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Rearrangements of Functions, Maximization of Convex Functionals, and Vortex Rings.
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This article is published in Mathematische Annalen.The article was published on 1987-01-01. It has received 172 citations till now. The article focuses on the topics: Convex analysis & Subderivative.read more
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Topological methods in hydrodynamics
Vladimir I. Arnold,Boris Khesin +1 more
TL;DR: A group theoretical approach to hydrodynamics is proposed in this article, where the authors consider the hydrodynamic geometry of diffeomorphism groups and the principle of least action implies that the motion of a fluid is described by geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy.
Journal ArticleDOI
Variational problems on classes of rearrangements and multiple configurations for steady vortices
TL;DR: In this article, the existence of at least four solutions for a problem on the steady configurations of a vortex in an ideal fluid was proved for a convex functional restricted to the class of rearrangements of a fixed Lp function.
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Global Nonlinear Stability for Steady Ideal Fluid Flow in Bounded Planar Domains
TL;DR: In this article, the authors 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.
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Eleven Great Problems of Mathematical Hydrodynamics
TL;DR: The key unsolved problems of mathematical fluid dynamics, their current state and outlook are discussed in this article, where global existence and uniquness theorems for basic boundary and initialboundary value problems in the theory of ideal and viscous incompressible fluids, the spectral problems in hydrodynamic stability theory for steady and time periodic flows, creation of secondary, tertiary, etc... flow regimes as a result of bifurcations and the asymptotics of vanishing viscosity.
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Steady symmetric vortex pairs and rearrangements
TL;DR: In this paper, an existence theorem for a steady planar flow of an ideal fluid, containing a bounded symmetric pair of vortices and approaching a uniform flow at infinity, was proved by maximizing the kinetic energy over all flows whose vorticity fields are rearrangements of a specified function.
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Elliptic Partial Differential Equations of Second Order
David Gilbarg,Neil S. Trudinger +1 more
TL;DR: In this article, Leray-Schauder and Harnack this article considered the Dirichlet Problem for Poisson's Equation and showed that it is a special case of Divergence Form Operators.
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Convex analysis and variational problems
Ivar Ekeland,Roger Téman +1 more
TL;DR: In this article, the authors consider non-convex variational problems with a priori estimate in convex programming and show that they can be solved by the minimax theorem.