of Applied Physics 83, 3521-3528 (1988)
22. GILBARG, D., TRUDINGER, N.S.: Elliptic partial differential equations of second order. Grundlehren der mathematischen Wissenschaften 224 2nd edition, Berlin, Springer-Verlag, 1983
23. GREENGARD, C., RAVIART, P.A.: A boundary value problem for the stationary Vlasov-Poisson equations: the plane diode. Comm. Pure Appl. Math. 43, 473-507 (1990)
24. GURTIN, M.E.: Configurational forces as basic concepts of continuum physics. Applied Mathematical Sciences 137, Springer-Verlag, New York 2000
25. HA, S.-Y., SLEMROD, M.: Global existence of plasma-ion sheaths and their dynamics. Commun. Math. Phys. 238, 149-186 (2003)
26. KIM, Y.-W., KIM, G.-H., HAN, S., LEE, Y., CHO, J., RHEE, S.Y.: Measurement of sheath expansion in plasma source ion implantation. Surface and Coatings Technology 136, 97-101 (2001)
27. LI, T.-T., YU, W.-C.: Boundary value problems for quasi-linear hyperbolic systems. Duke University Mathematics Series 5, Durham, N.C. p.74 (1985)
28. LIEBERMAN, M.A., LICHTENBERG, A.J.: Principles of plasma discharge and materials processing. A Wiley-Interscience Publication, New York: (1994)
29. NOURI, A. : An Euler-Poisson system in plasma. J. Austral. Math. Soc. Ser: B 41, 442-450 (2000)
30. OSHER, S., FEDKIW, R.: Level set methods and dynamic implicit surfaces. Applied Mathematical Sciences 153, Springer-Verlag, New York (2003)
31. RIEMANN, K.-U.: Theory of collisional presheath in an oblique magnetic field. Phys. Plasmas 1, 552-558 (1994)
32. RIEMANN, K.-U.: The Bohm criterion and sheath formation.J Phys. D: Appl. Phys. 24, 493-518 (1991)
33. RIEMANN, K.-U., DAUBE, Th.: Analytical model of the relaxation of a collisionless ion matrix sheath. J. Appl. Phys. 86, 1201-1207 (1999)
34. SCHOCHET, S.: Private communication
35. SCHOCHET, S.: The compressible Euler equations in a bounded domain: Existence of solutions and the incompressible limit. Commum. Math. Phys. 104, 49-75 (1986)
36. WIDNER, M., ALEXEFF, I., JONES, W.D.: Ion acoustic wave excitation and ion sheath evolution. Physics of Fluids 13, 2532-2540 (1970)