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).
Similar content being viewed by others
References
Ambrosio, L., Figalli, A.: Geodesics in the space of measure-preserving maps and plans. Preprint, 2007
Arnold V. (1966). Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. (French). Ann. Inst. Fourier (Grenoble) 16(fasc. 1): 319–361
Benamou J.-D. and Brenier Y. (2000). A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numer. Math. 84: 375–393
Brenier Y. (1989). The least action principle and the related concept of generalized flows for incompressible perfect fluids. J. Am. Math. Soc. 2: 225–255
Brenier Y. (1993). The dual least action problem for an ideal, incompressible fluid. Arch. Rational Mech. Anal. 122: 323–351
Brenier Y. (1997). A homogenized model for vortex sheets. Arch. Rational Mech. Anal. 138: 319–353
Brenier Y. (1999). Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations. Comm. Pure Appl. Math. 52: 411–452
Shnirelman, A.I.: The geometry of the group of diffeomorphisms and the dynamics of an ideal incompressible fluid (Russian). Mat. Sb. (N.S.) 128(170), no. 1, 82–109 (1985)
Shnirelman A.I. (1994). Generalized fluid flows, their approximation and applications. Geom. Funct. Anal. 4(5): 586–620
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00526-007-0123-8