On the kinetic Fokker-Planck equation in the half-space with absorbing barriers (Q2788638)

This paper is devoted to an investigation of the class of partial differential equations of Fokker-Planck type. These equations serve as mathematical models representing various diffusion processes and so called Brownian motion. They are applicable in different sciences, for instance, biology, ecology, chemistry, gas and stellar dynamics, statistical physics, economics and so on. The mathematical theory of the Brownian particles comes from the studies of Rayleigh (1890), and later Einstein (1905) and Smoluchowski (1906). In 1930, Uhlenbeck and Ornstein derived the master equation for the distribution of particles. In 1940, the mathematical model of the Brownian motion in a force field was generalized by Kramer.NEWLINENEWLINEIn the present paper, the authors consider a mathematical model of randomly accelerated particles in the three-dimensional half-space \(\mathbb{R}^3\) (\(x^3 > 0 \)). It is introduced a function \(f = f (x, v, t)\), that is the probabilistic distribution function of the particles w. r. t. their position \(x\) and velocity \(v\) at a point \(t\) of the time. Note that here each one of the particles changes its velocity randomly, and the particles after the collision with the boundary (called absorbing barrier) are discarded. Then, the governing equation is just the Fokker-Planck equation with the absorbing boundary condition, i.e. it is an initial and boundary value problem of the form: NEWLINE\[NEWLINE \partial_tf(x,v,t)+v\cdot \nabla_xf(x,v,t)=\Delta_{v}f(x,v,t), \;\;(x,v,t)\in\mathbb{R}_{+}^{3}\times\mathbb{R}^{3}\times (0,T), NEWLINE\]NEWLINE \((x,v,0)=f_0(x,v)\), for \((x,v)\in \mathbb{R}_{+}^{3}\times\mathbb{R}^{3}\) (initial condition), and \(f(x,v,t)=0\), for \(x\in\partial\mathbb{R}_{+}^{3}\), \(v_3>0\) (boundary condition). Here the initial function \(f_0\) is bounded and integrable, i.e. it is in the class \(L^1\cap\L^{\infty }(\mathbb{R}_{+}^{3}\times\mathbb{R}^{3})\). The main result is that there exists a unique weak solution \(f\in L^{\infty }([ 0 , T ] ; L^1\cap L^{\infty } (\mathbb{R}_{+}^{3}\times\mathbb{R}^{3}))\) of the above stated mixed problem. Moreover, the weak solution \(f (t)\) satisfies the bounds \(\| f(t)\|_{L^{\infty }}\leq \| f_0\|_{L^{\infty }} \), and \(\| f(t)\|_{L^{1}}\leq \| f_0\|_{L^{1}} \) for each \(t\in[0,T]\) (\(T>0\)). Next, a result of regularity of weak solutions is proved as well.