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On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations

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Abstract

In this paper we improve the regularity in time of the gradient of the pressure field arising in Brenier’s variational weak solutions (Comm Pure Appl Math 52:411–452, 1999) to incompressible Euler equations. This improvement is necessary to obtain that the pressure field is not only a measure, but a function in \(L^2_{\rm loc}\left((0,T);BV_{\rm loc}(D)\right)\) . In turn, this is a fundamental ingredient in the analysis made by Ambrosio and Figalli (2007, preprint) of the necessary and sufficient optimality conditions for the variational problem by Brenier (J Am Mat Soc 2:225–255, 1989; Comm Pure Appl Math 52:411–452, 1999).

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Correspondence to Luigi Ambrosio.

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Ambrosio, L., Figalli, A. On the regularity of the pressure field of Brenier’s weak solutions to incompressible Euler equations. Calc. Var. 31, 497–509 (2008). https://doi.org/10.1007/s00526-007-0123-8

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  • DOI: https://doi.org/10.1007/s00526-007-0123-8

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