A geometric view on the generalized Proudman-Johnson and $r$-Hunter-Saxton equations
| Autor: | Bauer, Martin, Lu, Yuxiu, Maor, Cy |
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| Rok vydání: | 2021 |
| Předmět: | |
| Zdroj: | Journal of Nonlinear Science (2022) 32:17 |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1007/s00332-021-09775-5 |
| Popis: | We show that two families of equations on the real line, the generalized inviscid Proudman--Johnson equation, and the $r$-Hunter--Saxton equation (recently introduced by Cotter et al.) coincide for a certain range of parameters. This gives a new geometric interpretation of these Proudman--Johnson equations as geodesic equations of right invariant homogeneous $W^{1,r}$-Finsler metrics on an appropriate diffeomorphism group on $\mathbb{R}$. Generalizing a construction of Lenells for the Hunter--Saxton equation, we analyze the $r$-Hunter--Saxton equation using an isometry from the diffeomorphism group to an appropriate subset of real-valued functions. Thereby we show that the periodic case is equivalent to the geodesic equation on the $L^r$-sphere in the space of functions, and the non-periodic case is equivalent to a geodesic flow on a flat space. This allows us to give explicit solutions to these equations in the non-periodic case, and answer several questions of Cotter et al. regarding their limiting behavior. Comment: version 3: corrected an error regarding the equivalence of the equations in the periodic case (there is equivalence only in the non-periodic case). Abstract and introduction were changed accordingly. version 2: minor changes |
| Databáze: | arXiv |
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