Weak solutions obtained by the vortex method for the 2D Euler equations are Lagrangian and conserve the energy. (English) Zbl 1467.35256
This paper studies a class of weak solutions of 2D unsteady incompressible Euler equations with initial data in \(L^2_{\mathrm{loc}}(\mathbb{R}^2)\). Such weak solutions are called VB-solutions and are obtained by the method of the vortex-blob approximation. First the authors show that VB-solutions are Lagrangian. Secondly under the assumption that the initial vorticity is in \(L^p(\mathbb{R}^2)\) with zero mean value and compact support for \(p>1\), the authors prove the conservation of the kinetic energy for the corresponding VB solution.
Reviewer: Yongqian Zhang (Shanghai)
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35Q31 | Euler equations |
| 35D30 | Weak solutions to PDEs |
| 35F55 | Initial value problems for systems of nonlinear first-order PDEs |
| 76B47 | Vortex flows for incompressible inviscid fluids |
Software:
SimEulerReferences:
| [1] | Ambrose, DM; Kelliher, JP; Lopes Filho, MC; Nussenzveig Lopes, HJ, Serfati solutions to the 2D Euler equations on exterior domain, J. Differ. Equ., 259, 4509-4560 (2015) · Zbl 1321.35144 · doi:10.1016/j.jde.2015.06.001 |
| [2] | Ambrosio, L., Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158, 227-260 (2004) · Zbl 1075.35087 · doi:10.1007/s00222-004-0367-2 |
| [3] | 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) · Zbl 0850.76491 |
| [4] | 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) · Zbl 1382.35199 |
| [5] | Bouchut, F.; Crippa, G., Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyperb. Differ. Equ., 10, 235-282 (2013) · Zbl 1275.35076 · doi:10.1142/S0219891613500100 |
| [6] | Bressan, A., Murray, R.: On self-similar solutions to the incompressible Euler equations (2020) http://personal.psu.edu/axb62/PSPDF/EulerSS50.pdf · Zbl 1442.35313 |
| [7] | Bressan, A., Shen, W.: A posteriori error estimates for self-similar solutions to the Euler Equations (2020) https://arxiv.org/abs/2002.01962 |
| [8] | Cheskidov, A.; Lopes Filho, MC; Nussenzveig Lopes, HJ; Shvydkoy, R., Energy conservation in Two-dimensional incompressible ideal fluids, Commun. Math. Phys, 348, 129-143 (2016) · Zbl 1351.35112 · doi:10.1007/s00220-016-2730-8 |
| [9] | Cheskidov, A., Luo, X.: Nonuniqueness of weak solutions for the transport equation at critical space regularity (2020) arXiv:2004.09538 |
| [10] | Crippa, G.; De Lellis, C., Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616, 15-46 (2008) · Zbl 1160.34004 |
| [11] | 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) · Zbl 1379.35224 · doi:10.1137/17M1130988 |
| [12] | Crippa, G.; Spirito, S., Renormalized solutions of the 2D Euler equations, Commun. Math. Phys., 339, 191-198 (2015) · Zbl 1320.35267 · doi:10.1007/s00220-015-2411-z |
| [13] | Delort, J-M, Existence de nappes de tourbillon en dimension deux, J. Am. Math. Soc., 4, 553-586 (1991) · Zbl 0780.35073 · doi:10.1090/S0894-0347-1991-1102579-6 |
| [14] | DiPerna, RJ; Lions, P-L, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98, 511-547 (1989) · Zbl 0696.34049 · doi:10.1007/BF01393835 |
| [15] | DiPerna, RJ; Majda, A., Concentrations in regularizations for 2-D incompressible flow, Commun. Pure Appl. Math., 40, 301-345 (1987) · Zbl 0850.76730 · doi:10.1002/cpa.3160400304 |
| [16] | Lichtenstein, L., Über einige Existenzproblem der Hydrodynamik homogener, unzusammendrückbarer, reibungsloser Flüssigkeiten und die Helmholtzschen Wirbelsätze, M. Z., 23, 89-154 (1925) · JFM 51.0658.01 · doi:10.1007/BF01506223 |
| [17] | 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) · Zbl 0829.35098 · doi:10.1002/cpa.3160480603 |
| [18] | Lopes Filho, MC; Mazzucato, AL; Nussenzveig Lopes, HJ, Weak solutions, renormalized solutions and enstrophy defects in 2D turbulence, Arch. Ration. Mech. Anal, 179, 3, 353-387 (2006) · Zbl 1138.76362 · doi:10.1007/s00205-005-0390-5 |
| [19] | Majda, AJ, Remarks on weak solutions for vortex sheets with a distinguished sign, Indiana Univ. Math. J., 42, 921-939 (1993) · Zbl 0791.76015 · doi:10.1512/iumj.1993.42.42043 |
| [20] | Majda, AJ; Bertozzi, AL, Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 0983.76001 |
| [21] | 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). doi:10.1016/j.anihpc.2020.03.002 · Zbl 1458.35363 |
| [22] | Modena, S.; Szèkelyhidi, L. Jr, Non-uniqueness for the transport equation with Sobolev vector fields, Ann. PDE, 4, 2, 18 (2018) · Zbl 1411.35066 · doi:10.1007/s40818-018-0056-x |
| [23] | Modena, S.; Szèkelyhidi, L. Jr, Non-renormalized solutions to the continuity equation, Calc. Var., 58, 208 (2019) · Zbl 1428.35005 · doi:10.1007/s00526-019-1651-8 |
| [24] | 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) · Zbl 0835.76012 |
| [25] | Stein, EM, Singular Integrals and Differentiability Properties of Functions (1970), Princeton: Princeton University Press, Princeton · Zbl 0207.13501 |
| [26] | Vecchi, I.; Wu, SJ, On \(L^1\)-vorticity for 2-D incompressible flow, Manuscripta Math., 78, 4, 403-412 (1993) · Zbl 0807.35115 · doi:10.1007/BF02599322 |
| [27] | 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 |
| [28] | 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 |
| [29] | 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) · Zbl 0008.06901 · doi:10.1007/BF01474610 |
| [30] | Yudovič, VI, Non-stationary flows of an ideal incompressible fluid, Ž. Vyčisl. Mat. i Mat. Fiz., 3, 1032-1066 (1963) · Zbl 0129.19402 |
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