Abstract
We construct new stationary weak solutions of the 3D Euler equation with compact support. The solutions, which are piecewise smooth and discontinuous across a surface, are axisymmetric with swirl. The range of solutions we find is different from, and larger than, the family of smooth stationary solutions recently obtained by Gavrilov and Constantin–La–Vicol; in particular, these solutions are not localizable. A key step in the proof is the construction of solutions to an overdetermined elliptic boundary value problem where one prescribes both Dirichlet and (nonconstant) Neumann data.
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Acknowledgements
M.D.-V. is supported by the grants MTM2016-75897-P, PID2019-105138GB-C21 (AEI/FEDER, Spain) and ED431C 2019/10, ED431F 2020/04 (Xunta de Galicia, Spain), and by the Ramón y Cajal program of the Spanish Ministry of Science. A.E. is supported by the ERC Starting Grant 633152. D.P.-S. is supported by the grant PID2019-106715GB-C21 (MINECO/FEDER) and Europa Excelencia EUR2019-103821 (MCIU). This work is supported in part by the ICMAT–Severo Ochoa grant SEV-2015-0554 and the CSIC grant 20205CEX001.
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Domínguez-Vázquez, M., Enciso, A. & Peralta-Salas, D. Piecewise Smooth Stationary Euler Flows with Compact Support Via Overdetermined Boundary Problems. Arch Rational Mech Anal 239, 1327–1347 (2021). https://doi.org/10.1007/s00205-020-01594-4
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DOI: https://doi.org/10.1007/s00205-020-01594-4