Abstract
AbstractWe show that certain singular structures (Hölderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocity and the results are shown to be sharp in the sense that slightly more singular structures cannot generally be propagated. For the 2D Euler equation, we prove that certain singular structures are preserved by the motion, e.g. a system of $$\log \log _+(1/|x|)$$
log
log
+
(
1
/
|
x
|
)
vortices (and those that are slightly less singular) travel with the fluid in a nonlinear fashion, up to bounded perturbations. We also give stability results for weak Euler solutions away from their singular set.
Funder
Division of Mathematical Sciences
Institute for Advanced Study
Alfred P. Sloan Foundation
Publisher
Springer Science and Business Media LLC
Reference27 articles.
1. Albritton, D., Brué, E., Colombo, M., De Lellis, C., Giri, V., Janisch, M., Kwon, H.: (2021)Instability and nonuniqueness for the $$2 d $$ Euler equations in vorticity form, after M. Vishik. arXiv preprint arXiv:2112.04943
2. Ambrosio, L., Bernard, P.: Uniqueness of signed measures solving the continuity equation for Osgood vector fields, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 19(3), 237–245 (2008)
3. Bahouri, H., Chemin, J.Y., Danchin: Fourier analysis and nonlinear partial differential equations (Vol. 343). Springer Science & Business Media (2011)
4. Bertozzi, A.L., Constantin, P.: Global regularity for vortex patches. Comm. Math. Phys. 152(1), 19–28 (1993)
5. Bressan, A., Murray, R.: On self-similar solutions to the incompressible Euler equations. J. Diff. Equ. 269(6), 5142–5203 (2020)
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