Martin Kohlmann

THE GEOMETRY OF THE TWO-COMPONENT CAMASSA-HOLM AND DEGASPERIS-PROCESI EQUATIONS

We use geometric methods to study two natural two-component generalizations of the periodic Camassa–Holm and Degasperis–Procesi equations. We show that these generalizations can be regarded as geodesic equations on the semidirect product of the diffeomorphism group of the circle Diff(S1) with some space of sufficiently smooth functions on the circle. Our goals are to understand the geometric properties of these two-component systems and to prove local well-posedness in various function spaces. Furthermore, we perform some explicit curvature calculations for the two-component Camassa–Holm equation, giving explicit examples of large subspaces of positive curvature.

The curvature of semidirect product groups associated with two-component Hunter–Saxton systems

In this paper, we study two-component versions of the periodic Hunter?Saxton equation and its ?-variant. Considering both equations as a geodesic flow on the semidirect product of the circle diffeomorphism group with a space of scalar functions on we show that both equations are locally well posed. The main result of this paper is that the sectional curvature associated with the 2HS is constant and positive and that 2?HS allows for a large subspace of positive sectional curvature. The issues of this paper are related to some of the results for 2CH and 2DP presented in Escher et al (2011 J. Geom. Phys. 61 436?52).

On a two-component π-Camassa–Holm system

Abstract A novel π -Camassa–Holm system is studied as a geodesic flow on a semidirect product obtained from the diffeomorphism group of the circle. We present the corresponding details of the geometric formalism for metric Euler equations on infinite-dimensional Lie groups and compare our results to what has already been obtained for the usual two-component Camassa–Holm equation. Our approach results in well-posedness theorems and explicit computations of the sectional curvature.

Global existence and blow-up for a weakly dissipative μDP equation

Abstract In this paper, we study a weakly dissipative variant of the periodic Degasperis–Procesi equation. We show the local well-posedness of the associated Cauchy problem in H s ( S ) , s > 3 / 2 , and discuss the precise blow-up scenario for s = 3 . We also present explicit examples for globally existing solutions and blow-up.

THE PERIODIC μ-b-EQUATION AND EULER EQUATIONS ON THE CIRCLE

In this paper, we study the μ-variant of the periodic b-equation and show that this equation can be realized as a metric Euler equation on the Lie group Diff ∞() if and only if b = 2 (for which it becomes the μ-Camassa–Holm equation). In this case, the inertia operator generating the metric on Diff ∞() is given by . In contrast, the μ-Degasperis–Procesi equation (obtained for b = 3) is not a metric Euler equation on Diff ∞() for any regular inertia operator . The paper generalizes some recent results of [13, 16, 24].

A new model for electrostatic MEMS with two free boundaries

Abstract A moving boundary problem with two free boundaries modeling a two-dimensional idealized MEMS device with pull-in instability is discussed. We use a fixed point argument to show that the model possesses stationary solutions for small source voltages. We also give a rigorous evidence that solutions of the model converge towards solutions of the associated small aspect ratio equation in the vanishing aspect ratio limit.

Geometric Aspects of the Periodic μ-Degasperis-Procesi Equation

We consider the periodic μDP equation (a modified version of the Degasperis-Procesi equation) as the geodesic flow of a right-invariant affine connection ∇ on the Frechet Lie group Diff∞(\(\mathbb{S}^1\)) of all smooth and orientation-preserving diffeomorphisms of the circle \(\mathbb{S}^1\,=\,\mathbb{R}/\mathbb{Z}\). On the Lie algebra C∞(\(\mathbb{S}^1\)) of Diff∞(\(\mathbb{S}^1\)), this connection is canonically given by the sum of the Lie bracket and a bilinear operator. For smooth initial data, we show the short time existence of a smooth solution of μDP which depends smoothly on time and on the initial data. Furthermore, we prove that the exponential map defined by ∇ is a smooth local diffeomorphism of a neighbourhood of zero in C∞(\(\mathbb{S}^1\)) onto a neighbourhood of the unit element in Diff∞(\(\mathbb{S}^1\)). Our results follow from a general approach on non-metric Euler equations on Lie groups, a Banach space approximation of the Frechet space C∞(\(\mathbb{S}^1\)), and a sharp spatial regularity result for the geodesic flow.

A note on multi-dimensional Camassa–Holm-type systems on the torus

We present a 2n-component nonlinear evolutionary PDE which includes the n-dimensional versions of the Camassa–Holm and the Hunter–Saxton systems as well as their partially averaged variations. Our goal is to apply Arnolds geometric formalism (Arnold 1966 Ann. Inst. Fourier (Grenoble) 16 319–61; Ebin and Marsden 1970 Ann. Math. 92 102–63) to this general equation in order to obtain results on well-posedness, conservation laws or stability of its solutions. Following the line of arguments of Kohlmann (2011 J. Phys. A: Math. Theor. 44 465205), we present geometric aspects of a two-dimensional periodic μ-b-equation on the diffeomorphism group of the torus in this context.

The two-dimensional periodic b-equation on the diffeomorphism group of the torus

In this paper, the two-dimensional periodic b-equation is discussed under geometric aspects, i.e. as a geodesic flow on the diffeomorphism group of the torus . In the framework of Arnold?s (1966 Ann. Inst. Fourier (Grenoble) 16 319) famous approach, we achieve some well-posedness results for the b-equation and we perform explicit curvature computations for the 2D Camassa?Holm equation, which is obtained for b = 2. Finally, we explain the special role of the choice b = 2 by giving a rigorous proof that b = 2 is the only case in which the associated geodesic flow is weakly Riemannian.

Dislocation Problems for Periodic Schrödinger Operators and Mathematical Aspects of Small Angle Grain Boundaries

We discuss two types of defects in two-dimensional lattices, namely (1) translational dislocations and (2) defects roduced by a rotation of the lattice in a half-space. For Lipschitz-continuous and ℤ2-periodic potentials, we first show that translational dislocations produce spectrum inside the gaps of the periodic problem; we also give estimates for the (integrated) density of the associated surface states.W e then study lattices with a small angle defect where we find that the gaps of the periodic problem fill with spectrum as the defect angle goes to zero.T o introduce our methods, we begin with the study of dislocation problems on the real line and on an infinite strip.F inally, we consider examples of muffin tin type.O ur overview refers to results in [HK1, HK2].