• Corpus ID: 248506058

Geometric Hydrodynamics in Open Problems

@inproceedings{Khesin2022GeometricHI,
  title={Geometric Hydrodynamics in Open Problems},
  author={Boris Khesin and Gerard Misiołek and Alexander Shnirelman},
  year={2022}
}
Geometric Hydrodynamics has flourished ever since the celebrated 1966 paper of V. Arnold. In this paper we present a collection of open problems along with several new constructions in fluid dynamics and a concise survey of recent developments and achievements in this area. The topics discussed include variational settings for different types of fluids, models for invariant metrics, the Cauchy and boundary value problems, partial analyticity of solutions to the Euler equations, their steady and… 

Figures from this paper

References

SHOWING 1-10 OF 150 REFERENCES

Geometric hydrodynamics and infinite-dimensional Newton’s equations

We revisit the geodesic approach to ideal hydrodynamics and present a related geometric framework for Newton’s equations on groups of diffeomorphisms and spaces of probability densities. The latter

On the Long Time Behavior of Fluid Flows

Geometric hydrodynamics via Madelung transform

A geometric framework is introduced revealing a closer link between hydrodynamics and quantum mechanics than previously recognized and the Madelung transform between the Schrödinger equation and Newton’s equations is a symplectomorphism of the corresponding phase spaces.

Shock waves for the Burgers equation and curvatures of diffeomorphism groups

We establish a simple relation between certain curvatures of the group of volume-preserving diffeomorphisms and the lifespan of potential solutions to the inviscid Burgers equation before the

Local Structure of The Set of Steady-State Solutions to The 2d Incompressible Euler Equations

It is well known that the incompressible Euler equations can be formulated in a very geometric language. The geometric structures provide very valuable insights into the properties of the solutions.

Vortex sheets and diffeomorphism groupoids

Microglobal Analysis of the Euler Equations

Abstract.The flow of the ideal incompressible fluid can be regarded as the motion along a geodesic on the group of volume preserving diffeomorphisms of the flow domain. Thus, we can define the

Remarks on a paper by Gavrilov: Grad–Shafranov equations, steady solutions of the three dimensional incompressible Euler equations with compactly supported velocities, and applications

We describe a method to construct smooth and compactly supported solutions of 3D incompressible Euler equations and related models. The method is based on localizable Grad–Shafranov equations and is

Particle dynamics inside shocks in Hamilton–Jacobi equations

  • K. KhaninA. Sobolevski
  • Mathematics
    Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
  • 2010
It is shown that, for any convex Hamiltonian, there exists a uniquely defined canonical global non-smooth coalescing flow that extends particle trajectories and determines the dynamics inside shocks.
...