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Discrete & Continuous Dynamical Systems - A

2000 , Volume 6 , Issue 3

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Extended gradient systems: Dimension one
Siniša Slijepčević
2000, 6(3): 503-518 doi: 10.3934/dcds.2000.6.503 +[Abstract](243) +[PDF](2633.1KB)
Abstract:
We propose a general theorey of formally gradient differential equations on unbounded one-dimensional domains, based on an energy-flow inequality, and on the study of the induced semiflow on the space of probability measures on the phase space. We prove that the $\omega$-limit set of each point contains an equilibrium, and that the $\omega$-limit set of $\mu$-almost every point in the phase space consists of equilibria, where $\mu$ is any Borel probability measure invariant for spatial translation.
On pairs of differential $1$-forms in the plane
R.D.S. Oliveira and F. Tari
2000, 6(3): 519-536 doi: 10.3934/dcds.2000.6.519 +[Abstract](209) +[PDF](267.9KB)
Abstract:
We classify pairs of germs of differential $1$-forms $(\alpha, beta)$ in the plane, where $\alpha$, $beta$ are either regular or have a singularity of type saddle/node/focus. The main tools used here are singularity theory and the method of polar blowing up. We also present a desingularization theorem for pairs of germs of differential $1$-forms in the plane.
A note on boundary layer effects in periodic homogenization with Dirichlet boundary conditions
Micol Amar
2000, 6(3): 537-556 doi: 10.3934/dcds.2000.6.537 +[Abstract](188) +[PDF](276.2KB)
Abstract:
This paper focus on the properties of boundary layers in periodic homogenization of Dirichlet boundary value problems. We consider here the case of Dirichlet problems in rectangular domains which have an oscillating boundary, emphasizing the influence of boundary layers on interior error estimates.
The maximum principle in infinite dimension
H. O. Fattorini
2000, 6(3): 557-574 doi: 10.3934/dcds.2000.6.557 +[Abstract](219) +[PDF](233.7KB)
Abstract:
We provide a counterexample to the maximum principle for the minimum norm problem and establish several relations between this problem and the time optimal problem. The system is linear, infinite dimensional, with point target and "full" control.
Existence of bounded trajectories via upper and lower solutions
Luisa Malaguti and Cristina Marcelli
2000, 6(3): 575-590 doi: 10.3934/dcds.2000.6.575 +[Abstract](238) +[PDF](219.1KB)
Abstract:
The paper deals with the boundary value problem on the whole line

$u'' - f(u,u') + g(u) = 0 $

$u(\infty,1) = 0$,   $ u(+\infty) = 1$     $(P)$

where $g : R \to R$ is a continuous non-negative function with support $[0,1]$, and $f : R^2\to R$ is a continuous function. By means of a new approach, based on a combination of lower and upper-solutions methods and phase-plane techniques, we prove an existence result for $(P)$ when $f$ is superlinear in $u'$; by a similar technique, we also get a non-existence one. As an application, we investigate the attractivity of the singular point $(0,0)$ in the phase-plane $(u,u')$. We refer to a forthcoming paper [13] for a further application in the field of front-type solutions for reaction diffusion equations.

Dynamic theory for incompressible Smectic-A liquid crystals: Existence and regularity
Chun Liu
2000, 6(3): 591-608 doi: 10.3934/dcds.2000.6.591 +[Abstract](334) +[PDF](198.2KB)
Abstract:
We study a system that was proposed in [3] in order to model the dynamic of Smectic-A liquid crystals. We establish the energy dissipative relation of the system and prove the existence of global weak solutions. A higher order energy estimate is also established for the existence of the classical solutions and the regularity of the weak solutions. Some regularity and stability results are also discussed in this paper.
Center-focus and isochronous center problems for discontinuous differential equations
B. Coll, A. Gasull and R. Prohens
2000, 6(3): 609-624 doi: 10.3934/dcds.2000.6.609 +[Abstract](244) +[PDF](258.2KB)
Abstract:
The study of the center focus problem and the isochronicity problem for differential equations with a line of discontinuities is usually done by computing the whole return map as the composition of the two maps associated to the two smooth differential equations. This leads to large formulas which usually are treated with algebraic manipulators. In this paper we approach to this problem from a more theoretical point of view. The results that we obtain relate the order of degeneracy of the critical point of the discontinuous differential equations with the order of degeneracy of the two smooth component differential equations. Finally we apply them to some families of examples.
Asymptotic smoothing effect for weakly damped forced Korteweg-de Vries equations
Olivier Goubet
2000, 6(3): 625-644 doi: 10.3934/dcds.2000.6.625 +[Abstract](179) +[PDF](243.5KB)
Abstract:
Weakly damped forced KdV equation provides a dissipative semigroup on $L_x^2$. We prove that this semigroup enjoys an asymptotic smoothing effect, i.e. that all solutions converge towards a set of smoother solutions, when time goes to infinity.
On rigidity properties of contact time changes of locally symmetric geodesic flows
Jeffrey Boland
2000, 6(3): 645-650 doi: 10.3934/dcds.2000.6.645 +[Abstract](168) +[PDF](1014.2KB)
Abstract:
In analogy with the geometric $1/4$-pinching and entropy rigidiity of compact negatively curved locally symmetric spaces, we study in this note the dynamical rigidity of contact time changes of the geodesic flow for these spaces.
A PDE approach to stochastic invariance
Hitoshi Ishii, Paola Loreti and Maria Elisabetta Tessitore
2000, 6(3): 651-664 doi: 10.3934/dcds.2000.6.651 +[Abstract](224) +[PDF](187.3KB)
Abstract:
We study an invariance property for a controlled stochastic differential equation and give a few of its characterizations in connection with the corresponding Hamilton-Jacobi-Bellman equation.
Exact boundary controllability of a coupled system
Mohammed Aassila
2000, 6(3): 665-672 doi: 10.3934/dcds.2000.6.665 +[Abstract](221) +[PDF](957.4KB)
Abstract:
We give a sufficient condition under which a coupled vibrating system is exactly controllable in a square area.
A uniqueness condition for hyperbolic systems of conservation laws
Alberto Bressan and Marta Lewicka
2000, 6(3): 673-682 doi: 10.3934/dcds.2000.6.673 +[Abstract](286) +[PDF](203.5KB)
Abstract:
Consider the Cauchy problem for a hyperbolic $n\times n$ system of conservation laws in one space dimension:

$u_t + f(u)_x = 0$,   $u(0,x)=\bar u (x).$       $(CP)$

Relying on the existence of a continuous semigroup of solutions, we prove that the entropy admissible solution of $(CP)$ is unique within the class of functions $u=u(t,x)$ which have bounded variation along a suitable family of space-like curves.

Symmetry results for functions yielding best constants in Sobolev-type inequalities
Bernhard Kawohl
2000, 6(3): 683-690 doi: 10.3934/dcds.2000.6.683 +[Abstract](379) +[PDF](179.0KB)
Abstract:
I report on symmetry results for functions which yield sharp constants in various Sobolev-type inequalities. One of these results relies on a surprising convexity property.
Multipliers of homoclinic orbits on surfaces and characteristics of associated invariant sets
V. Afraimovich and T.R. Young
2000, 6(3): 691-704 doi: 10.3934/dcds.2000.6.691 +[Abstract](213) +[PDF](241.9KB)
Abstract:
Suppose that $f$ is a surface diffeomorphism with a hyperbolic fixed point $\mathcal O$ and this fixed point has a transversal homoclinic orbit. It is well known that in a vicinity of this type of homoclinic there are hyperbolic invariants sets. We introduce smooth invariants for the homoclinic orbit which we call the multipliers. As an application, we study the influence of the multipliers on numerical invariants of the hyperbolic invariant sets as the vicinity becomes small.
The Schrödinger equation with singular time-dependent potentials
Holger Teismann
2000, 6(3): 705-722 doi: 10.3934/dcds.2000.6.705 +[Abstract](258) +[PDF](292.7KB)
Abstract:
The aim of this note is to extend the theory of (linear) Schrödinger equations with time-dependent potentials developed by K. Yajima [26, 27] to slightly more singular potentials. This is done by proving that the well-known Strichartz estimates for the Schrödinger group remain valid if the usual Lebesgue spaces$^1$ are replaced by the Lorentz spaces $L^{p,2}$. Moreover, the regularity of the solutions can be described more precisely by utilizing a generalized Leibniz rule for fractional derivatives.
A quantized henon map
John Erik Fornæss and Brendan Weickert
2000, 6(3): 723-740 doi: 10.3934/dcds.2000.6.723 +[Abstract](227) +[PDF](240.4KB)
Abstract:
We quantize the classical Henon map on $\mathbb R^2$, obtaining a unitary map on $L^2 (\mathbb R)$ whose dynamics we study, developing analogies to the classical dynamics.
On pairs of foliations defined by vector fields in the plane
Jorge Sotomayor and Michail Zhitomirskii
2000, 6(3): 741-749 doi: 10.3934/dcds.2000.6.741 +[Abstract](240) +[PDF](176.1KB)
Abstract:
We obtain a smooth and analytic local classification of pairs of foliations of the plane assuming that one of the foliations is defined by a nonsingular vector field and the other foliation is defined by a singular vector field.

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