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Laplacian with singular drift in a critical borderline case

Damir Kinzebulatov

Abstract

We establish well-posedness and regularity for parabolic diffusion equation in the case when the singularities of a general drift reach the critical magnitude. The latter dictates the need to work in an Orlicz space situated between all $L^p$ and $L^\infty$.

Laplacian with singular drift in a critical borderline case

Abstract

We establish well-posedness and regularity for parabolic diffusion equation in the case when the singularities of a general drift reach the critical magnitude. The latter dictates the need to work in an Orlicz space situated between all and .
Paper Structure (6 sections, 1 theorem, 82 equations)

This paper contains 6 sections, 1 theorem, 82 equations.

Table of Contents

  1. Introduction and result
  2. Introduction
  3. Main result. Critical case $\delta=4$
  4. Forward Kolmogorov equation and absence of blow up
  5. Further remarks
  6. Proof of Theorem \ref{['thm1']}

Key Result

Theorem 1

Let $b \in \mathbf{F}_\delta=\mathbf{F}_\delta(\Pi^d)$, $0<\delta \leq 4$. The following are true: (i) There exists a strongly continuous quasi contraction Markov semigroup $e^{-t\Lambda(b)}$ on $L_\Phi$, such that, for every $f \in C^\infty$, The generator $\Lambda(b)$ of the semigroup is the appropriate operator realization of the formal operator $-\Delta + b \cdot \nabla$ in $L_\Phi$, so $u(

Theorems & Definitions (1)

  • Theorem 1