Propagation of singularities by Osgood vector fields and for 2D inviscid incompressible fluids

We 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 t...

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Bibliographic Details
Published inMathematische annalen Vol. 387; no. 3-4; pp. 1691 - 1718
Main Authors Drivas, Theodore D., Elgindi, Tarek M., La, Joonhyun
Format Journal Article
LanguageEnglish
Published Berlin/Heidelberg Springer Berlin Heidelberg 01.12.2023
Springer Nature B.V
Subjects
Euler-Lagrange equation
Fields (mathematics)
Fluid flow
Incompressible flow
Incompressible fluids
Mathematics
Mathematics and Statistics
Perturbation
Singularity (mathematics)
Velocity distribution
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Summary:We 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 | ) 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.
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ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-022-02498-2