zbMATH — the first resource for mathematics

Global nonlinear stability for steady ideal fluid flow in bounded planar domains. (English) Zbl 1064.76053
Summary: We prove stability of steady flows of an ideal fluid in a bounded, simply connected, planar region, that are strict maximisers or minimisers of kinetic energy on an isovortical surface. The proof uses conservation of energy and transport of vorticity for solutions of the vorticity equation with initial data in \(L^{p}\) for \(p > 4/3\). A related stability theorem using conservation of angular momentum in a circular domain is also proved.

76E30 Nonlinear effects in hydrodynamic stability
35Q35 PDEs in connection with fluid mechanics
Full Text: DOI
[1] Arnol?d, V.I.: Conditions for nonlinear stability of stationary plane curvilinear flows of an ideal fluid. Soviet Math. Doklady 162, 773-777 (1965); Translation of Dokl. Akad. Nauk SSSR 162, 975-998 (1965) · Zbl 0141.43901
[2] Arnol?d, V.I.: Variational principles for three-dimensional steady-state flows of an ideal fluid. J. Appl. Math. Mech. 29, 1002-1008 (1965); Translation of Prikl. Mat. Mekh. 29, 846-851 (1965) · Zbl 0163.19807 · doi:10.1016/0021-8928(65)90119-X
[3] Arnol?d, V.I.: On an a priori estimate in the theory of hydrodynamic stability. Am. Math. Soc. Transl. (2) 79, 267-269 (1969); Translation of Izv. Vyssh. Uchebn. Zaved. Mat. 1966, 3-5 (1966)
[4] Bouchut, F.: Renormalized solutions to the Vlasov equation with coefficients of bounded variation. Arch. Rational Mech. Anal. 157, 75-90 (2001) · Zbl 0979.35032 · doi:10.1007/PL00004237
[5] Burton, G.R.: Rearrangements of functions, maximization of convex functionals, and vortex rings. Math. Ann. 276, 225-253 (1987) · Zbl 0592.35049 · doi:10.1007/BF01450739
[6] Burton, G.R.: Variational problems on classes of rearrangements and multiple configurations of steady vortices. Ann. Inst. H. Poincaré - Anal. Non Linéaire 6, 295-319 (1989) · Zbl 0677.49005
[7] Burton, G.R., McLeod, J.B.: Maximisation and minimisation on classes of rearrangements. Proc. Roy. Soc. Edin. Sect. A 119, 287-300 (1991) · Zbl 0736.49006
[8] DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511-547 (1989) · Zbl 0696.34049 · doi:10.1007/BF01393835
[9] Ryff, J.V.: Majorized functions and measures. Indag. Math. 30, 431-437 (1968) · Zbl 0164.15903
[10] Thomson, W. (Lord Kelvin): Maximum and minimum energy in vortex motion. In: Mathematical and Physical Papers, volume 4, Cambridge University Press, 1910 pp. 172-183
[11] 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
[12] Yudovich, V.I.: Non-stationary flow of an ideal incompressible liquid. U.S.S.R. Comput. Math. and Math. Phys. 3, 1407-1456 (1963); Translation of Zh. Vychisl. Mat. i Mat. Fiz. 6, 1032-1066 (1963) · Zbl 0147.44303 · doi:10.1016/0041-5553(63)90247-7
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.