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January 2022
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2021-GdT ITER - C. Bardos
Mardi 19 janvier à 11h : Claude Bardos (Université de Paris)
Quasilinear Approximation of the Vlasov Equation
The object of this talk is to report on a joint ongoing program with Nicolas Besse Prof. Observatoire de la Cote d’Azur et Université de Nice.
The quasi linear approximation for solutions of the Vlasov equation is a very popular tool in Plasma Physic cf. [1] which proposes, for the quantity :
q ( ∫f (x, v, t)dx) , (1)
the solution of a parabolic, linear or non linear evolution equation
∂_t q(t, v) − ∇_v (D(q, t ; v)∇_v q) = 0 . (2)
Since the Vlasov equation is a hamiltonian reversible dynamic while (2) is not reversible whenever D(q, t, v) ≠ 0 the problem is subtle. Hence I will do the following things :
1. Give some sufficient conditions, in particular in relation with Landau damping that will imply that D(q, t, v) ≃ 0 .
2. Building on contributions of [3] and coworkers show the validity of the approximation (2) for large time and for a family of convenient randomized solutions.
3. In the spirit of a Chapman Enskog approximation formalize the very classical physicist approach for a short time approximation under analyticity assumptions.
References
[1] Plasma Physic in the 20th century as told by players. Guest editors.:P. Diamond, U. Frisch and Y. Pomeau. Vol 43 , 4-5, December 2018.
[2] N.A. Krall, A.W. Trivelpiece Principles of plasma physics, McGraw-Hill, 1973.
[3] F. Poupaud, A. Vasseur, Classical and quantum transport in random media, J. Math. Pures Appl. 82 711–748 (2003).