Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Explicit Formula of the Infinite Energy Solutions for the Exterior 2D Div-Curl Problem with Dirichlet Boundary Condition

Aleksei Gorshkov

TL;DR

The paper analyzes the 2D exterior div-curl problem with prescribed vorticity $w$, divergence $\rho$, boundary data, and flow at infinity. It develops explicit Biot–Savart-type representations for the velocity in exterior domains, first for the disk via Fourier modes and then for general planar exterior domains using a Riemann mapping, together with precise solvability (orthogonality) conditions. It proves existence and uniqueness under appropriate function-space assumptions and provides $L^2$, $H^1$, and $L^\infty$ estimates, enabling numerical reduction to a Poisson problem. Numerical experiments validate the method for flows around simple geometries and discuss limitations and potential extensions to multiple solids and 3D analogues.

Abstract

In the paper we study the 2D div-curl problem in the exterior domain which models the flow with given vorticity, divergency, boundary condition at infinity, and Dirichlet condition on the solid surface. We will find the relations on vorticity and divergence for uniqueness solvability of the problem and deduce the explicit formula.

Explicit Formula of the Infinite Energy Solutions for the Exterior 2D Div-Curl Problem with Dirichlet Boundary Condition

TL;DR

The paper analyzes the 2D exterior div-curl problem with prescribed vorticity , divergence , boundary data, and flow at infinity. It develops explicit Biot–Savart-type representations for the velocity in exterior domains, first for the disk via Fourier modes and then for general planar exterior domains using a Riemann mapping, together with precise solvability (orthogonality) conditions. It proves existence and uniqueness under appropriate function-space assumptions and provides , , and estimates, enabling numerical reduction to a Poisson problem. Numerical experiments validate the method for flows around simple geometries and discuss limitations and potential extensions to multiple solids and 3D analogues.

Abstract

In the paper we study the 2D div-curl problem in the exterior domain which models the flow with given vorticity, divergency, boundary condition at infinity, and Dirichlet condition on the solid surface. We will find the relations on vorticity and divergence for uniqueness solvability of the problem and deduce the explicit formula.
Paper Structure (9 sections, 3 theorems, 88 equations, 4 figures)

This paper contains 9 sections, 3 theorems, 88 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Div-curl problem in the exterior of the disk
  3. Div-curl problem in the exterior planar domain
  4. Uniqueness solvability of the planar div-curl problem
  5. Numerical results
  6. Discussion
  7. Conclusion
  8. Data availability statements
  9. Statements and Declarations

Key Result

Theorem 1

[$L_2$-estimate of the $\nabla \mathbf{v}$] Let $\Omega = B_{r_0}$ be an exterior of the disk, $\rho, w \in L_{2}(B_{r_0})$ and the relations (intcond) are satisfied for $k\in \mathbb{N} \cup \{0\}$. Then the solution $\mathbf{v}$ of the problem (intro:1)-(intro:4) is given by explicit formula (intr

Figures (4)

  • Figure 1: Velocity field around rectangle. Flow data was generated in program AGVortex.
  • Figure 2: Vortical flow around rectangle. Flow data was generated in program AGVortex.
  • Figure 3: Velocity field around line segment. Flow data was generated in program AGVortex.
  • Figure 4: Vortical flow around line segment. Flow data was generated in program AGVortex.

Theorems & Definitions (6)

  • Theorem 1
  • proof
  • Theorem 2: $L_2$-estimates of the velocity
  • proof
  • Theorem 3
  • proof