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Article: Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces
Title | Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces |
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Authors | |
Issue Date | 2015 |
Citation | Inventiones Mathematicae, 2015, v. 201, n. 1, p. 97-157 How to Cite? |
Abstract | For the d-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space (Formula presented.), (Formula presented.). The borderline case (Formula presented.) was a folklore open problem. In this paper we consider the physical dimension d=2 and show that if we perturb any given smooth initial data in (Formula presented.) norm, then the corresponding solution can have infinite (Formula presented.) norm instantaneously at (Formula presented.). In a companion paper [1] we settle the 3D and more general cases. The constructed solutions are unique and even (Formula presented.)-smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems. |
Persistent Identifier | http://hdl.handle.net/10722/327046 |
ISSN | 2023 Impact Factor: 2.6 2023 SCImago Journal Rankings: 4.321 |
ISI Accession Number ID |
DC Field | Value | Language |
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dc.contributor.author | Bourgain, Jean | - |
dc.contributor.author | Li, Dong | - |
dc.date.accessioned | 2023-03-31T05:28:25Z | - |
dc.date.available | 2023-03-31T05:28:25Z | - |
dc.date.issued | 2015 | - |
dc.identifier.citation | Inventiones Mathematicae, 2015, v. 201, n. 1, p. 97-157 | - |
dc.identifier.issn | 0020-9910 | - |
dc.identifier.uri | http://hdl.handle.net/10722/327046 | - |
dc.description.abstract | For the d-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space (Formula presented.), (Formula presented.). The borderline case (Formula presented.) was a folklore open problem. In this paper we consider the physical dimension d=2 and show that if we perturb any given smooth initial data in (Formula presented.) norm, then the corresponding solution can have infinite (Formula presented.) norm instantaneously at (Formula presented.). In a companion paper [1] we settle the 3D and more general cases. The constructed solutions are unique and even (Formula presented.)-smooth in some cases. To prove these results we introduce a new strategy: large Lagrangian deformation induces critical norm inflation. As an application we also settle several closely related open problems. | - |
dc.language | eng | - |
dc.relation.ispartof | Inventiones Mathematicae | - |
dc.title | Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces | - |
dc.type | Article | - |
dc.description.nature | link_to_subscribed_fulltext | - |
dc.identifier.doi | 10.1007/s00222-014-0548-6 | - |
dc.identifier.scopus | eid_2-s2.0-84931577367 | - |
dc.identifier.volume | 201 | - |
dc.identifier.issue | 1 | - |
dc.identifier.spage | 97 | - |
dc.identifier.epage | 157 | - |
dc.identifier.isi | WOS:000356732000002 | - |