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Mikhail V. Korobkov

Publications and source records attributed to Mikhail V. Korobkov.

15 recordsLinked to original sources
arXiv2026

On flows generated by square-integrable vector fields

For square-integrable divergence-free vector field $\boldsymbol{v}$ on $\mathbb{R}^d$ we prove that the following properties are equivalent: 1) the operator $A_0 ρ= \boldsymbol{v} \cdot \nabla ρ$ (where $ρ\in C^\infty_c(\mathbb{R}^d)$) is essentially skew-adjoint on $L^2(\mathbb{R}^d)$; 2) square-integrable (with respect to spatial variables) generalized solutions of the continuity equation are renormalized; 3) generalized square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique both forward an backward in time. We also construct a compactly supported bounded divergence-free vector field $\boldsymbol{v}\colon \mathbb{R}^3 \to \mathbb{R}^3$ for which square-integrable (with respect to spatial variables) solutions of the Cauchy problem for the corresponding continuity equation are unique forward, but not backward in time.

math.AP↗
arXiv2026

On the uniqueness and structural stability of Couette-Poiseuille flow in a channel for arbitrary values of the flux

We establish uniqueness and structural stability of a class of parallel flows in a 2D straight, infinite channel, under perturbations with either globally or locally bounded Dirichlet integrals. The significant feature of our result is that it does not require any restriction on the size of the flux characterizing the flow. Precisely, by extending and refining an approach initially introduced by J.B. McLeod, we demonstrate the continuous invertibility of the linearized operator at a generic Couette-Poiseuille solution that does not exhibit flow reversal. We then deduce local uniqueness of these solutions as well as their nonlinear structural stability under small external forces. Moreover, we prove the uniqueness of certain class of Couette-Poiseuille solutions ``in the large," within the set of solutions possessing natural symmetry. Finally, we bring an example showing that, in general, if the flow reversal assumption is violated, the linearized operator is no longer invertible.

math.AP↗
arXiv2025

The Nelson conjecture and chain rule property

Let $p\ge 1$ and let $\boldsymbol{v} \colon \mathbb R^d \to \mathbb R^d$ be a compactly supported vector field with $\boldsymbol{v} \in L^p(\mathbb R^d)$ and $\operatorname{div} \boldsymbol{v} = 0$ (in the sense of distributions). It was conjectured by Nelson that it $p=2$ then the operator $\mathsf{A}(ρ) := \boldsymbol{v} \cdot \nabla ρ$ with the domain $D(\mathsf A)=C_0^\infty(\mathbb R^d)$ is essentially skew-adjoint on $L^2(\mathbb R^d)$. A counterexample to this conjecture for $d\ge 3$ was constructed by Aizenmann. From recent results of Alberti, Bianchini, Crippa and Panov it follows that this conjecture is false even for $d=2$. Nevertheless, we prove that for $d=2$ the condition $p\ge 2$ is necessary and sufficient for the following chain rule property of $\boldsymbol{v}$: for any $ρ\in L^\infty(\mathbb R^2)$ and any $β\in C^1(\mathbb R)$ the equality $\operatorname{div}(ρ\boldsymbol{v}) = 0$ implies that $\operatorname{div}(β(ρ) \boldsymbol{v}) = 0$. Furthermore, for $d=2$ we prove that $\boldsymbol{v}$ has the renormalization property if and only if the stream function (Hamiltonian) of $\boldsymbol{v}$ has the weak Sard property, and that both of the properties are equivalent to uniqueness of bounded weak solutions to the Cauchy problem for the corresponding continuity equation. These results generalize the criteria established for $d=2$ and $p=\infty$ by Alberti, Bianchini and Crippa.

math.AP↗
arXiv2025

The steady Navier-Stokes equations in a system of unbounded channels with sources and sinks

The steady motion of a viscous incompressible fluid in a junction of unbounded channels with sources and sinks is modeled through the Navier-Stokes equations under inhomogeneous Dirichlet boundary conditions. In contrast to many previous works, the domain is not assumed to be simply-connected and the fluxes are not assumed to be small. In this very general setting, we prove the existence of a solution with a uniformly bounded Dirichlet integral in every compact subset. This is a generalization of the classical Ladyzhenskaya-Solonnikov result obtained under the additional assumption of zero boundary conditions. For small data of the problem we also prove the unique solvability and attainability of Couette-Poiseuille flows at infinity. The main novelty of our approach is the proof of the corresponding Leray-Hopf-type inequality by Leray's reductio ad absurdum argument (since the standard Hopf cutoff extension procedure does not work for general boundary data). For this contradiction approach, we use some fine properties of weak solutions to the Euler system based on Morse-Sard-type theorems in Sobolev spaces obtained by Bourgain, Korobkov & Kristensen.

math.AP↗
arXiv2023

Distortion of Hausdorff measures under Orlicz--Sobolev maps

A comprehensive theory of the effect of Orlicz-Sobolev maps, between Euclidean spaces, on subsets with zero or finite Hausdorff measure is offered. Arbitrary Orlicz-Sobolev spaces embedded into the space of continuous function and Hausdorff measures built upon general gauge functions are included in our discussion. An explicit formula for the distortion of the relevant gauge function under the action of these maps is exhibited in terms of the Young function defining the Orlicz-Sobolev space. New phenomena and features, related to the flexibility in the definition of the degree of integrability of weak derivatives of maps and in the notion of measure of sets, are detected. Classical results, dealing with standard Sobolev spaces and Hausdorff measures, are recovered, and their optimality is shown to hold in a refined stronger sense. Special instances available in the literature, concerning Young functions and gauge functions of non-power type, are also reproduced and, when not sharp, improved.

math.AP↗
arXiv2021

Leray's plane stationary solutions at small Reynolds numbers

In the celebrated paper by Jean Leray, published in JMPA journal in 1933, the invading domains method was proposed to construct D-solutions for the stationary Navier-Stokes flow around obstacle problem. In two dimensions, whether Leray's D-solution achieves the prescribed limiting velocity at spatial infinity became a major open problem since then. In this paper, we solve this problem at small Reynolds numbers. The proof builds on a novel blow-down argument which rescales the invading domains to the unit disc, and the ideas developed in a recent paper [Korobkov-Pileckas-Russo2020], where the nontriviality of Leray solutions in the general case was proved, and [Korobkov-Ren-2021], where the uniqueness result for small Reynolds number was established.

math.AP↗
arXiv2019

On some universal Morse-Sard type Theorem

The classical Morse--Sard theorem claims that for a mapping $v:\mathbb R^n\to\mathbb R^{m+1}$ of class $C^k$ the measure of critical values $v(Z_{v,m})$ is zero under condition $k\ge n-m$. Here the critical set, or $m$-critical set is defined as $Z_{v,m} = \{ x \in \mathbb R^n : \, {\rm rank}\,\nabla v(x)\le m \}$. Further Dubovitski\uı in 1957 and independently Federer and Dubovitski\uı in 1967 found some elegant extensions of this theorem to the case of other (e.g., lower) smoothness assumptions. They also established the sharpness of their results within the $C^k$ category. Here we formulate and prove a \textit{bridge theorem} that includes all the above results as particular cases: namely, if a function $v:\mathbb R^n\to\mathbb R^d$ belongs to the Holder class $C^{k,α}$, $0\leα\le1$, then for every $q>m$ the identity $$\mathcal H^μ(Z_{v,m}\cap v^{-1}(y))=0$$ holds for $\mathcal H^q$-almost all $y\in\mathbb R^d$, where $μ=n-m-(k+α)(q-m)$. The result is new even for the classical $C^k$-case (when $α=0$); a similar result is established for the Sobolev classes of mappings $W^k_p(\mathbb R^n,\mathbb R^d)$ with minimal integrability assumptions $p=\max(1,n/k)$, i.e., it guarantees in general only the continuity (not everywhere differentiability) of a mapping. However, using some $N$-properties for Sobolev mappings, established in our previous paper, we obtained that the sets of nondifferentiability points of Sobolev mappings are fortunately negligible in the above bridge theorem. We cover also the case of fractional Sobolev spaces. The proofs of the most results are based on our previous joint papers with J. Bourgain and J. Kristensen (2013, 2015).

math.AP↗
arXiv2019

A bridge between Dubovitskii - Federer theorems and the coarea formula

The Morse-Sard theorem requires that a mapping $v:R^n \to R^m$ is of class $C^k$, $k>n-m$. In 1957 Dubovitski\uı generalized this result by proving that almost all level sets for a $C^k$ mapping have $H^s$-negligible intersection with its critical set, where $s=\max(n-m-k+1,0)$. Here the critical set, or $m$-critical set is defined as $Z_{v,m} = \{ x \in R^n : {\rm rank} \nabla v(x) < m \}$. Another generalization was obtained independently by Dubovitski\uı and Federer in 1966, namely for $C^k$ mappings $v:R^n\to R^d$ and integers $m\le d$ they proved that the set of $m$-critical values $v(Z_{v,m})$ is $H^{b}$-negligible for $b= m-1+\frac{n-m+1}{k}$. They also established the sharpness of these results within the $C^k$ category. Here we prove that Dubovitski\uı's theorem can be generalized to the case of continuous mappings of the Sobolev-Lorentz class $W^{k}_{p,1}(R^n,R^d )$, $p=\frac{n}k$ (this is the minimal integrability assumption that guarantees the continuity of mappings). In this situation the mappings need not be everywhere differentiable and in order to handle the set of nondifferentiability points, we establish for such mappings an analog of the Luzin $N$-property with respect to lower dimensional Hausdorff content. Finally, we formulate and prove a~${\rm bridge\ theorem}$ that includes all the above results as particular cases. This result is new also for smooth mappings but is presented here in the general Sobolev context. The proofs of the results are based on our previous joint papers with J.~Bourgain (2013, 2015). Note, that in this paper some result concerning the Coarea formula was not formulated accurately. Now we put an Addendum consisting of three parts: first, we describe the accurate formulation of this result, then we give some historical remarks, and finally its relation to other results of the paper.

math.AP↗
arXiv2017

On the steady Navier--Stokes equations in 2D exterior domains

We study the boundary value problem for the stationary Navier--Stokes system in two dimensional exterior domain. We prove that any solution of this problem with finite Dirichlet integral is uniformly bounded. Also we prove the existence theorem under zero total flux assumption.

math.AP↗
arXiv2017

On Luzin N-property and uncertainty principle for the Sobolev mappings

We study Luzin N-property with respect to the Hausdorff measures for Sobolev spaces W^k_p(R^n,R^d). We prove that such N-property holds except for one critical dimensional value t_*=n-(k-1)p; for this critical value the N-property fails in general, and we constructed the corresponding nontrivial counterexample (based on the theory of lacunary Fourier series). Nevertheless, this N-property holds if we assume in addition that the highest k-derivatives belongs to the Lorentz space L_{p,1} instead of L_p. We extend these results to the case of fractional Sobolev spaces as well. Also, we establish some Fubini type theorems for $N$-properties and discuss their applications to the Morse--Sard theorem and its recent extensions.

math.AP↗
arXiv2015

On the Morse-Sard property and level sets of $W^{n,1}$ Sobolev functions on ${\mathbb R}^n$

We establish Luzin N and Morse--Sard properties for functions from the Sobolev space $W^{n,1}({\mathbb R}^{n})$. Using these results we prove that almost all level sets are finite disjoint unions of $C^1$--smooth compact manifolds of dimension $n-1$. These results remain valid also within the larger space of functions of bounded variation $BV_{n}({\mathbb R}^{n})$. For the proofs we establish and use some new results on Luzin--type approximation of Sobolev and BV--functions by $C^k$--functions, where the exceptional sets have small Hausdorff content.

math.AP↗
arXiv2014

Rigidity Conditions for the Boundaries of Submanifolds in a Riemannian Manifold

Developing A.D. Aleksandrov's ideas, the first-named author of this article proposed the following approach to study of rigidity problems for the boundary of a $C^0$-submanifold in a smooth Riemannian manifold: Let $Y_1$ be a 2-dimensional compact connected $C^0$-submanifold with nonempty boundary in a 2-dimensional smooth connected Riemannian manifold $(X,g)$ without boundary satisfying the condition $$ρ_{Y_1}(x,y) = \liminf_{x' \to x, y' \to y, x',y' \in \mathop{\rm Int} Y_1} \{[l(γ_{x',y',\mathop{\rm Int} Y_1})]\} < \infty,$$ if $x,y \in Y_1$. Here $\inf[l(γ_{x',y',\mathop{\rm Int} Y_1})]$ is the infimum of the length of smooth paths joining $x'$ and $y'$ in the interior $\mathop{\rm Int} Y_1$ of $Y_1$. In the present paper, we first establish that $ρ_{Y_1}$ is a metric on $Y_1$. Suppose further that $Y_1$ is strictly convex in the metric $ρ_{Y_1}$. Consider another 2-dimensional compact connected $C^0$-submanifold $Y_2$ of $X$ with boundary satisfying the condition $ρ_{Y_2}(x,y) < \infty$, $x,y \in Y_2$, and assume that $\partial Y_1$ and $\partial Y_2$ are isometric in the metrics $ρ_{Y_j}$, $j = 1,2$. There appears the following natural question: Under which additional conditions are the boundaries $\partial Y_1$ and $\partial Y_2$ of $Y_1$ and $Y_2$ isometric in the metric $ρ_X$ of the ambient manifold $X$? The paper is devoted to the detailed discussions of this question. In it, we in particular obtain a number of new results concerning the rigidity problems for the boundaries of $C^0$-submanifolds in a Riemannian manifold. The case of $\dim Y_j = \dim X = n$, $n > 2$, is also considered.

math.MG↗
arXiv2011

On the flux problem in the theory of steady Navier-Stokes equations with nonhomogeneous boundary conditions

We study the nonhomogeneous boundary value problem for Navier--Stokes equations of steady motion of a viscous incompressible fluid in a two--dimensional bounded multiply connected domain $Ω=Ω_1\setminus\barΩ_2, \;\barΩ_2\subset Ω_1$. We prove that this problem has a solution if the flux $\F$ of the boundary value through $\partialΩ_2$ is nonnegative. The proof of the main result uses the Bernoulli law for a weak solution to the Euler equations and the one-side maximum principle for the total head pressure corresponding to this solution.

math-ph↗
arXiv2010

On the Morse-Sard Property and Level Sets of Sobolev and BV Functions

We establish Luzin $N$ and Morse-Sard properties for $BV_2$-functions defined on open domains in the plane. Using these results we prove that almost all level sets are finite disjoint unions of Lipschitz arcs whose tangent vectors are of bounded variation. In the case of $W^{2,1}$-functions we strengthen the conclusion and show that almost all level sets are finite disjoint unions of $C^1$-arcs whose tangent vectors are absolutely continuous.

math.AP↗