EXPONENTIAL CONVERGENCE TOWARD EQUILIBRIUM FOR HOMOGENEOUS FOKKER-PLANCK-TYPE EQUATIONS

We consider homogeneous solutions of the Vlasov-Fokker-Planck equation in plasma theory proving that they reach the equilibrium with a time exponential rate in various norms. By Csiszar-Kullback inequality, str ong L-1-convergence is a consequence of the 'sharp' exponential decay of relative entropy and relative Fisher information. To prove exponent ial strong decay in Sobolev spaces H-k, k greater than or equal to 0, We take into account the smoothing effect of the Fokker-Planck kernel. Finally, we prove that in a metric for probability distributions rece ntly introduced in [9] and studied in [4, 14] the decay towards equili brium is exponential at a rate depending on the number of moments boun ded initially. Uniform bounds on the solution in various norms are the n combined, by interpolation inequalities, with the convergence in thi s weak metric, to recover the optimal rate of decay in Sobolev spaces. (C) 1998 by B. G. Teubner Stuttgart-John Wiley & Sons, Ltd.