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July 2017 Feller evolution families and parabolic equations with form-bounded vector fields
Damir Kinzebulatov
Osaka J. Math. 54(3): 499-516 (July 2017).

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 $L^p$.

Citation

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Damir Kinzebulatov. "Feller evolution families and parabolic equations with form-bounded vector fields." Osaka J. Math. 54 (3) 499 - 516, July 2017.

Information

Published: July 2017
First available in Project Euclid: 7 August 2017

zbMATH: 1377.35128
MathSciNet: MR3685589

Subjects:
Primary: 35K10 , 60G12

Rights: Copyright © 2017 Osaka University and Osaka City University, Departments of Mathematics

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Vol.54 • No. 3 • July 2017
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