Weak Solutions Obtained by the Vortex Method for the 2D Euler Equations are Lagrangian and Conserve the Energy

• Ambrose, D.M., Kelliher, J.P., Lopes Filho, M.C., Nussenzveig Lopes, H.J.: Serfati solutions to the 2D Euler equations on exterior domain. J. Differ. Equ. 259, 4509–4560 (2015)

  Article  MathSciNet  Google Scholar 

• Ambrosio, L.: Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158, 227–260 (2004)

  Article  MathSciNet  Google Scholar 

• Beale, J. T.: The approximation of weak solutions to the 2-D Euler equations by vortex elements. Multidymensional Hyperbolic Problems and Computations, In: Glimm, J., Majda, A. (eds.) IMA Vol. Math. Appl. 29, 23–37 (1991)

• Bohun, A., Bouchut, F., Crippa, G.: Lagrangian solutions to the 2D Euler system with \(L^1\) vorticity and infinite energy. Nonlinear Anal. Theory, Methods Appl. 132, 160–172 (2016)

• Bouchut, F., Crippa, G.: Lagrangian flows for vector fields with gradient given by a singular integral. J. Hyperb. Differ. Equ. 10, 235–282 (2013)

  Article  MathSciNet  Google Scholar 

• Bressan, A., Murray, R.: On self-similar solutions to the incompressible Euler equations (2020) http://personal.psu.edu/axb62/PSPDF/EulerSS50.pdf

• Bressan, A., Shen, W.: A posteriori error estimates for self-similar solutions to the Euler Equations (2020) https://arxiv.org/abs/2002.01962

• Cheskidov, A., Lopes Filho, M.C., Nussenzveig Lopes, H.J., Shvydkoy, R.: Energy conservation in Two-dimensional incompressible ideal fluids. Commun. Math. Phys 348, 129–143 (2016)

  Article  MathSciNet  Google Scholar 

• Cheskidov, A., Luo, X.: Nonuniqueness of weak solutions for the transport equation at critical space regularity (2020) arXiv:2004.09538

• Crippa, G., De Lellis, C.: Estimates and regularity results for the DiPerna-Lions flow. J. Reine Angew. Math. 616, 15–46 (2008)

  MathSciNet  MATH  Google Scholar 

• Crippa, G., Nobili, C., Seis, C., Spirito, S.: Eulerian and Lagrangian solutions to the continuity and Euler equations with \(L^1\) vorticity. SIAM J. Math. Anal. 49(5), 3973–3998 (2017)

  Article  MathSciNet  Google Scholar 

• Crippa, G., Spirito, S.: Renormalized solutions of the 2D Euler equations. Commun. Math. Phys. 339, 191–198 (2015)

  Article  MathSciNet  Google Scholar 

• Delort, J.-M.: Existence de nappes de tourbillon en dimension deux. J. Am. Math. Soc. 4, 553–586 (1991)

  Article  MathSciNet  Google Scholar 

• DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

  Article  MathSciNet  Google Scholar 

• DiPerna, R.J., Majda, A.: Concentrations in regularizations for 2-D incompressible flow. Commun. Pure Appl. Math. 40, 301–345 (1987)

  Article  MathSciNet  Google Scholar 

• Lichtenstein, L.: Über einige Existenzproblem der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsätze. M. Z. 23, 89–154 (1925)

  Article  Google Scholar 

• Liu, J.-G., Xin, Z.: Convergence of vortex methods for weak solutions to the 2-D Euler equations with vortex sheets data. Commun. Pure Appl. Math. 48, 611–628 (1995)

  Article  MathSciNet  Google Scholar 

• Lopes Filho, M.C., Mazzucato, A.L., Nussenzveig Lopes, H.J.: Weak solutions, renormalized solutions and enstrophy defects in 2D turbulence. Arch. Ration. Mech. Anal 179(3), 353–387 (2006)

  Article  MathSciNet  Google Scholar 

• Majda, A.J.: Remarks on weak solutions for vortex sheets with a distinguished sign. Indiana Univ. Math. J. 42, 921–939 (1993)

  Article  MathSciNet  Google Scholar 

• Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, vol. 27. Cambridge University Press, Cambridge (2002)

  Google Scholar 

• Modena, S., Sattig, G.: Convex integration solutions to the transport equation with full dimensional concentration. Ann. Inst. H. Poincarè Anal. Non Linèaire, in press (2020). https://doi.org/10.1016/j.anihpc.2020.03.002

• Modena, S., Szèkelyhidi Jr., L.: Non-uniqueness for the transport equation with Sobolev vector fields. Ann. PDE 4(2), 18 (2018)

  Article  MathSciNet  Google Scholar 

• Modena, S., Szèkelyhidi Jr., L.: Non-renormalized solutions to the continuity equation. Calc. Var. 58, 208 (2019)

  Article  MathSciNet  Google Scholar 

• Serfati, P.: Solutions \(C^\infty \) en temps, \(n\)-log Lipschitz bornées en espace et équation d’Euler. C. R. Acad. Sci. Paris Sér. I Math. 320, 555–558 (1995)

  MathSciNet  MATH  Google Scholar 

• Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

  MATH  Google Scholar 

• Vecchi, I., Wu, S.J.: On \(L^1\)-vorticity for 2-D incompressible flow. Manuscripta Math. 78(4), 403–412 (1993)

  Article  MathSciNet  Google Scholar 

• Vishik, M.: Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part I (2018a) arXiv:1805.09426

• Vishik, M.: Instability and non-uniqueness in the Cauchy problem for the Euler equations of an ideal incompressible fluid. Part II (2018b) arXiv:1805.09440

• Wolibner, W.: Un théorème sur l’existence du mouvement plan d’un fluide parfait, homogene, incompressible, pendant un temps infiniment long. Math. Z. 37, 698–726 (1933)

  Article  MathSciNet  Google Scholar 

• Yudovič, V.I.: Non-stationary flows of an ideal incompressible fluid. Ž. Vyčisl. Mat. i Mat. Fiz. 3, 1032–1066 (1963)

  MathSciNet  Google Scholar 

Download references