Osaka Journal of Mathematics

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

Damir Kinzebulatov

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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 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


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References

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Osaka University and Osaka City University, Departments of Mathematics

Osaka Journal of Mathematics

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