Geometric hydrodynamics in open problems
DOI10.1007/s00205-023-01848-xOpenAlexW4322008064MaRDI QIDQ2686986
Gerard Misiołek, Alexander I. Shnirelman, Boris A. Khesin
Publication date: 1 March 2023
Published in: Archive for Rational Mechanics and Analysis (Search for Journal in Brave)
Full work available at URL: https://arxiv.org/abs/2205.01143
Smoothness and regularity of solutions to PDEs (35B65) Attractors (35B41) PDEs in connection with fluid mechanics (35Q35) Variational methods applied to PDEs (35A15) Analyticity in context of PDEs (35A20) Groups of diffeomorphisms and homeomorphisms as manifolds (58D05) Existence problems for PDEs: global existence, local existence, non-existence (35A01) Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics (76N10) Research exposition (monographs, survey articles) pertaining to fluid mechanics (76-02) Singularity in context of PDEs (35A21) Existence, uniqueness, and regularity theory for incompressible inviscid fluids (76B03) Euler equations (35Q31) Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness (35A02) Geophysical flows (76U60)
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Cites Work
- Régularité du problème de Kelvin–Helmholtz pour l'équation d'Euler 2d
- Knotted structures in high-energy Beltrami fields on the torus and the sphere
- Continuity of the solution map of the Euler equations in Hölder spaces and weak norm inflation in Besov spaces
- Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 1. Fundamentals
- SPLITTING OF THE SEPARATRICES AND THE NONEXISTENCE OF FIRST INTEGRALS IN SYSTEMS OF DIFFERENTIAL EQUATIONS OF HAMILTONIAN TYPE WITH TWO DEGREES OF FREEDOM
- Inertial energy dissipation for weak solutions of incompressible Euler and Navier-Stokes equations
- Compressible helical flows
- Conjugate points in $\mathcal {D}_{\mu }(T^2)$
- Symplectic structures and dynamics on vortex membranes
- Quantum mechanics writ large
- Geometric hydrodynamics via Madelung transform
- On the regularity of the composition of diffeomorphisms
- Geometric hydrodynamics and infinite-dimensional Newton’s equations
- Canonical scale separation in two-dimensional incompressible hydrodynamics
- A Casimir preserving scheme for long-time simulation of spherical ideal hydrodynamics
- Time-analyticity of Lagrangian particle trajectories in ideal fluid flow
- Ill-posedness examples for the quasi-geostrophic and the Euler equations
- Classification of Casimirs in 2D hydrodynamics
- On the Euler equations of incompressible fluids
- The degree of knottedness of tangled vortex lines
- Some topological aspects of fluid dynamics
- Classification of Coadjoint Orbits for Symplectomorphism Groups of Surfaces
- The \(N\)-vortex problem. Analytical techniques
- On dynamics of Lagrangian trajectories for Hamilton-Jacobi equations
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Unnamed Item
- Beltrami fields with a nonconstant proportionality factor are rare
- Sticky particle dynamics with interactions
- On stationary solutions of two-dimensional Euler equation
- Local structure of the set of steady-state solutions to the 2D incompressible Euler equations
- KAM theory and the 3D Euler equation
- Conjugate points on the symplectomorphism group
- On the motion of a curve by its binormal curvature
- Coadjoint orbits of symplectic diffeomorphisms of surfaces and ideal hydrodynamics
- Galoisian obstructions to integrability of Hamiltonian systems. II.
- Knots and links in steady solutions of the Euler equation
- Microglobal analysis of the Euler equations
- Integrability of point-vortex dynamics via symplectic reduction: a survey
- On the volumorphism group, the first conjugate point is always the hardest
- Analyticity of Lagrangian trajectories for well posed inviscid incompressible fluid models
- Inviscid damping and the asymptotic stability of planar shear flows in the 2D Euler equations
- Loss of smoothness and energy conserving rough weak solutions for the \(3d\) Euler equations
- Remarks on the breakdown of smooth solutions for the 3-D Euler equations
- The Hamiltonian structure for dynamic free boundary problems
- On nonstationary flows of viscous and ideal fluids in \(L^ p_ s({\mathbb{R}}^ 2)\)
- Divergence-free fields: Energy and asymptotic crossing number
- Wandering solutions of Euler's D-2 equation
- Curvatures of left invariant metrics on Lie groups
- Hydrodynamics in Besov spaces
- Mathematical theory of incompressible nonviscous fluids
- Generalized fluid flows, their approximation and applications
- Topological methods in hydrodynamics
- On the universality of the incompressible Euler equation on compact manifolds
- Continuum dynamics on manifolds: application to elasticity of residually-stressed bodies
- The geometry of axisymmetric ideal fluid flows with swirl
- A steady Euler flow with compact support
- For ideal fluids, Eulerian and Lagrangian instabilities are equivalent
- Vortex sheets and diffeomorphism groupoids
- Nonlinear Grassmannians as coadjoint orbits
- Incompressible Euler system and analytic microlocal regularity
- Uniqueness theorem for the basic nonstationary problem in the dynamics on an ideal incompressible fluid
- Holomorphic structures with weak spatial regularity in fluid mechanics
- Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics
- Linking, twisting, writing, and helicity on the 3-sphere and in hyperbolic 3-space
- On the initial value problem for the Madelung hydrodynamic equations
- Weak solutions with decreasing energy of incompressible Euler equations
- Remarks on a paper by Gavrilov: Grad-Shafranov equations, steady solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications
- Vortex motion of the Euler and lake equations
- The golden ratio and hydrodynamics
- On maximally mixed equilibria of two-dimensional perfect fluids
- On Moffatt's magnetic relaxation equations
- Conjugate and cut points in ideal fluid motion
- Chaos in the incompressible Euler equation on manifolds of high dimension
- Existence of a conjugate point in the incompressible Euler flow on an ellipsoid
- Arnold stability and Misiołek curvature
- Beltrami fields with hyperbolic periodic orbits enclosed by knotted invariant tori
- Global, local and dense non-mixing of the 3D Euler equation
- Finite-mode analogs of 2D ideal hydrodynamics: Coadjoint orbits and local canonical structure
- Lie groupoids and algebroids applied to the study of uniformity and homogeneity of material bodies
- \(L^\infty\) ill-posedness for a class of equations arising in hydrodynamics
- Geometry of the Madelung transform
- Strong illposedness of the incompressible Euler equation in integer \(C^m\) spaces
- Existence of knotted vortex tubes in steady Euler flows
- On the uniqueness of the solution of the flow problem with a given vortex
- Shock waves for the Burgers equation and curvatures of diffeomorphism groups
- The Madelung transform as a momentum map
- Mechanisms of Lagrangian analyticity in fluids
- Perestroikas of shocks in optimal control
- Lax pair formulation for the Euler equation
- Singularities of the exponential map on the volume-preserving diffeomorphism group
- Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces
- Sur la géométrie différentielle des groupes de Lie de dimension infinite et ses applications à l'hydrodynamique des fluides parfaits
- On classical solutions of the two-dimensional non-stationary Euler equation
- Groups of diffeomorphisms and the motion of an incompressible fluid
- Fredholm properties of Riemannian exponential maps on diffeomorphism groups
- On the loss of smoothness of the solutions of the Euler equations and the inherent instability of flows of an ideal fluid
- Asymptotic Massey products, induced currents and Borromean torus links
- Vorticity and Incompressible Flow
- THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION
- Vortex dynamics in $\mathbb {R}^4$R4
- Helicity is the only integral invariant of volume-preserving transformations
- Fourier Analysis and Nonlinear Partial Differential Equations
- An Optimal Transport View of Schrödinger's Equation
- Existence of Solution for the Euler Equations in a Critical Besov Space (ℝn)
- On the Analyticity of Particle Trajectories in the Ideal Incompressible Fluid
- Vortex dynamics on a Möbius strip
- Vortex dynamics for the two dimensional non homogeneous Gross-Pitaevskii equation
- Euler equations for incompressible ideal fluids
- Loss of Smoothness and Inherent Instability of 2D Inviscid Fluid Flows
- Lax Pairs and Darboux Transformations for Euler Equations
- Particle dynamics inside shocks in Hamilton–Jacobi equations
- The WKB method for conjugate points in the volumorphism group
- Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. Part 2. Stability considerations
- Commutator estimates and the euler and navier-stokes equations
- Minimal geodesics on groups of volume-preserving maps and generalized solutions of the Euler equations
- FINITE GENERATION OF GROUPS OF DIFFEOMORPHISMS
- Evolution of singularities, generalized Liapunov function and generalized integral for an ideal incompressible fluid
- A new look at the derivation of the Schrödinger equation from Newtonian mechanics