arXiv subjects

Martin Kohlmann

Publications and source records attributed to Martin Kohlmann.

At least 19 recordsLinked to original sources

arXiv2015

Curvature computations for a two-component Camassa-Holm equation with vorticity

In the present paper, a two-component Camassa-Holm (2CH) system with vorticity is studied as a geodesic flow on a suitable Lie group. The paper aims at presenting various details of the geometric formalism and a major result is the computation of the sectional curvature $K$ of the underlying configuration manifold. As a further result, we show that there are directions for which $K$ is strictly positive and bounded away from zero.

math.AP↗

arXiv2014

Bound States for Nano-Tubes with a Dislocation

As a model for an interface in solid state physics, we consider two real-valued potentials $V^{(1)}$ and $V^{(2)}$ on the cylinder or tube $S=\mathbb R \times (\mathbb R/\mathbb Z)$ where we assume that there exists an interval $(a_0,b_0)$ which is free of spectrum of $-\Delta+V^{(k)}$ for $k=1,2$. We are then interested in the spectrum of $H_t = -\Delta + V_t$, for $t \in \mathbb R$, where $V_t(x,y) = V^{(1)}(x,y)$, for $x > 0$, and $V_t(x,y) = V^{(2)}(x+t,y)$, for $x < 0$. While the essential spectrum of $H_t$ is independent of $t$, we show that discrete spectrum, related to the interface at $x = 0$, is created in the interval $(a_0, b_0)$ at suitable values of the parameter $t$, provided $-\Delta + V^{(2)}$ has some essential spectrum in $(-\infty, a_0]$. We do not require $V^{(1)}$ or $V^{(2)}$ to be periodic. We furthermore show that the discrete eigenvalues of $H_t$ are Lipschitz continuous functions of $t$ if the potential $V^{(2)}$ is locally of bounded variation.

math-ph↗

arXiv2014

The abstract quasilinear Cauchy problem for a MEMS model with two free boundaries

In this paper, we reformulate a mathematical model for the dynamics of an idealized electrostatically actuated MEMS device with two elastic membranes as an initial value problem for an abstract quasilinear evolution equation. Applying the Contraction Mapping Theorem, it is shown that the model is locally well-posed in time for any value of the source voltage of the device. In addition it is proven that the MEMS model considered here possesses global solutions for small source voltages whereas for large source voltages solutions of the model have a finite maximal existence time. Furthermore, we comment on the relationship of our model to its stationary version and to its small aspect ratio limit by showing that there exists a unique exponentially stable steady state and by proving convergence towards a solution of the narrow gap model in the vanishing aspect ratio limit. Our results extend the discussion of the elliptic-parabolic MEMS model presented in [Martin Kohlmann: On an elliptic-parabolic MEMS model with two free boundaries, 2014] leading to a Cauchy problem for a semilinear abstract evolution equation.

math.AP↗

arXiv2014

A new model for electrostatic MEMS with two free boundaries

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.

math.AP↗

arXiv2014

On an elliptic-parabolic MEMS model with two free boundaries

We discuss an evolution free boundary problem of mixed type with two free boundaries modeling an idealized electrostatically actuated MEMS device. While the electric potential is the solution of an elliptic equation, the dynamics of the membranes' displacement is modeled by two parabolic equations. It is shown that the model is locally well-posed in time and that solutions exist globally for small source voltages whereas non-existence holds for large voltage values. Moreover, our model possesses a steady state solution that is asymptotically stable. Finally, we show that in the vanishing aspect ratio limit, solutions of the model converge towards solutions of the associated small aspect ratio problem.

math.AP↗

arXiv2013

The two-component Camassa-Holm system in weighted $L_p$ spaces

We present some new persistence results for the non-periodic two-component Camassa-Holm (2CH) system in weighted $L_p$ spaces. Working with moderate weight functions that are commonly used in time-frequency analysis, the paper generalizes some recent persistence results for the Camassa-Holm equation [L. Brandolese, Int. Math. Res. Notices 22 (2012) 5161-81] to its supersymmetric extension. As an application we discuss the spatial asymptotic profile of solutions to 2CH.

math.AP↗

arXiv2011

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 Arnold's [V.I. Arnold, Sur la g\'eom\'etrie diff\'erentielle des groupes de Lie de dimension infinie et ses applications \`a l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16 (1966) 319-361], [D.G. Ebin and J.E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid. Ann. of Math. 92(2) (1970) 102-163] geometric formalism 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 the paper [M. Kohlmann, The two-dimensional periodic $b$-equation on the diffeomorphism group of the torus. J. Phys. A.: Math. Theor. 44 (2011) 465205 (17 pp.)] we present geometric aspects of a two-dimensional periodic $\mu$-$b$-equation on the diffeomorphism group of the torus in this context.

math.AP↗

arXiv2011

The periodic two-dimensional $\mu$-$b$-equation as an EPDiff equation

We introduce a periodic two-dimensional $\mu$-$b$-equation and a periodic two-dimensional two-component $(\mu)$-Camassa-Holm equation which we study as geodesic flows on the diffeomorphism group of the torus and a semidirect product respectively. The paper explains the derivation of these equations within V.I. Arnold's (1966) general framework, some analogies to recently discussed related equations and gives a self-contained presentation of the geometric aspects. As an application, we obtain well-posedness results and some explicit curvature computations.

math.AP↗

arXiv2011

The Burgers equation and the Korteweg-de Vries equation with quadratic nonlinearity

We study generalized variants of the Burgers equation and the KdV equation on the circle. The main goal of the paper is to show that both extensions can be recast as geodesic equations on a suitable diffeomorphism group of the circle and the corresponding Bott-Virasoro group respectively. As a consequence we obtain that the initial value problem for the Burgers equation with an additional quadratic term is well-posed on a scale of Sobolev spaces on the circle.

math.AP↗

arXiv2011

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 $\T=S^1\times S^1$. In the framework of Arnold's [V.I. Arnold, Sur la g\'eom\'etrie diff\'erentielle des groupes de Lie de dimension infinie et ses applications \`a l'hydrodynamique des fluides parfaits. Ann. Inst. Fourier (Grenoble) 16 (1966) 319-361] 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.

math.AP↗

arXiv2011

A note on a multi-layer tumor growth model

A two-dimensional free boundary model for the growth of multi-layer tumors has been proposed in [S. Cui, J. Escher: ARMA 191 (2009) 173-193] where the authors derive well-posedness in a functional analytic setting, the stationary solutions and their asymptotic stability. In this note, we consider once again the cancer growth model of [S. Cui, J. Escher: ARMA 191 (2009) 173-193] but coping without periodic boundary conditions for the $x$ variable which is reasonable from the point of view of modeling. The paper points out that the generalized model allows for the same stationary solutions as compared to the model with periodic boundary conditions.

math.AP↗

arXiv2011

Necrotic tumor growth: an analytic approach

The present paper deals with a free boundary problem modeling the growth process of necrotic multi-layer tumors. We prove the existence of flat stationary solutions and determine the linearization of our model at such an equilibrium. Finally, we compute the solutions of the stationary linearized problem and comment on bifurcation.

math.AP↗

arXiv2011

Analysis of a mathematical model for the growth of cancer cells

In this paper, a two-dimensional model for the growth of multi-layer tumors is presented. The model consists of a free boundary problem for the tumor cell membrane and the tumor is supposed to grow or shrink due to cell proliferation or cell dead. The growth process is caused by a diffusing nutrient concentration $\sigma$ and is controlled by an internal cell pressure $p$. We assume that the tumor occupies a strip-like domain with a fixed boundary at $y=0$ and a free boundary $y=\rho(x)$, where $\rho$ is a $2\pi$-periodic function. First, we prove the existence of solutions $(\sigma,p,\rho)$ and that the model allows for peculiar stationary solutions. As a main result we establish that these equilibrium points are locally asymptotically stable under small perturbations.

math.AP↗

arXiv2011

Well-posedness and blow-up for a two-component Degasperis-Procesi equation with infinitely fast propagating solutions

In this paper, a two-component variant of the Degasperis-Procesi equation on the real line is discussed. Applying Kato's theory, we first prove the local well-posedness for the equation under consideration in $H^s\times H^{s-1}$, for $s\geq 2$. Second we establish the precise blow-up scenario. For compactly supported initial data, we show that the associated solution does not have compact support for any positive time; the localized initial disturbance propagates with an infinite speed. Although the solution is no longer compactly supported we prove that it decays at an exponentially fast rate for the duration of its existence.

math.AP↗

arXiv2011

On initial boundary value problems for variants of the Hunter-Saxton equation

The Hunter-Saxton equation serves as a mathematical model for orientation waves in a nematic liquid crystal. The present paper discusses a modified variant of this equation, coming up in the study of critical points for the speed of orientation waves, as well as a two-component extension. We establish well-posedness and blow-up results for some initial boundary value problems for the modified Hunter-Saxton equation and the two-component Hunter-Saxton system.

math.AP↗

arXiv2011

On a two-component $\pi$-Camassa--Holm system

A novel $\pi$-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.

math.AP↗

arXiv2011

Dislocation problems for periodic Schr\"odinger 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 produced by a rotation of the lattice in a half-space. For Lipschitz-continuous and $\Z^2$-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. We 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. To introduce our methods, we begin with the study of dislocation problems on the real line and on an infinite strip. Finally, we consider examples of muffin tin type. Our overview refers to results in [HK1, HK2].

math-ph↗

arXiv2010

Global existence and blow-up for a weakly dissipative $\mu$DP equation

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.

math-ph↗