Stochastic equations with time-dependent singular drift

Kinzebulatov, Damir; R., Madou, K.

doi:10.1016/j.jde.2022.07.042

Cited by 8 publications

(2 citation statements)

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“…2. Theorem 1 can be extended to non-homogeneous parabolic equation with form-bounded right-hand side, moreover, the corresponding gradient estimates can be localized, which, together with Theorem 2, allows to prove the following result (see [3] for details). Theorem 3.…”

Section: Introduction and Main Resultsmentioning

confidence: 93%

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On the theory of the Kolmogorov operator in the spaces $L^p$ and $C_\infty$

Kinzebulatov¹,

Semenov²

2020

Annali Scuola Normale Superiore - Classe Di Scienze

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We establish the basic results concerning the problem of constructing operator realizations of the formal differential expression ∇ · a · ∇ − b · ∇ with measurable matrix a and vector field b having critical-order singularities as the generators of Markov semigroups in L p and C∞. 2000 Mathematics Subject Classification. 31C25, 47B44 (primary), 35D70 (secondary). Key words and phrases. Markov semigroups, form-bounded vector fields, regularity of solutions, Feller semigroups. The research of D.K. is supported by the Natural Sciences and Engineering Research Council of Canada and the Fonds de recherche du Québec -Nature et technologies. 1 2 2− d−2 d √ δ , ∞[. 1 1 The maximal interval of quasi bounded solvability for A − V , with 0 ≤ V ≤ δA + c(δ), 0 < δ < 1, isÎm := ]r(δ), r ′ (δ)[, r ′ (δ) := 2 1− √ 1−δ d d−2 . This was proved in 1995 by Yu.A. Semënov, based on ideas set forth in [31]. The fact that the semigroup associated with the Schrödinger operator −∆ − V , V ∈ L d 2 ,∞ , can be extended to a C0 semigroup on L r (R d ) for every r ∈Îm was first observed in [17]. * → reads: if b ∈ Ws (s > 1), then b ∈ F δ 2 with δ = δ( b 2 Ws ) < ∞.

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Section: Introduction and Main Resultsmentioning

confidence: 93%

“…The vector fields in class F δ,g are called form-bounded. This class contains the well known critical Ladyzhenskaya-Prodi-Serrin class, as well as vector fields that can have stronger singularities, see examples in [3,4].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%