Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach. (Q2574514)

The density of particles \(f(v,t)\geq 0\) with velocity \(v\in \mathbb R^d\) at time \(t\in \mathbb R^+\) is described by the Landau equation \[ \frac {\partial f}{\partial t} (v,t)= \frac {1}{2}\sum _{i,j=1}^d \frac {\partial }{\partial v_i}\left \{\int _{\mathbb R^d}dv_*a_{ij}(v-v_*)\left [f(v_*,t)\frac {\partial f}{\partial v_j}(v,t)-f(v,t)\frac {\partial f}{\partial v_{*j}}(v_*,t)\right ]\right \}, \] where the matrix \(a\) has the form \(a_{i,j}=h(| z| ^2)(| z| ^2\delta _{ij} - z_iz_j)\), and \(h\) is a continuous function evolving between \(m,M>0\). Existence of a weak function-solution of class \(\mathcal C^\infty \) is proven by means of Malliavin calculus in case when the initial datum is a probability measure.