## Osaka Journal of Mathematics

- Osaka J. Math.
- Volume 54, Number 3 (2017), 499-516.

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

#### 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 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 $L^p$.

#### Article information

**Source**

Osaka J. Math., Volume 54, Number 3 (2017), 499-516.

**Dates**

First available in Project Euclid: 7 August 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ojm/1502092825

**Mathematical Reviews number (MathSciNet)**

MR3685589

**Zentralblatt MATH identifier**

1377.35128

**Subjects**

Primary: 35K10: Second-order parabolic equations 60G12: General second-order processes

#### Citation

Kinzebulatov, Damir. Feller evolution families and parabolic equations with form-bounded vector fields. Osaka J. Math. 54 (2017), no. 3, 499--516. https://projecteuclid.org/euclid.ojm/1502092825