A MINIMIZATION FORMULATION OF A BI-KINETIC SHEATH

The mathematical description of the interaction between a plasma and a solid surface is a major issue that still remains challenging. In this paper, we model this interaction as a stationary and bi-kinetic Vlasov-Poisson-Ampere boundary value problem with boundary conditions that are consistent with the physics. In particular, we show that the wall potential can be determined from the ampibolarity of the particle flows as the unique solution of a non linear equation. Based on variational techniques, our analysis establishes the well-posedness of the model, provided that the incoming ion distribution satisfies a moment condition that generalizes the historical Bohm criterion of plasma physics. Quantitative estimates are also given, together with numerical illustrations that validate the robustness of our approach.

[1]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[2]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[3]  R. Marshak.  The Variational Method for Asymptotic Neutron Densities , 1947 .

[4]  R. K. Wakerling,et al.  The characteristics of electrical discharges in magnetic fields , 1949 .

[5]  J. G. Laframboise.  Theory of spherical and cylindrical Langmuir probes in a collisionless , 1966 .

[6]  A. W. Trivelpiece,et al.  Introduction to Plasma Physics , 1976 .

[7]  D. Kinderlehrer,et al.  An introduction to variational inequalities and their applications , 1980 .

[8]  R. Płuciennik.  The superposition operator in Musielak-Orlicz spaces of vector-valued functions , 1987 .

[9]  P. Lions,et al.  Ordinary differential equations, transport theory and Sobolev spaces , 1989 .

[10]  Claude Greengard,et al.  A boundary-value problem for the stationary vlasov-poisson equations: The plane diode , 1990 .

[11]  K. Riemann,et al.  The Bohm criterion and sheath formation , 1991 .

[12]  Otared Kavian,et al.  Introduction à la théorie des points critiques : et applications aux problèmes elliptiques , 1993 .

[13]  Pierre-Henri Maire.  Etablissement et comparaison de modeles fluides pour un plasma faiblement ionise quasi-neutre. Determination des conditions aux limites a la paroi , 1996 .

[14]  SOME REMARKS ON A STATIONARY VLASOV–POISSON SYSTEM WITH SOURCE TERM ARISING IN ION BEAM NEUTRALIZATION , 2001 .

[15]  Giovanni Manfredi,et al.  Numerical study of plasma–wall transition in an oblique magnetic field , 2001 .

[16]  Philippe Ghendrih,et al.  The Plasma Boundary of Magnetic Fusion Devices , 2001 .

[17]  T. E. Sheridan.  Solution of the plasma-sheath equation with a cool Maxwellian ion source , 2001 .

[18]  YAN GUO,et al.  The Dynamics of a Plane Diode , 2004, SIAM J. Math. Anal..

[19]  Thomas Lachand-Robert,et al.  Some properties of monotone rearrangement with applications to elliptic equations in cylinders , 2004 .

[20]  M. Slemrod,et al.  A Geometric Level-Set Formulation of a Plasma-Sheath Interface , 2004, math-ph/0409040.

[21]  S. Devaux,et al.  Magnetized plasma–wall transition—consequences for wall sputtering and erosion , 2008 .

[22]  C. Hegna,et al.  Reply to comment on ‘Kinetic theory of the presheath and the Bohm criterion’ , 2010, 1212.4226.

[23]  Ronald F. Boisvert,et al.  NIST Handbook of Mathematical Functions , 2010 .

[24]  L. Evans,et al.  Partial Differential Equations , 1941 .

[25]  D. Tskhakaya,et al.  Magnetized plasma-wall transition layer with cold ions , 2010, Journal of Plasma Physics.

[26]  R. Khanal,et al.  A kinetic trajectory simulation model for magnetized plasma sheath , 2012 .

[27]  H. Kohno,et al.  Radio-frequency sheath-plasma interactions with magnetic field tangency points along the sheath surface , 2013 .