2023/06/16 by Martin Bauer, Bauer, Martin, Stephen C. Preston +3
Mathematics · Physics and Astronomy · #Advanced Differential Equations and Dynamical Systems #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2306.09748
openalex publication_date 2023/06/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we introduce a new blowup criterion for (generalized) Euler-Arnold equations on \mathbb Rn. Our method is based on treating the equation in Lagrangian coordinates, where it is an ODE on the diffeomorphism group, and comparison with the Liouville equation; in contrast to the usual comparison approach at a single point, we apply comparison in an infinite dimensional function space. We thereby show that the Jacobian of the Lagrangian flow map of the solution reaches zero in finite time, which corresponds to C1-blowup of the velocity field solution. We demonstrate the applicability of our result by proving blowup of smooth solutions to some higher-order versions of the EPDiff equation in all dimensions n≥ 3. Previous results on blowup of higher dimensional EPDiff equations were only for versions where the geometric description corresponds to a Sobolev metric of order zero or one. In these situations the behavior does not depend on the dimension and thus already solutions to the one-dimensional version were exhibiting blowup. In the present paper blowup is proved even in situations where the one-dimensional equation has global solutions, such as the EPDiff equation corresponding to a Sobolev metric of order two.