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## Feller evolution families and parabolic equations with form-bounded vector fields

1407.4861

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### Abstract

We show that the weak solutions of parabolic equation $$\partial_t u - \Delta u + b(t,x) \cdot \nabla u=0$$, $$(t,x) \in (0,\infty) \times \mathbb R^d$$, $$d \geqslant 3$$, for $$b(t,x)$$ in a wide class of time-dependent vector fields capturing critical singularities both in time and in spatial variables, constitute a Feller evolution family and, thus, determine a Feller process. Our proof uses a Moser-type iterative procedure and 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.

### Author and article information

###### Journal
2014-07-17
2015-09-06

35K10, 60G12
The argument in the proof of Lemma 4 expanded, reference list updated
math.AP math.PR
 ScienceOpen disciplines: Analysis, Probability