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Damir Kinzebulatov

Publications and source records attributed to Damir Kinzebulatov.

18 recordsLinked to original sources
arXiv2026

Kinetic SDEs and kinetic Fokker-Planck equations with critical Morrey-type drifts, and local blow-ups

We develop a Sobolev theory for the kinetic Fokker-Planck equation subject to a singular Morrey-form $L^\infty+M^q_K$ force field which can produce local blow-ups due to strong attraction to a submanifold. Our approach is based on the fractional representation for the Duhamel series of the equation and on establishing kinetic Adams-type estimates. Accordingly, we construct weak solutions of Brownian-driven kinetic SDEs in a critical Morrey set-up, obtain kinetic Krylov-type estimates and a strongly continuous Feller propagator, and establish uniqueness criteria for a large class of weak solutions.

math.PR↗
arXiv2026

Non-local SDEs, critical drifts and local blow-ups

The paper is concerned with $α$-stable SDEs with singular time-inhomogeneous general drift. Our drifts satisfy a condition that is close to the minimal possible scaling-invariance and can introduce local blow-ups of the type arising in some particle systems with strong attracting interactions.

math.PR↗
arXiv2024

On particle systems and critical strengths of general singular interactions

For finite interacting particle systems with strong repulsing-attracting or general interactions, we prove global weak well-posedness almost up to the critical threshold of the strengths of attracting interactions (independent of the number of particles), and establish other regularity results, such as a heat kernel bound in the regions where strongly attracting particles are close to each other. Our main analytic instruments are a variant of De Giorgi's method in $L^p$ with appropriately chosen large $p$, and an abstract desingularization theorem.

math.PR↗
arXiv2024

Non-local parabolic equations with singular (Morrey) time-inhomogeneous drift

We obtain Sobolev regularity estimates for solutions of non-local parabolic equations with locally unbounded drift satisfying some minimal assumptions. These results yield Krylov bound for the corresponding Feller stable process as well as some a priori regularity estimates on solutions of McKean-Vlasov equations. A key element of our arguments is a parabolic operator norm inequality that we prove using some ideas of Adams and Krylov.

math.AP↗
arXiv2024

Laplacian with singular drift in a critical borderline case

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 $L^\infty$.

math.AP↗
arXiv2022

On divergence-free (form-bounded type) drifts

We develop regularity theory for elliptic Kolmogorov operator with divergence-free drift in a large class (or, more generally, drift having singular divergence). A key step in our proofs is "Caccioppoli's iterations", used in addition to the classical De Giorgi's iterations and Moser's method.

math.AP↗
arXiv2022

On the vanishing of Green's function, desingularization and Carleman's method

The subject of the present paper is the phenomenon of vanishing of the Green function of the operator $-Δ+ V$ on $\mathbb R^3$ at the points where a potential $V$ has positive critical singularities. More precisely, imposing minimal assumptions on $V$ (i.e. the form-boundedness), we obtain an upper bound on the order of vanishing of the Green function. As a by-product of our proof, we improve the existing results on the strong unique continuation for eigenfunctions of $-Δ+ V$ in dimension $d=3$.

math.AP↗
arXiv2021

Sharp solvability for singular SDEs

The attracting inverse-square drift provides a prototypical counterexample to solvability of singular SDEs: if the coefficient of the drift is larger than a certain critical value, then no weak solution exists. We prove a positive result on solvability of singular SDEs where this critical value is attained from below (up to strict inequality) for the entire class of form-bounded drifts. This class contains e.g. the inverse-square drift, the critical Ladyzhenskaya-Prodi-Serrin class. The proof is based on a $L^p$ variant of De Giorgi's method.

math.PR↗
arXiv2021

Stochastic transport equation with singular drift

We prove existence, uniqueness and Sobolev regularity of weak solution of the Cauchy problem of the stochastic transport equation with drift in a large class of singular vector fields containing, in particular, the $L^d$ class, the weak $L^d$ class, as well as some vector fields that are not even in $L_{\rm loc}^{2+\varepsilon}$ for any $\varepsilon>0$.

math.PR↗
arXiv2018

$\mathcal W^{α, p}$ and $C^{0,γ}$ regularity of solutions to $(μ- Δ+ b \cdot \nabla)u=f$ with form-bounded vector fields

We consider the operator $-Δ+b \cdot \nabla$ with $b:\mathbb R^d \rightarrow \mathbb R^d$ ($d \geq 3$) in the class of form-bounded vector fields (containing vector fields having critical-order singularities), and characterize quantitative dependence of the $\mathcal W^{1+\frac{2}{q},p}$ ($2 \leq p < q$) and the $C^{0,γ}$ regularity of solutions to the corresponding elliptic equation in $L^p$ on the value of the form-bound of $b$.

math.AP↗
arXiv2016

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

We show that the weak solutions of parabolic equation $\partial_t u - Δu + b(t,x) \cdot \nabla u=0$, $(t,x) \in (0,\infty) \times \mathbb R^d$, $d \geqslant 3$, 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 $L^p$.

math.AP↗
arXiv2015

Strong Feller processes with measure-valued drifts

We construct a strong Feller process associated with $-Δ+ σ\cdot \nabla$, with drift $σ$ in a wide class of measures (weakly form-bounded measures, e.g. combining weak $L^d$ and Kato class measure singularities), by exploiting a quantitative dependence of the smoothness of the domain of an operator realization of $-Δ+ σ\cdot \nabla$ generating a holomorphic $C_0$-semigroup on $L^p(\mathbb R^d)$, $p>d-1$, on the value of the form-bound of $σ$. Our method admits extension to other types of perturbations of $-Δ$ or $(-Δ)^{\fracα{2}}$, e.g. to yield new $L^p$-regularity results for Schrödinger operators with form-bounded measure potentials.

math.AP↗
arXiv2015

A new approach to the $L^p$-theory of $-Δ+ b\cdot\nabla$, and its applications to Feller processes with general drifts

We develop a detailed regularity theory of $-Δ+b\cdot\nabla$ in $L^p(\mathbb R^d)$, for a wide class of vector fields. The $L^p$-theory allows us to construct associated strong Feller process in $C_\infty(\mathbb R^d)$. Our starting object is an operator-valued function, which, we prove, coincides with the resolvent of an operator realization of $-Δ+ b\cdot \nabla$, the generator of a holomorphic $C_0$-semigroup on $L^p(\mathbb R^d)$. Then the very form of the operator-valued function yields crucial information about smoothness of the domain of the generator.

math.AP↗
arXiv2007

On well-posedness of the linear Cauchy problem with the distributional right-hand side and discontinuous coefficients

We prove the well-posedness of the Cauchy problem for the linear differential system of the form $x^{\prime}-A(t)x=f$, where $f$ is a distribution and $A$ possesses at most first-kind discontinuities together with all its derivatives defined almost everywhere. The left-hand side of this system contains the product of a distribution and, in general, a discontinuous function, which is undefined in the classical space of the distributions with the smooth test functions $\mathcal D'$, so the Cauchy problem has no solution in $\mathcal D'$. In what follows, we cosider this system in the space of distributions with the discontinuous test functions, whose elements admit continuous and associative multiplication by functions possessing at most first-kind discontinuities (together with all their derivatives defined almost everywhere), and show that there exists the unique solution of the Cauchy problem which depends continuously on $f$.

math.FA↗