Feller evolution families and parabolic equations with form-bounded vector fields
Damir Kinzebulatov · Institutional Repositories DataBase (IRDB) · 2017
e show that the weak solutions of parabolic equation ∂ₜu − Δu + b(t, x) · ∇u = 0, (t, x) ∈ (0,∞) × ℝ^d, d?≽ 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^∞ to L^p.