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Piecewise Smooth Stationary Euler Flows with Compact Support Via Overdetermined Boundary Problems

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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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References

  1. Chae, D., Constantin, P.: Remarks on a Liouville-type theorem for Beltrami flows. Int. Math. Res. Not. 10012–10016, 2015

  2. Choffrut, A., Székelyhidi, L.: Weak solutions to the stationary incompressible Euler equations. SIAM J. Math. Anal. 46, 4060–4074, 2014

    Article  MathSciNet  Google Scholar 

  3. Constantin, P., La, J., Vicol, V.: Remarks on a paper by Gavrilov: Grad–Shafranov equations, steady solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications. Geom. Funct. Anal. 29, 1773–1793, 2019

    Article  MathSciNet  Google Scholar 

  4. Delay, E., Sicbaldi, P.: Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete Contin. Dyn. Syst. 35, 5799–5825, 2015

    Article  MathSciNet  Google Scholar 

  5. Domínguez-Vázquez, M., Enciso, A., Peralta-Salas, D.: Solutions to the overdetermined boundary problem for semilinear equations with position-dependent nonlinearities. Adv. Math. 351, 718–760, 2019

    Article  MathSciNet  Google Scholar 

  6. Fraenkel, L.E., Berger, M.S.: A global theory of steady vortex rings in an ideal fluid. Acta Math. 132, 14–51, 1974

    Article  MathSciNet  Google Scholar 

  7. Gavrilov, A.V.: A steady Euler flow with compact support. Geom. Funct. Anal. 29, 190–197, 2019

    Article  MathSciNet  Google Scholar 

  8. Jiu, Q., Xin, Z.: Smooth approximations and exact solutions of the 3D steady axisymmetric Euler equations. Comm. Math. Phys. 287, 323–350, 2009

    Article  ADS  MathSciNet  Google Scholar 

  9. Nadirashvili, N.: Liouville theorem for Beltrami flow. Geom. Funct. Anal. 24, 916–921, 2014

    Article  MathSciNet  Google Scholar 

  10. Pacard, F., Sicbaldi, P.: Extremal domains for the first eigenvalue of the Laplace-Beltrami operator. Ann. Inst. Fourier 59, 515–542, 2009

    Article  MathSciNet  Google Scholar 

  11. Serrin, J.: A symmetry problem in potential theory. Arch. Rational Mech. Anal. 43, 304–318, 1971

    Article  ADS  MathSciNet  Google Scholar 

Download references

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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Authors and Affiliations

  1. Departamento de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

    Miguel Domínguez-Vázquez

  2. Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Científicas, Madrid, Spain

    Alberto Enciso & Daniel Peralta-Salas

Authors
  1. Miguel Domínguez-Vázquez
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  2. Alberto Enciso
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  3. Daniel Peralta-Salas
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Correspondence to Alberto Enciso.

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Communicated by V. Vicol

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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

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