[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.

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About baotangquoc

Lecturer School of Applied Mathematics and Informatics Hanoi University of Science and Technology No 1, Dai Co Viet Street, Hanoi
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