Boundary value problems for two dimensional steady incompressible fluids. (English) Zbl 1478.35169

Summary: In this paper we study the solvability of different boundary value problems for the two dimensional steady incompressible Euler equation and for the magneto-hydrostatic equation. Two main methods are currently available to study those problems, namely the Grad-Shafranov method and the vorticity transport method. We describe for which boundary value problems these methods can be applied. The obtained solutions have non-vanishing vorticity.


35Q31 Euler equations
76W05 Magnetohydrodynamics and electrohydrodynamics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35M12 Boundary value problems for PDEs of mixed type
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
Full Text: DOI arXiv


[1] Alber, H. D., Existence of three dimensional, steady, inviscid, incompressible flows with nonvanishing vorticity, Math. Ann., 292, 493-528 (1992) · Zbl 0772.35049
[2] Amari, T.; Boulmezaoud, T.; Milkić, Z., An iterative method for the reconstruction of the solar coronal magnetic field. Method for regular solutions, Astron. Astrophys., 350, 1051-1059 (1999)
[3] Arnold, V. I.; Khesin, B. A., Topological Methods in Hydrodynamics, Vol. 125 (1999), Springer Science & Business Media
[4] Bineau, M., On the existence of force-free magnetic fields, Commun. Pure Appl. Math., 27, 77-84 (1972) · Zbl 0226.35016
[5] Buffoni, B.; Wahlén, E., Steady three-dimensional rotational flows: an approach via two stream functions and Nash-Moser iteration, Anal. PDE, 12, 1225-1258 (2019) · Zbl 1406.35244
[6] Coddington, E. A.; Levinson, N., Theory of Ordinary Differential Equations (1955), McGraw Hill Publishing · Zbl 0064.33002
[7] Constantin, P.; Drivas, T.; Ginsberg, D., Flexibility and rigidity in steady fluid motion (2020) · Zbl 1467.76013
[8] Constantin, P.; Drivas, T.; Ginsberg, D., On quasisymmetric plasma equilibria sustained by small force, J. Plasma Phys., 87 (2020)
[9] 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) · Zbl 1397.35192
[10] Evans, L., Partial Differential Equations (2010), American Mathematical Society
[11] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Classics in Mathematics (2001), Springer-Verlag: Springer-Verlag Berlin · Zbl 1042.35002
[12] Goedbloed, J. P.; Poedts, S., Principles of Magnetohydrodynamics with Applications to Laboratory and Astrophysical Plasmas (2010), Cambridge University Press
[13] Goedbloed, J. P.; Poedts, S., Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas (2010), Cambridge University Press
[14] Grad, H.; Rubin, H., Hydromagnetic equilibria and force-free fields, (Proceedings of the 2nd UN Conf. on the Peaceful Uses of Atomic Energy, Vol. 31 (1958), IAEA: IAEA Geneva), 190
[15] Grad, H., Toroidal containment of a plasma, Phys. Fluids, 10, 1, 137-154 (1967)
[16] Hamel, F.; Nadirashvili, N., Shear flows of an ideal fluid and elliptic equations in unbounded domains, Commun. Pure Appl. Math., 70, 3, 590-608 (2017) · Zbl 1360.35155
[17] Hamel, F.; Nadirashvili, N., Circular flows for the Euler equations in two-dimensional annular domains (2019)
[18] Molinet, L., On the existence of inviscid compressible steady flows through a three- dimensional bounded domain, Adv. Differ. Equ., 4, 493-528 (1999) · Zbl 0961.35129
[19] Priest, E., Magnetohydrodynamics of the Sun (2014), Cambridge University Press
[20] Safranov, V. D., Plasma Equilibrium in a Magnetic Field, Reviews of Plasma Physics, vol. 2, 103 (1966), Consultants Bureau: Consultants Bureau New York
[21] Seth, D. S., Steady three-dimensional ideal flows with nonvanishing vorticity in domains with edges, J. Differ. Equ., 274, 345-381 (2021) · Zbl 1455.35184
[22] Seth, D. S., On Steady Ideal Flows with Nonvanishing Vorticity in Cylindrical Domains (2016), Faculty of Science, Centre of Mathematical Sciences, Lund University, Master Thesis
[23] Tang, C.; Xin, Z., Existence of solutions for three dimensional stationary incompressible Euler equations with nonvanishing vorticity, Chin. Ann. Math., Ser. B, 30, 803-830 (2009) · Zbl 1191.35176
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