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

2024/05/07 by Misha Chernobai, Chernobai, Misha, Tim Shilkin +1
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2405.04302

openalex publication_date 2024/05/07 · openalex created_date 2024/05/10 · openalex updated_date 2026/07/28

Abstract

In this paper we study the Dirichlet problem for a scalar elliptic equation in a bounded Lipschitz domain Ω⊂ \mathbb R³ with a singular drift of the form b₀= b-α\frac x'|x'|² where x'=(x₁,x₂,0), α∈ \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∈ \mathbb R³:~|x'|=0~\.

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