Numerical Study of Reported Multiple Solutions to a Cauchy Problem for the 2d Incompressible Euler Equations
Abstract
Bressan and Shen (2021) jointly with Bressan and Murray (2020) proposed a simple example of a Cauchy problem on ℝ2, characterized by two key parameters, intended to yield nonunique weak solutions of the incompressible Euler equations. They suggest that a limiting process applied to two specific approximations of their example initial data yields two distinct solutions. Here we provide evidence coming from a carefully designed numerical procedure to support if, or when, this intriguing claim is valid. Some of the numerical challenges encountered in the present study are: (i) the proposed initial data is unbounded in a neighborhood of the origin, (ii) the proposed problem is posed on all of ℝ2, (iii) several essential symmetries present in the true solution must be captured, (iv) the proposed example depends on two parameters which may be critically linked. We demonstrate that an extremely large computational domain is mathematically necessary to capture the core symmetries of the exact free-space problem. Accordingly we implement a nested-grid strategy specifically designed for its particular suitability and efficacy for the problem at hand. Our strategy also allows us to achieve extreme grid refinement in the region where solution accuracy is most critical. Within our nested-grid framework, we design an underlying discretization scheme that is high-order, robust, and fast. We consider a broad collection of allowable parameter pairs, but among all of them we find only one that may satisfy the suggested claim.
Related
- Nonuniqueness of Solutions to 2D Incompressible Euler Equations with Unbounded Vorticity
- Stability Analysis of Bounded Solutions to 2-D Incompressible Navier-Stokes Equations in L^2_uloc Spaces
- Self-Similar Solutions of the 2D Incompressible Euler Equations via a Lagrangian Construction
- Asymptotic profiles for solutions to the incompressible Navier-Stokes equations in the critical Fourier-Herz spaces
- Homothetic Self-Similar Solutions to the Incompressible Navier-Stokes Equations
BibTeX
@misc{tesdall2026numerical,
title = {{Numerical Study of Reported Multiple Solutions to a Cauchy Problem for the 2d Incompressible Euler Equations}},
author = {Allen Tesdall and Richard Sanders},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11694},
eprint = {2609.11694},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11694},
}
RIS
TY - UNPB
TI - Numerical Study of Reported Multiple Solutions to a Cauchy Problem for the 2d Incompressible Euler Equations
AU - Allen Tesdall
AU - Richard Sanders
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11694
UR - https://arxiv.org/abs/2609.11694
AB - Bressan and Shen (2021) jointly with Bressan and Murray (2020) proposed a simple example of a Cauchy problem on ℝ^2, characterized by two key parameters, intended to yield nonunique weak solutions of the incompressible Euler equations. They suggest that a limiting process applied to two specific approximations of their example initial data yields two distinct solutions. Here we provide evidence coming from a carefully designed numerical procedure to support if, or when, this intriguing claim is valid. Some of the numerical challenges encountered in the present study are: (i) the proposed initial data is unbounded in a neighborhood of the origin, (ii) the proposed problem is posed on all of ℝ^2, (iii) several essential symmetries present in the true solution must be captured, (iv) the proposed example depends on two parameters which may be critically linked. We demonstrate that an extremely large computational domain is mathematically necessary to capture the core symmetries of the exact free-space problem. Accordingly we implement a nested-grid strategy specifically designed for its particular suitability and efficacy for the problem at hand. Our strategy also allows us to achieve extreme grid refinement in the region where solution accuracy is most critical. Within our nested-grid framework, we design an underlying discretization scheme that is high-order, robust, and fast. We consider a broad collection of allowable parameter pairs, but among all of them we find only one that may satisfy the suggested claim.
ER -