×

zbMATH — the first resource for mathematics

Existence of three-dimensional, steady, inviscid, incompressible flows with nonvanishing vorticity. (English) Zbl 0772.35049
The author studies the flow of an inviscid incompressible medium through a bounded, simply connected domain of \(\mathbb{R}^ 3\). He is particularly interested in constructing solutions with nonvanishing vorticity. In general the expectation is that these type of flows are unstable and this instability introduces difficulties into the existence proof.
The author proves that if there exists a solution of a particular boundary value problem with sufficiently small vorticity, then there exists a neighbourhood of this solution and flows with nonvanishing vorticity in this neighbourhood with special stability properties.
Reviewer: F.Rosso (Firenze)

MSC:
35Q35 PDEs in connection with fluid mechanics
76B47 Vortex flows for incompressible inviscid fluids
35B35 Stability in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Bourguignon, J., Brezis, H.: Remarks on the Euler equation. J. Funct. Anal.15, 341-363 (1974) · Zbl 0279.58005
[2] Ebin, D., Marsden, J.: Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. Math.92, 102-163 (1970) · Zbl 0211.57401
[3] H?lder, E.: ?ber die unbeschr?nkte Fortsetzbarkeit einer stetigen ebenen Bewegung in einer unbegrenzten inkompressiblen Fl?ssigkeit. Math. Z.37, 727-738 (1933) · Zbl 0008.06902
[4] Kato, T.: Non-stationary flows of viscous and ideal fluids in ?3. J. Funct. Anal.9, 296-305 (1972) · Zbl 0229.76018
[5] Kazhikhov, A.V.: Note on the formulation of the problem of flow through a bounded region using equations of perfect fluid. Prikl. Mat. Mekh.44, 947-950 (1980); English translation in: J. Appl. Math. Mech.44, 672-674 (1980) · Zbl 0468.76004
[6] Kazhikhov, A.V., Ragulin, V.V.: Nonstationary leakage problem for an ideal fluid in a bounded domain (in Russian). Dokl. Akad. Nauk SSSR250, 1344-1347 (1980) · Zbl 0445.76016
[7] Kochin, N.E.: About the existence theorem for hydrodynamics (in Russian). Prikl. Mat. Mekh.20, 153-172 (1956)
[8] Ne?as, J.: Les M?thods directes en th?orie des ?quations elliptiques. Paris: Masson 1967
[9] Picard, R.: On the low frequency asymptotics in electromagnetic theory. J. Reine Angew. Math.354, 50-73 (1985) · Zbl 0541.35049
[10] Serrin, J.: Mathematical principles of classical fluid mechanics. In: Fl?gge, S. (ed.) Handbuch der Physik, Band VIII/1 Str?mungsmechanik 1. Berlin: Springer 1959 · Zbl 0086.20001
[11] Temam, R.: On the Euler equations of incompressible perfect fluids. J. Funct. Anal.20, 32-43 (1975) · Zbl 0309.35061
[12] Weber, C.: Regularity theorems for Maxwell’s equations. Math. Methods Appl. Sci.3, 523-536 (1981) · Zbl 0477.35020
[13] Werner, P.: Randwertprobleme f?r die zeitunabh?ngigen Maxwellschen Gleichungen mit variablen Koeffizienten. Arch. Ration. Mech. Anal.18, 167-195 (1965) · Zbl 0142.37501
[14] Wolibner, W.: Un th?or?me sur l’existence du mouvement plan d’un fluide parfait, homog?ne incompressible, pendant un temps infiniment long. Math. Z.37, 698-726 (1933) · JFM 59.1447.02
[15] Yudovich, V.I.: Twodimensional nonstationary leakage problem for an ideal incompressible fluid in a given domain (in Russian). Mat. Sb.64, 562-588 (1964)
[16] Zajaczkowski, W.M.: Local solvability of a nonstationary leakage problem for an ideal incompressible fluid 1 (in Russian). Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova92, 39-56 (1980) · Zbl 0463.76018
[17] Zajaczkowski, W.M.: Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 3. Math. Methods Appl. Sci.4, 1-14 (1982) · Zbl 0582.76024
[18] Zajaczkowski, W.M.: Local solvability of a nonstationary leakage problem for an ideal incompressible fluid, 2. Pac. J. Math.113(1), 229-255 (1984)
[19] Zajaczkowski, W.M.: Solvability of an initial boundary value problem for the Euler equations in twodimensional domain with corners. Math. Methods Appl. Sci.6, 1-22 (1984) · Zbl 0549.76015
[20] Zajaczkowski, W.M.: Ideal incompressible fluid motion in domains with edges I. Bull. Pol. Acad. Sci., Techn. Sci.33, 183-194 (1985) · Zbl 0603.76022
[21] Zajaczkowski, W.M.: Ideal incompressible fluid motion in domains with edges 2. Bull. Pol. Acad. Sci., Math.33, 332-338 (1985)
[22] Zajaczkowski, W.M.: Some leakage problems for ideal incompressible fluid motion in domains with edges. Banach Cent. Publ.19, 383-397 (1987)
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.