ISSN:
1078-0947
eISSN:
1553-5231
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Discrete & Continuous Dynamical Systems - A
July 2000 , Volume 6 , Issue 3
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2000, 6(3): 503-518
doi: 10.3934/dcds.2000.6.503
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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.
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.
2000, 6(3): 519-536
doi: 10.3934/dcds.2000.6.519
+[Abstract](1161)
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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.
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.
2000, 6(3): 537-556
doi: 10.3934/dcds.2000.6.537
+[Abstract](1315)
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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.
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.
2000, 6(3): 557-574
doi: 10.3934/dcds.2000.6.557
+[Abstract](1550)
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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.
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.
2000, 6(3): 575-590
doi: 10.3934/dcds.2000.6.575
+[Abstract](1524)
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Abstract:
The paper deals with the boundary value problem on the whole line
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.
2000, 6(3): 591-608
doi: 10.3934/dcds.2000.6.591
+[Abstract](1968)
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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.
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.
2000, 6(3): 609-624
doi: 10.3934/dcds.2000.6.609
+[Abstract](1508)
+[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.
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.
2000, 6(3): 625-644
doi: 10.3934/dcds.2000.6.625
+[Abstract](1295)
+[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.
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.
2000, 6(3): 645-650
doi: 10.3934/dcds.2000.6.645
+[Abstract](1066)
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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.
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.
2000, 6(3): 651-664
doi: 10.3934/dcds.2000.6.651
+[Abstract](1342)
+[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.
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.
2000, 6(3): 665-672
doi: 10.3934/dcds.2000.6.665
+[Abstract](1435)
+[PDF](957.4KB)
Abstract:
We give a sufficient condition under which a coupled vibrating system is exactly controllable in a square area.
We give a sufficient condition under which a coupled vibrating system is exactly controllable in a square area.
2000, 6(3): 673-682
doi: 10.3934/dcds.2000.6.673
+[Abstract](1772)
+[PDF](203.5KB)
Abstract:
Consider the Cauchy problem for a hyperbolic $n\times n$ system of conservation laws in one space dimension:
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.
2000, 6(3): 683-690
doi: 10.3934/dcds.2000.6.683
+[Abstract](2035)
+[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.
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.
2000, 6(3): 691-704
doi: 10.3934/dcds.2000.6.691
+[Abstract](1445)
+[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.
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.
2000, 6(3): 705-722
doi: 10.3934/dcds.2000.6.705
+[Abstract](1661)
+[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.
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.
2000, 6(3): 723-740
doi: 10.3934/dcds.2000.6.723
+[Abstract](1425)
+[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.
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.
2000, 6(3): 741-749
doi: 10.3934/dcds.2000.6.741
+[Abstract](1621)
+[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.
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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