Structure of Green’s function of elliptic equations and helical vortex patches for 3D incompressible Euler equations. (English) Zbl 1533.35247
Summary: We develop a new structure of the Green’s function of a second-order elliptic operator in divergence form in a 2D bounded domain. Based on this structure and the theory of rearrangement of functions, we construct concentrated traveling-rotating helical vortex patches to 3D incompressible Euler equations in an infinite pipe. By solving an equation for vorticity
\[
w=\frac{1}{\varepsilon^2}f_\varepsilon \left( \mathcal{G}_{K_H}w-\frac{\alpha }{2}|x|^2|\ln \varepsilon |\right) \text{ in } \Omega
\]
for small \(\varepsilon >0\) and considering a certain maximization problem for the vorticity, where \(\mathcal{G}_{K_H}\) is the inverse of an elliptic operator \(\mathcal{L}_{K_H}\) in divergence form, we get the existence of a family of concentrated helical vortex patches, which tend asymptotically to a singular helical vortex filament evolved by the binormal curvature flow. We also get nonlinear orbital stability of the maximizers in the variational problem under \(L^p\) perturbation when \(p\ge 2 \).
MSC:
| 35Q31 | Euler equations |
| 76B47 | Vortex flows for incompressible inviscid fluids |
| 76U05 | General theory of rotating fluids |
| 35J15 | Second-order elliptic equations |
| 35J08 | Green’s functions for elliptic equations |
| 35C07 | Traveling wave solutions |
| 35B50 | Maximum principles in context of PDEs |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B35 | Stability in context of PDEs |
| 35A15 | Variational methods applied to PDEs |
| 35A21 | Singularity in context of PDEs |
References:
| [1] | Abidi, H.; Sakrani, S., Global well-posedness of helicoidal Euler equations, J. Funct. Anal., 271, 8, 2177-2214 (2016) · Zbl 1446.35112 · doi:10.1016/j.jfa.2016.06.007 |
| [2] | Ao, W.; Liu, Y.; Wei, J., Clustered travelling vortex rings to the axisymmetric three-dimensional incompressible Euler flows, Phys. D, 434 (2022) · Zbl 1495.76025 · doi:10.1016/j.physd.2022.133258 |
| [3] | Arnold, VI, On an a priori estimate in the theory of hydrodynamical stability, Am. Math. Soc. Transl., 79, 267-269 (1969) · Zbl 0191.56303 |
| [4] | Benvenutti, M., Nonlinear stability for stationary helical vortices, Nonlinear Differ. Equ. Appl. (NoDEA), 27, 2, 15 (2020) · Zbl 1437.35554 · doi:10.1007/s00030-020-0620-4 |
| [5] | Bronzi, AC; Lopes Filho, MC; Nussenzveig Lopes, HJ, Global existence of a weak solution of the incompressible Euler equations with helical symmetry and \(L^p\) vorticity, Indiana Univ. Math. J., 64, 1, 309-341 (2015) · Zbl 1319.35171 · doi:10.1512/iumj.2015.64.5467 |
| [6] | Burchard, A.; Guo, Y., Compactness via symmetrization, J. Funct. Anal., 214, 1, 40-73 (2004) · Zbl 1065.49006 · doi:10.1016/j.jfa.2004.04.005 |
| [7] | Burton, GR, Rearrangements of functions, maximization of convex functionals, and vortex rings, Math. Ann., 276, 2, 225-253 (1987) · Zbl 0592.35049 · doi:10.1007/BF01450739 |
| [8] | Burton, GR, Global nonlinear stability for steady ideal fluid flow in bounded planar domains, Arch. Ration. Mech. Anal., 176, 149-163 (2005) · Zbl 1064.76053 · doi:10.1007/s00205-004-0339-0 |
| [9] | Burton, GR; Nussenzveig Lopes, HJ; Lopes Filho, MC, Nonlinear stability for steady vortex pairs, Commun. Math. Phys., 324, 445-463 (2013) · Zbl 1278.35188 · doi:10.1007/s00220-013-1806-y |
| [10] | Caffarelli, LA; Friedman, A., Asymptotic estimates for the plasma problem, Duke Math. J., 47, 705-742 (1980) · Zbl 0466.35033 · doi:10.1215/S0012-7094-80-04743-2 |
| [11] | Cao, D.; Liu, Z.; Wei, J., Regularization of point vortices for the Euler equation in dimension two, Arch. Ration. Mech. Anal., 212, 179-217 (2014) · Zbl 1293.35223 · doi:10.1007/s00205-013-0692-y |
| [12] | Da Rios, L.S.: Sul moto d’un liquido indefinito con un filetto vorticoso di forma qualunque. Rend. Circ. Mat. Palermo (1884-1940) 22(1), 117-135 (1906) · JFM 37.0764.01 |
| [13] | Dávila, J.; del Pino, M.; Musso, M.; Wei, J., Gluing methods for vortex dynamics in Euler flows, Arch. Ration. Mech. Anal., 235, 3, 1467-1530 (2020) · Zbl 1439.35382 · doi:10.1007/s00205-019-01448-8 |
| [14] | Dávila, J.; del Pino, M.; Musso, M.; Wei, J., Travelling helices and the vortex filament conjecture in the incompressible Euler equations, Calc. Var. Partial Differ. Equ., 61, 119 (2022) · Zbl 1490.35262 · doi:10.1007/s00526-022-02217-4 |
| [15] | Dekeyser, J.; Van Schaftingen, J., Vortex motion for the lake equations, Commun. Math. Phys., 375, 1459-1501 (2020) · Zbl 1443.76117 · doi:10.1007/s00220-020-03742-z |
| [16] | de Valeriola, S.; Van Schaftingen, J., Desingularization of vortex rings and shallow water vortices by semilinear elliptic problem, Arch. Ration. Mech. Anal., 210, 2, 409-450 (2013) · Zbl 1294.35083 · doi:10.1007/s00205-013-0647-3 |
| [17] | Dutrifoy, A., Existence globale en temps de solutions hélicoïdales des équations d’Euler, C. R. Acad. Sci. Paris Sér. I Math., 329, 7, 653-656 (1999) · Zbl 0948.76009 · doi:10.1016/S0764-4442(00)80019-1 |
| [18] | Ettinger, B.; Titi, ES, Global existence and uniqueness of weak solutions of three-dimensional Euler equations with helical symmetry in the absence of vorticity stretching, SIAM J. Math. Anal., 41, 1, 269-296 (2009) · Zbl 1303.76006 · doi:10.1137/08071572X |
| [19] | Fraenkel, LE, On steady vortex rings of small cross-section in an ideal fluid, Proc. R. Soc. Lond. A., 316, 29-62 (1970) · Zbl 0195.55101 · doi:10.1098/rspa.1970.0065 |
| [20] | Fraenkel, LE; Berger, MS, A global theory of steady vortex rings in an ideal fluid, Acta Math., 132, 13-51 (1974) · Zbl 0282.76014 · doi:10.1007/BF02392107 |
| [21] | Gilbarg, D.; Trudinger, NS, Elliptic Partial Differential Equations of Second Order. Classics in Mathematics (2001), Berlin: Springer, Berlin · Zbl 0361.35003 · doi:10.1007/978-3-642-61798-0 |
| [22] | Greub, W.H.: Linear Algebra, 3rd edn, Die Grundlehren der Mathematischen Wissenschaften, Band 97. Springer, New York, xiii+434 pp (1967) · Zbl 0147.27408 |
| [23] | Grüter, M.; Widman, KO, The Green function for uniformly elliptic equations, Manuscr. Math., 37, 303-342 (1982) · Zbl 0485.35031 · doi:10.1007/BF01166225 |
| [24] | Helmholtz, H., On integrals of the hydrodynamics equations which express vortex motion, J. Reine Angew. Math., 55, 25-55 (1858) · ERAM 055.1448cj |
| [25] | Jerrard, RL; Seis, C., On the vortex filament conjecture for Euler flows, Arch. Ration. Mech. Anal., 224, 1, 135-172 (2017) · Zbl 1371.35205 · doi:10.1007/s00205-016-1070-3 |
| [26] | Jerrard, RL; Smets, D., On the motion of a curve by its binormal curvature, J. Eur. Math. Soc. (JEMS), 17, 6, 1487-1515 (2015) · Zbl 1327.53086 · doi:10.4171/jems/536 |
| [27] | Jiu, Q.; Li, J.; Niu, D., Global existence of weak solutions to the three-dimensional Euler equations with helical symmetry, J. Differ. Equ., 262, 10, 5179-5205 (2017) · Zbl 1360.35158 · doi:10.1016/j.jde.2017.01.019 |
| [28] | Kenig, C.E., Ni, W.-M.: On the elliptic equation \(Lu+k+Kexp[2u]=0\). Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12(4), 191-224 (1985) · Zbl 0593.35044 |
| [29] | Lamb, H., Hydrodynamics, Cambridge Mathematical Library (1932), Cambridge: Cambridge University Press, Cambridge · JFM 58.1298.04 |
| [30] | Levi-Civita, T., Sull’attrazione esercitata da una linea materiale in punti prossimi alla linea stessa, Rend. R. Acc. Lincei, 17, 3-15 (1908) · JFM 39.0830.01 |
| [31] | Levi-Civita, T.: Attrazione newtoniana dei tubi sottili e vortici filiformi. Annali della Scuola Normale Superiore di Pisa Classe di Scienze Ser. 2 1(3), 229-250 (1932) · JFM 58.0873.02 |
| [32] | Lieb, EH; Loss, M., Analysis, 2nd edn, Graduate Studies in Mathematics (2001), Providence: American Mathematical Society, Providence · Zbl 0966.26002 |
| [33] | Lin, CC, On the motion of vortices in two dimension—I. Existence of the Kirchhoff-Routh function, Proc. Natl. Acad. Sci. U.S.A., 27, 570-575 (1941) · Zbl 0063.03560 · doi:10.1073/pnas.27.12.570 |
| [34] | Littman, W.; Stampacchia, G.; Weinberger, HF, Regular points for elliptic equations with discontinuous coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 17, 3, 43-77 (1963) · Zbl 0116.30302 |
| [35] | Majda, A.; Bertozzi, A., Vorticity and Incompressible Flow (2002), Cambridge: Cambridge University Press, Cambridge · Zbl 0983.76001 |
| [36] | Marchioro, C.; Pulvirenti, M., Mathematical Theory of Incompressible Nonviscous Fluids (1994), Berlin: Springer, Berlin · Zbl 0789.76002 · doi:10.1007/978-1-4612-4284-0 |
| [37] | Pomponio, A.; Secchi, S., On a class of singularly perturbed elliptic equations in divergence form: existence and multiplicity results, J. Differ. Equ., 207, 2, 229-266 (2004) · Zbl 1129.35403 · doi:10.1016/j.jde.2004.06.015 |
| [38] | Ricca, RL, Rediscovery of Da Rios equations, Nature, 352, 6336, 561-562 (1991) · doi:10.1038/352561a0 |
| [39] | Ricca, RL, The contributions of Da Rios and Levi-Civita to asymptotic potential theory and vortex filament dynamics, Fluid Dyn. Res., 18, 5, 245-268 (1996) · Zbl 1006.01505 · doi:10.1016/0169-5983(96)82495-6 |
| [40] | Smets, D.; Van Schaftingen, J., Desingularization of vortices for the Euler equation, Arch. Ration. Mech. Anal., 198, 3, 869-925 (2010) · Zbl 1228.35171 · doi:10.1007/s00205-010-0293-y |
| [41] | Tang, Y., Nonlinear stability of vortex patches, Trans. Am. Math. Soc., 304, 617-637 (1987) · Zbl 0636.76019 · doi:10.1090/S0002-9947-1987-0911087-X |
| [42] | Taylor, JL; Kim, S.; Brown, RM, The Green function for elliptic systems in two dimensions, Commun. Partial Differ. Equ., 38, 9, 1574-1600 (2013) · Zbl 1279.35021 · doi:10.1080/03605302.2013.814668 |
| [43] | Turkington, B.: On steady vortex flow in two dimensions. I, II. Commun. Partial Differ. Equ. 8, 999-1030, 1031-1071 (1983) · Zbl 0523.76015 |
| [44] | Wan, Y-H; Pulvirenti, M., Nonlinear stability of circular vortex patches, Commun. Math. Phys., 99, 435-450 (1985) · Zbl 0584.76062 · doi:10.1007/BF01240356 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
