ИСТИНА

Vishik–Lyusternik’s method and the inverse problem for plasma equilibrium in a tokamak Alexandre Demidov Moscow State University, Moscow 119992, Russia alexandre.demidov@mtu-net.ru Control over thermonuclear fusion reactions (including suppression of instabilities of the plasma discharge) depends essentially on how well the information about the current density through plasma is taken into account. In the case cylindrical approximation (when the tokamak (toroidal magnetic) chamber and the resulting plasma discharge are modeled in the form of infinite cylinders S £ R and ! £ R with simply connected cross-sections S b R2 and ! b S), the required current distribution is given by the mapping fu : ! 3 (x; y) 7! f ¡ u(x; y) ¢ ¸ 0; where the required functions u 2 C2(!) and f are as follows: @2u(x; y) @x2 + @2u(x; y) @y2 = f ¡ u(x; y) ¢ in ! ; and u = 0 on ° = @!; sup P2° ¯¯¯ @u @º (P) ¡ ©(P) ¯¯¯ · ¸ sup P2° ¯¯¯ ©(P) ¯¯¯ ; Z ° @u @º d° = 1 = the total current : Here, ° = ! n ! is the boundary of the domain !, ¸ ¸ 0 is small parameter, º is the outward unit normal to the curve ° = @! (with respect to the domain !). Both the function © and the curve ° = @! (and hence the domain !) may be regarded as known: they are determinable from measurements of the magnetic field at the tokamak chamber @S: Within the class of affine functions f : u 7! f(u) = au+b, Vishik–Lyusternik’s method is capable of showing that ¯¯¯¯¯ @u @º ¯¯¯¯ s2° ¡ à 1 j°j ¡ k(s) ¡ j°j¡1 R ° k(s) ds 2j°j p a !¯¯¯¯¯ · C°(a) p a ; C°(a) ! 0 as a ! 1; ¯¯¯¯¯ d da @u @º ¯¯¯¯ s2° ¡ k(s) ¡ j°j¡1 R ° k(s) ds 4j°j a3=2 ¯¯¯¯¯ · C°(a) a3=2 ; C°(a) ! 0 as a ! 1; where k(s) is the curvature of ° at s 2 °. Using these asymptotic relations, it follows that there is only one affine distribution fu (for a large class of domains !) if ¸ = 0. However, for any arbitrarily small ¸ > 0, there is an infinite number ffju gj2N of distributions, for which kfj1 u k ¿ kfj2 u k, j1 6= j2, where kfju k = max (x;y)2! ¯¯ fju (x; y) ¯¯: It is shown that all these different distributions are necessarily members of a sequence converging to the ±-function supported on ° (the so-called skinned current). Hence these distributions are not essentially different from the physical standpoint. Two truly physically essentially different current distributions f1 u and f2 u are found in the class of polynomials f : u 7! f(u) = P3 m=0 amum of third degree (see Russian J. Math. Physics, 17 (1), 56–65 (2010) and Asymptotic Analysis, 74 (1), 95–121 (2011)).