Abstract
Strong Beltrami fields, that is, vector fields in three dimensions whose curl is the product of the field itself by a constant factor, have long played a key role in fluid mechanics and magnetohydrodynamics. In particular, they are the kind of stationary solutions of the Euler equations where one has been able to show the existence of vortex structures (vortex tubes and vortex lines) of arbitrarily complicated topology. On the contrary, there are very few results about the existence of generalized Beltrami fields, that is, divergence-free fields whose curl is the field times a non-constant function. In fact, generalized Beltrami fields (which are also stationary solutions to the Euler equations) have been recently shown to be rare, in the sense that for “most” proportionality factors there are no nontrivial Beltrami fields of high enough regularity (e.g., of class \({C^{6,\alpha}}\)), not even locally. Our objective in this work is to show that, nevertheless, there are “many” Beltrami fields with non-constant factor, even realizing arbitrarily complicated vortex structures. This fact is relevant in the study of turbulent configurations. The core results are an “almost global” stability theorem for strong Beltrami fields, which ensures that a global strong Beltrami field with suitable decay at infinity can be perturbed to get “many” Beltrami fields with non-constant factor of arbitrarily high regularity and defined in the exterior of an arbitrarily small ball, and a “local” stability theorem for generalized Beltrami fields, which is an analogous perturbative result which is valid for any kind of Beltrami field (not just with a constant factor) but only applies to small enough domains. The proof relies on an iterative scheme of Grad–Rubin type. For this purpose, we study the Neumann problem for the inhomogeneous Beltrami equation in exterior domains via a boundary integral equation method and we obtain Hölder estimates, a sharp decay at infinity and some compactness properties for these sequences of approximate solutions. Some of the parts of the proof are of independent interest.
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Amari T., Boulmezaoud T.Z., Mikić Z.: An iterative method for the reconstruction of the solar coronal magnetic field. I. Method for regular solutions. Astron Astrophys. 350, 1051–1059 (1999)
Arnold V.I.: Sur la topologie des écoulements stationnaires des fluides parfaits. C. R. Acad. Sci. Paris. 261, 17–20 (1965)
Arnold V.I.: Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits. Ann. Inst. Fourier. 16, 319–361 (1966)
Backus G.E., Chandasekhar S.: On Cowling’s theorem on the impossibility of self-maintained axisymmetric homogeneous dynamos. Proc. Nat. Acad. Sci. 42, 105–109 (1956)
Bineau M.: On the existence of force-free magnetic fields. Commun. Pure Appl. Math. 27, 77–84 (1972)
Calderón A.P.: Cauchy integral in Lipschitz curves and related operators. Proc. Nat. Acad. Sci. 74(4), 1324–1327 (1977)
Chae D., Constantin P.: Remarks on a Liouville-type theorem for Beltrami flows. IMRN. 20, 10012–10016 (2015)
Coifman R.R., Mcintosh A., Meyer Y.: L’integrale de Cauchy définit un operateur borné sur L 2 pour les courbes lipschitziennes. Ann. Math. 116(2), 361–387 (1982)
Colton D.L., Kress R.: Integral Equation Methods in Scattering Theory. Wiley, New York (1983)
Colton D.L., Kress R.: Inverse Acoustic and Electromagnetic Scattering. Springer, Berlin (1992)
Constantin P.: An Eulerian–Lagrangian approach for incompressible fluids: local theory. J. Am. Math. Soc. 14(2), 263–278 (2001)
Constantin P., Majda A.: The Beltrami spectrum for incompressible fluid flows. Commun. Math. Phys. 115, 435–456 (1988)
David G.: Opérateurs intégraux singuliers sur certaines courbes du plan complexe. Ann. Sci. Éc. Norm. Supér. 17, 157–189 (1984)
Dahlberg B.: On estimates of harmonic measure. Arch. Ration. Mech. Anal. 65, 272–288 (1977)
Dahlberg B.: On the Poisson integral for Lipschitz and C* domains. Studia Math. 66, 13–24 (1979)
Dahlberg B., Kenig C.: Hardy spaces and the Neumann problem in L p for Laplace’s equation in Lipschitz domains. Ann. Math. 125, 437–465 (1987)
Dombre T., Frisch U., Greene J.M., Hénon M., Mehr A., Soward A.M.: Chaotic streamlines in the ABC flows. J. Fluid Mech. 167, 353–391 (1986)
Enciso A., Peralta-Salas D.: Existence of knotted vortex tubes in steady Euler flows. Acta Math. 214, 61–134 (2015)
Enciso A., Peralta-Salas D.: Knots and links in steady solutions of the Euler equation. Ann. Math. 175, 345–367 (2012)
Enciso A., Peralta-Salas D.: Beltrami fields with a nonconstant proporcionality factor are rare. Arch. Ration. Mech. Anal. 220, 243–260 (2016)
Fabes E.B., Jodeit M., Lewis J.E.: Double layer potentials for domains with corners and edges. Indiana Univ. Math. J. 26, 95–114 (1977)
Fabes E.B., Jodeit M., Riviére N.M.: Potential techniques for boundary value problems on C 1 domains. Acta Math. 141, 165–186 (1978)
Gilbarg D., Trudinger N.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001)
Giraud G.: Sur certains problèmes non linéaires de Neumann et sur certains problèmes non linéaires mixtes. Ann. Sci. Éc. Norm. Supér. 49, 1–104 (1932)
Heinemann, U.: Die regularisierende Wirkung der Randintegraloperatoren der klassischen Potentialtheorie in den Räumen hölderstetiger Funktionen. Diploma Thesis, University of Bayreuth (1992)
Hofmann S., Kenig C., Mayboroda S., Pipher J.: Square function/non-tangential maximal function estimates and the Dirichlet problem for non-symmetric elliptic operators. J. Am. Math. Soc. 28, 483–529 (2015)
Kaiser R., Neudert M., von Wahl W.: On the existence of force-free magnetic fields with small nonconstant \({\alpha}\) in exterior domains. Commun. Math. Phys. 211, 111–136 (2000)
Kleckner D., Irvine W.T.M.: Creation and dynamics of knotted vortices. Nat. Phys. 9, 253–258 (2013)
Kress R.: A boundary integral equation method for a Neumann boundary problem for force fields. J. Eng. Math. 15, 29–48 (1981)
Li C., McIntosh A., Semmes S.: Convolution singular integrals on Lipschitz surfaces. J. Am. Math. Soc. 5(3), 455–481 (1992)
Littman W.: Decay at infinity of solutions to partial differential equations with constant coefficients. Trans. AMS 123, 449–459 (1966)
Miranda C.: Partial Differential Equations of Elliptic Type. Springer, Berlin (1970)
Miranda C.: Sulle proprieta di regolarita di certe trasformazioni integrali. Mem. Acc. Lincei. 7, 303–336 (1965)
Moffatt H.K.: A Tsinober, Helicity in laminar and turbulent flow. Annu. Rev. Fluid Mech. 24, 281–312 (1992)
Moffatt H.K.: Helicity and singular structures in fluid dynamics. PNAS. 111, 3663–3670 (2014)
Nadirashvili N.: Liouville theorem for Beltrami flow. Geom. Funct. Anal. 24, 916–921 (2014)
Nédélec J.C.: Acoustic and Electromagnetic Equations. Applied Mathematical Sciences. Springer, Berlin (2001)
Neudert M., von Wahl W.: Asymptotic Behaviour of the div-curl problem in exterior domains. Adv. Differ. Equ. 6, 1347–1376 (2001)
Pelz R., Yakhot V., Orszag S.A., Shtilman L., Levich E.: Velocity-vorticity patterns in turbulent flow. Phys. Rev. Lett. 54, 2505–2509 (1985)
Poyato, D.: Generalized Beltrami fields in exterior domains. An application to the theory of knotted structures in stationary solutions of the 3-D Euler equation (Original title: “Campos de Beltrami generalizados en dominios exteriores. Una aplicación a la teoría de estructuras anudadas en soluciones estacionarias de la ecuación de Euler 3-D”), Master Thesis, University of Granada (2015)
Semmes S.W.: A criterion for the boundedness of singular integrals on hypersurfaces. Trans. Am. Math. Soc. 311, 501–513 (1989)
Stein E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
von Wahl, W.: Vorlesung über das Au\({\beta}\)enraumproblem für die instationären Gleichungen von Navier-Stokes, Rudolph-Lipschitz-Vorsenlug, Sondeforschungsbereich 256: Nichtlineare Partielle Differentialgleichungen 11 (1989)
Wiegelmann T., Sakurai T.: Solar force-free magnetic fields. Living Rev. Solar Phys. 9, 5 (2012)
Wilcox C.H.: An expansion theorem for electromagnetic fields. Commun. Pure Appl. Math. 9(2), 115–134 (1956)
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Enciso, A., Poyato, D. & Soler, J. Stability Results, Almost Global Generalized Beltrami Fields and Applications to Vortex Structures in the Euler Equations. Commun. Math. Phys. 360, 197–269 (2018). https://doi.org/10.1007/s00220-017-3063-y
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DOI: https://doi.org/10.1007/s00220-017-3063-y