Gerard Misiołek

Euler equations on homogeneous spaces and Virasoro orbits

Abstract We show that the following three systems related to various hydrodynamical approximations: the Korteweg–de Vries equation, the Camassa–Holm equation, and the Hunter–Saxton equation, have the same symmetry group and similar bihamiltonian structures. It turns out that their configuration space is the Virasoro group and all three dynamical systems can be regarded as equations of the geodesic flow associated to different right-invariant metrics on this group or on appropriate homogeneous spaces. In particular, we describe how Arnolds approach to the Euler equations as geodesic flows of one-sided invariant metrics extends from Lie groups to homogeneous spaces. We also show that the above three cases describe all generic bihamiltonian systems which are related to the Virasoro group and can be integrated by the translation argument principle: they correspond precisely to the three different types of generic Virasoro orbits. Finally, we discuss interrelation between the above metrics and Kahler structures on Virasoro orbits as well as open questions regarding integrable systems corresponding to a finer classification of the orbits.

Classical solutions of the periodic Camassa—Holm equation

Abstract. We study the periodic Cauchy problem for the Camassa—Holm equation and prove that it is locally well-posed in the space of continuously differentiable functions on the circle. The approach we use consists in rewriting the equation and deriving suitable estimates which permit application of o.d.e. techniques in Banach spaces. We also describe results in fractional Sobolev Hs spaces and in Appendices provide a direct well-posedness proof for arbitrary real s > 3/2 based on commutator estimates of Kato and Ponce as well as include a derivation of the equation on the diffeomorphism group of the circle together with related curvature computations.

Non-Uniform Dependence for the Periodic CH Equation

We show that the solution map of the periodic CH equation is not uniformly continuous in Sobolev spaces with exponent greater than 3/2. This extends earlier results to the whole range of Sobolev exponents for which local well-posedness of CH is known. The crucial technical tools used in the proof of this result are a sharp commutator estimate and a multiplier estimate in Sobolev spaces of negative index.

GENERALIZED HUNTER-SAXTON EQUATION AND THE GEOMETRY OF THE GROUP OF CIRCLE DIFFEOMORPHISMS

We study an equation lying ‘mid-way’ between the periodic Hunter–Saxton and Camassa–Holm equations, and which describes evolution of rotators in liquid crystals with external magnetic field and self-interaction. We prove that it is an Euler equation on the diffeomorphism group of the circle corresponding to a natural right-invariant Sobolev metric. We show that the equation is bihamiltonian and admits both cusped and smooth traveling-wave solutions which are natural candidates for solitons. We also prove that it is locally well-posed and establish results on the lifespan of its solutions. Throughout the paper we argue that despite similarities to the KdV, CH and HS equations, the new equation manifests several distinctive features that set it apart from the other three.

Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions

We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct “peakon” and “multi-peakon” solutions for all λ ≠ 0, 1, and “shock-peakons” for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold’s approach to Euler equations on Lie groups.

Symplectic Reduction for Semidirect Products and Central Extensions.

This paper proves a symplectic reduction by stages theorem in the context of geometric mechanics on symplectic manifolds with symmetry groups that are group extensions. We relate the work to the semidirect product reduction theory developed in the 1980’s by Marsden, Ratiu, Weinstein, Guillemin and Sternberg as well as some more recent results and we recall how semidirect product reduction finds use in examples, such as the dynamics of an underwater vehicle. We shall start with the classical cases of commuting reduction (first appearing in Marsden and Weinstein [1974]) and present a new proof

The cauchy problem for a shallow water type equation

We prove existence and uniqueness of local and global solutions of the periodic Cauchy problem for a higher order shallow water type equation under low regularity initial data. Using Fourier analysis we first prove local estimates in appropriate spaces and then use a contraction mapping argument and a conserved norm to get global existence.

Euler and Navier–Stokes equations on the hyperbolic plane

We show that nonuniqueness of the Leray–Hopf solutions of the Navier–Stokes equation on the hyperbolic plane ℍ2 observed by Chan and Czubak is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on ℍn whenever n ≥ 3. We also describe the corresponding general Hamiltonian framework of hydrodynamics on complete Riemannian manifolds, which includes the hyperbolic setting.

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 appearance of shocks. We show that shock formation corresponds to a focal point of the group of volume-preserving diffeomorphisms regarded as a submanifold of the full diffeomorphism group and, consequently, to a conjugate point along a geodesic in the Wasserstein space of densities. This relates the ideal Euler hydrodynamics (via Arnold’s approach) to shock formation in the multidimensional Burgers equation and the Kantorovich-Wasserstein geometry of the space of densities.

Exponential maps of Sobolev metrics on loop groups

We find conditions for the exponential map on a weak riemannian Hilbert manifold to be a nonlinear Fredholm map of index zero and apply the result to left-invariant Sobolev metrics on loop groups.