Vlasov Equation: From Derivation to Quasilinear Approximationand weak turbulence.Claude BardosFebruary 25, 2021This talk is devoted to the quasi linear approximation for solutions of the Vlasov equation avery popular tool in Plasma Physic cf. [4] which proposes, for the quantity:q(t,v)=SRdvf(x,v,t)dx),(1)the solution ofa parabolic, linear or non linear evolution equation∂tq(t,v)− ∇v(D(q,t;v)∇vq)=0(2)Since the Vlasov equation is an hamiltonian reversible dynamic while (2) is not reversible wheneverD(q,t,v)~=0 the problem is subtle. Hence I did the following things :1. Give some sufficient conditions, in particular in relation with the Landau damping that wouldimplyD(q,t,v)≃0.a situation where the equation (2) withD(q,t;v)=0 does not providesa meaning full approximation.2. Building on contributions of [7] and coworkers show the validity of the approximation (2) forlarge time and for a family of convenient randomized solutions. This is justified by the factthat the assumed randomness law is in agreement which what is observed by numerical orexperimental observations ( cf. [1]).3. In the spirit of a Chapman Enskog approximation formalize the very classical physicist ap-proach ( cf. [6] pages 514-532) one can show [3] that under analyticity assumptions thisapproximation is valid for short time. As in [6] one of the main ingredient of this construc-tion is based on the spectral analysis of the linearized equation and as such it makes a linkwith a classical analysis of instabilities in plasma physic.RemarksIn some sense the two approaches are complementary . The short time is purely deterministicand the stochastic is based on the intuition that over longer time the randomness will take overof course the transition remains from the first regime to the second remains a challenging openproblem. The similarity with the transition to turbulence in fluid mechanic is striking . It isunderlined by the fact that the tensorlim→0D(t,v)=lim→0SdxSt20dσE(t,x+σv)⊗E(t−2σ,x)which involves the electric fields here plays the role of the Reynolds stress tensor.2 Obtaining, for some macroscopic description, a space homogenous equation for the velocitydistribution is a very natural goal. Here the Vlasov equation is used as an intermediate step inthe derivation. And more generally it appears as an example of weak turbulence. In particulardefining what would be the physical natural probability seems related to the derivation of e of theLenard-Balescu equation as done in [5]. --------------------------------------------------------------------------------------------------------------------------------------------------------------------- References: [1] C. Bardos, N. BesseDiffusion limit of the Vlasov equation in the weak turbulent regime.submitted.[2] C. Bardos, N. BesseAbout the derivation of the quasilinear approximation in plasma physicsarXiv:2011.08085.[3] C. Bardos, N. BesseIn preparation[4] P. Diamond, U. Frisch and Y. Pomeau Guest editorsPlasma Physic in the 20 century astold by players.(201843.[5] M. Duerinckx, L.Saint -RaymondLenard-Balescu correction to mean-field theory .arXiv:1911.10151.[6] N.A. Krall, A.W. TrivelpiecePrinciples of plasma physics( 1973 ) McGraw-Hill.[7] F. Poupaud, A. Vasseur,Classical and quantum transport in random media, J. Math. PuresAppl.82(2003), 711–748 .