In this article, we will study unbounded solutions of the 2D incompressible Euler equations. One of the motivating factors for this is that the usual functional framework for the Euler equations (e.g. based on finite energy conditions, such as $L^2$) does not respect some of the symmetries of the problem, such as Galileo invariance.
Our first result, global existence and uniqueness of solutions for initial data with square-root growth $O(|x|^{{1}/{2} - \varepsilon })$ and bounded vorticity, is based on two key ingredients. Firstly an integral decomposition of the pressure, and secondly examining local energy balance leading to solution estimates in weighted $L^2$ spaces. We also prove continuity of the initial data to solution map by a substantial adaptation of Yudovich’s uniqueness argument.
The second result is global existence and uniqueness of solutions for initial data with sublinear growth $O(|x|^{1-\varepsilon })$. This is work in progress.
Dimitri Cobb; Herbert Koch. Unbounded solutions to the Euler equation. Séminaire Laurent Schwartz — EDP et applications (2025), Exposé no. 11, 13 p.. doi: 10.5802/slsedp.184
@article{SLSEDP_2025-2026____A3_0,
author = {Dimitri Cobb and Herbert Koch},
title = {Unbounded solutions to the {Euler} equation},
journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications},
note = {talk:11},
pages = {1--13},
year = {2025-2026},
publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique},
doi = {10.5802/slsedp.184},
language = {en},
url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.184/}
}
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