arXiv subjects

Misha Chernobai

Publications and source records attributed to Misha Chernobai.

7 recordsLinked to original sources

arXiv2025

Existence and regularity for perturbed Stokes system with critical drift in 2D

We consider a perturbed Stokes system with critical divergence-free drift in a bounded Lipschitz domain in $R^2$, with sufficiently small Lipschitz constant L. It extends our previous work in $\Bbb R^n, n\ge 3$, to two-dimensional case. For large drift in weak $L^2$ space, we prove unique existence of q-weak solutions for force in $L^q$ with q close to 2. Moreover, for drift in $L^2(\Bbb R^2)$ we prove the unique existence of $W^{1,2}$ solutions for arbitrarily large L. Using similar methods we can also prove analogous results for scalar equations with divergence-free drifts in weak $L^2$ space.

math.AP↗

arXiv2024

Existence and regularity for perturbed Stokes system with critical drift

We consider the existence and $L^q$ gradient estimates for perturbed Stokes systems with divergence-free critical drift in a bounded Lipschitz domain in $\mathbb{R}^n$, $n \ge 3$. The first two results assume the drift is either in $L^n$ or sufficiently small in weak $L^n$. The third result assumes the drift is in weak $L^n$ without smallness, and obtain results for $q$ close to 2.

math.AP↗

arXiv2024

Existence of global weak solutions to NSE in weighted spaces

We obtain a global existence result for the three-dimensional Navier-Stokes equations with a large class of initial data allowing growth at spatial infinity. Our work is a continuation of the results by T.-P. Tsai, Z. Bradshaw, I. Kukavica and proves global existence of suitable weak solutions with initial data in different weighted spaces as well as eventual regularity.

math.AP↗

arXiv2024

On the existence and uniqueness of weak solutions to elliptic equations with a singular drift

In this paper we study the Dirichlet problem for a scalar elliptic equation in a bounded Lipschitz domain $Ω\subset \mathbb R^3$ with a singular drift of the form $b_0= b-α\frac {x'}{|x'|^2}$ where $x'=(x_1,x_2,0)$, $α\in \mathbb R$ is a parameter and $b$ is a divergence free vector field having essentially the same regularity as the potential part of the drift. Such drifts naturally arise in the theory of axially symmetric solutions to the Navier-Stokes equations. For $α<0$ the divergence of such drifts is positive which potentially can ruin the uniqueness of solutions. Nevertheless, for $α<0$ we prove existence and Hölder continuity of a unique weak solution which vanishes on the axis $Γ:=\{ ~x\in \mathbb R^3:~|x'|=0~\}$.

math.AP↗

arXiv2023

Global Navier-Stokes flows in intermediate spaces

We construct global weak solutions of the three dimensional incompressible Navier-Stokes equations in intermediate spaces between the space of uniformly locally square integrable functions and Herz-type spaces which involve weighted integrals centered at the origin. Our results bridge the existence theorems of Lemarié-Rieusset and of Bradshaw, Kukavica and Tsai. An application to eventual regularity is included which generalizes the prior work of Bradshaw, Kukavica and Tsai as well as Bradshaw, Kukavica and Ozanski.

math.AP↗

arXiv2022

Scalar elliptic equations with a singular drift

We investigate the weak solvability and properties of weak solutions to the Dirichlet problem for a scalar elliptic equation $-Δu + b^{(α)}\cdot \nabla u= f$ in a bounded domain $Ω\subset {\mathbb R^2}$ containing the origin, where $f \in W^{-1}_q(Ω) $ with $q>2$ and $b^{(α)}:=b-α\frac{x}{|x|^2}$, $b$ is a divergence-free vector field and $α\in {\mathbb R}$ is a parameter.

math.AP↗

arXiv2022

Elliptic equations with a singular drift from a weak Morrey space

In this paper we prove the existence and uniqueness of weak solutions to the Dirichlet problem for an elliptic equation with a drift $b$ satisfying $\operatorname{div} b\le 0$ in $\Omega$. We assume $b$ belongs to some weak Morrey class which includes in the 3D case, in particular, drifts having a singularity along the axis $x_3$ with the asymptotic $b(x)\sim c/r$, where $r=\sqrt{x_1^2+x_2^2}$.

math.AP↗