Year: 2014
Type: article
Abstract: We show that the weak solutions of parabolic equation $\partial_t u - Δu + b(t,x) \cdot \nabla u=0$, $(t,x) \in (0,\infty) \times \mathbb R^d$, $d \geqslant 3$, for $b(t,x)$ in a wide class of time-dependent vector fields capturing critical order singularities, constitute a Feller evolution family and, thus, determine a Feller process. Our proof uses an a priori estimate on the $L^p$-norm of the gradient of solution in terms of the $L^q$-norm of the gradient of initial function, and an iterative procedure that moves the problem of convergence in $L^\infty$ to $L^p$.
Author: Kinzebulatov, Damir
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1819840666