## [28.4.2016] Global classical solutions of the Vlasov-Fokker-Planck equation with local alignment forces

Young-Pil Choi studied in this paper the existence and large time behaviour of solution to the following nonlinear Vlasov-Fokker-Planck equation

$\partial_t F + v\cdot \nabla_xF + \nabla_v\cdot((u_F - v)F) = \Delta_vF$

where $F = F(x,v,t), x, v\in \mathbb R^d, t\geq 0$ and

$u_F(x,t) = \dfrac{\int_{\mathbb R^d}vF(x,v,t)dv}{\int_{\mathbb R^d}F(x,v,t)dv}.$

Assuming the initial data $F_0$ close enough to the global Maxwellian

$M = M(v) = \dfrac{1}{(2\pi)^{d/2}}exp\left(-\dfrac{|v|^2}{2}\right)$

the author proved that the solution to the mentioned problem exists globally in the classical sense, and converges to the Maxwellian $M$ with an algebraic rate, i.e.

$\|f(t)\|_{H^s} \leq C(\|f_0\|_{H^s} + \|f_0\|_{L^2_v(L^1)})(1+t)^{-d/4}$

where

$F = M + \sqrt{M}f$.

Moreover, if the spatial is periodic, then the convergence is exponential.