×

zbMATH — the first resource for mathematics

Compactness and stability for planar vortex-pairs with prescribed impulse. (English) Zbl 07269183
Summary: Concentration-compactness is used to prove compactness of maximising sequences for a variational problem governing symmetric steady vortex-pairs in a uniform planar ideal fluid flow, where the kinetic energy is to be maximised and the constraint set comprises the set of all equimeasurable rearrangements of a given function (representing vorticity) that have a prescribed impulse (linear momentum). A form of orbital stability is deduced.
MSC:
49S05 Variational principles of physics (should also be assigned at least one other classification number in Section 49-XX)
76B47 Vortex flows for incompressible inviscid fluids
76E07 Rotation in hydrodynamic stability
49N25 Impulsive optimal control problems
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arnol′d, V. I., Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid, Sov. Math. Dokl.. Sov. Math. Dokl., Dokl. Akad. Nauk SSSR, 162, 975-998 (1965), translation of
[2] Brooke Benjamin, T., The alliance of practical and analytical insights into the nonlinear problems of fluid mechanics, (Applications of Methods of Functional Analysis to Problems in Mechanics. Applications of Methods of Functional Analysis to Problems in Mechanics, Lecture Notes in Mathematics, vol. 503 (1976), Springer-Verlag: Springer-Verlag Berlin), 8-29 · Zbl 0369.76048
[3] Burton, G. R., Steady symmetric vortex pairs and rearrangements, Proc. R. Soc. Edinb., Sect. A, 108, 269-290 (1988) · Zbl 0658.76016
[4] Burton, G. R., Variational problems on classes of rearrangements and multiple configurations of steady vortices, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 6, 295-319 (1989) · Zbl 0677.49005
[5] Burton, G. R., Isoperimetric properties of Lamb’s circular vortex-pair, J. Math. Fluid Mech., 7, S68-S80 (2005) · Zbl 1064.76021
[6] Burton, G. R.; Nussenzveig Lopes, H. J.; Lopes Filho, M. C., Nonlinear stability for steady vortex pairs, Commun. Math. Phys., 324, 445-463 (2013) · Zbl 1278.35188
[7] Douglas, R. J., Rearrangements of functions on unbounded domains, Proc. R. Soc. Edinb., Sect. A, 124, 621-644 (1994) · Zbl 0818.49010
[8] Friedlander, S. J.; Shnirelman, A. I., Instability of steady flows of an ideal incompressible fluid, (Mathematical Fluid Mechanics. Mathematical Fluid Mechanics, Adv. Math. Fluid Mech. (2001), Birkhauser: Birkhauser Basel), 143-172 · Zbl 0984.35129
[9] Friedlander, S. J.; Yudovich, V. I., Instabilities in fluid motion, Not. Am. Math. Soc., 46, 11, 1358-1367 (1999) · Zbl 0948.76003
[10] Kolmogorov, A. N.; Fomin, S. V., Introductory Real Analysis (1975), Dover Publications: Dover Publications New York
[11] Lieb, E. H., Existence and uniqueness of the minimising solution of Choquard’s nonlinear equation, Stud. Appl. Math., 57, 93-105 (1977) · Zbl 0369.35022
[12] Lieb, E. H.; Loss, M., Analysis, Graduate Studies in Mathematics, vol. 14 (2001), American Mathematical Society: American Mathematical Society Providence RI · Zbl 0966.26002
[13] Lions, P.-L., The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 1, 109-145 (1984) · Zbl 0541.49009
[14] Royden, H. L., Real Analysis (1988), Macmillan/Collier Macmillan: Macmillan/Collier Macmillan New York/London · Zbl 0704.26006
[15] Ryff, J. V., Majorized functions and measures, Indag. Math., 30, 431-437 (1968) · Zbl 0164.15903
[16] Yudovich, V. I., Non-stationary flow of an ideal incompressible liquid, USSR Comput. Math. Math. Phys.. USSR Comput. Math. Math. Phys., Ž. Vyčisl. Mat. Mat. Fiz., 6, 1032-1066 (1963), translation of
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.