[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}


F = M + \sqrt{M}f.

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

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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