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Discrete & Continuous Dynamical Systems - A

1996 , Volume 2 , Issue 1

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Partial regularity of the dynamic system modeling the flow of liquid crystals
Fanghua Lin and Chun Liu
1996, 2(1): 1-22 doi: 10.3934/dcds.1996.2.1 +[Abstract](469) +[PDF](222.9KB)
Abstract:
Here we established the partial regularity of suitable weak solutions to the dynamical systems modelling the flow of liquid crystals. It is a natural generalization of an earlier work of Caffarelli-Kohn-Nirenberg on the Navier-Stokes system with some simplifications due to better estimates on the pressure term.
Evolution equations governed by the subdifferential of a convex composite function in finite dimensional spaces
Sophie Guillaume
1996, 2(1): 23-52 doi: 10.3934/dcds.1996.2.23 +[Abstract](336) +[PDF](330.1KB)
Abstract:
Under quite general assumptions, we prove existence, uniqueness and regularity of a solution $U$ to the evolution equation $-U'(t)\in\partial(g\circ F)(U(t))$, $U(0)=u_0$, where $g:\mathbb{R}^q\rightarrow\mathbb{R}\cup\{+\infty\}$ is a closed proper convex function, $F:\mathbb{R}^p\rightarrow \mathbb{R}^q$ is a continuously differentiable mapping whose gradient is Lipschitz continuous on bounded subsets and $u_0\in\dom (g\circ F)$. We also study the asymptotic behavior of $U$ and give an application to nonlinear mathematical programming.
Time dependent Volterra integral inclusions in Banach spaces
Sergiu Aizicovici, Yimin Ding and N. S. Papageorgiou
1996, 2(1): 53-63 doi: 10.3934/dcds.1996.2.53 +[Abstract](383) +[PDF](193.7KB)
Abstract:
A nonlinear Volterra inclusion associated to a family of time-dependent $m$-accretive operators, perturbed by a multifunction, is considered in a Banach space. Existence results are established for both nonconvex and convex valued perturbations. The class of extremal solutions is also investigated.
Normal forms for quasiperiodic evolutionary equations
Shui-Nee Chow, Kening Lu and Yun-Qiu Shen
1996, 2(1): 65-94 doi: 10.3934/dcds.1996.2.65 +[Abstract](420) +[PDF](258.2KB)
Abstract:
In this paper, we study the normal forms and analytic conjugacy for a class of analytic quasiperiodic evolutionary equations including parabolic equations and Schrödinger equations. We first obtain a normal form theory. Then as a special case of the normal form theory, we show that if the frequency and the eigenvalues satisfy certain small divisor conditions then the nonlinear equation is locally analytically conjugated to a linear equation. In other words, the normal form is a linear equation.
Upper bound on the dimension of the attractor for nonhomogeneous Navier-Stokes equations
Alain Miranville and Xiaoming Wang
1996, 2(1): 95-110 doi: 10.3934/dcds.1996.2.95 +[Abstract](441) +[PDF](222.2KB)
Abstract:
Our aim in this article is to derive an upper bound on the dimension of the attractor for Navier-Stokes equations with nonhomogeneous boundary conditions. In space dimension two, for flows in general domains with prescribed tangential velocity at the boundary, we obtain a bound on the dimension of the attractor of the form $c\mathcal{R} e^{3/2}$, where $\mathcal{R} e$ is the Reynolds number. This improves significantly on previous bounds which were exponential in $\mathcal{R} e$.
On chain continuity
Ethan Akin
1996, 2(1): 111-120 doi: 10.3934/dcds.1996.2.111 +[Abstract](349) +[PDF](186.2KB)
Abstract:
A number of recent papers examine for a dynamical system $f: X \rightarrow X$ the concept of equicontinuity at a point. A point $x \in X$ is an equicontinuity point for $f$ if for every $\epsilon > 0$ there is a $\delta > 0$ so that the orbit of initial points $\delta$ close to $x$ remains at all times $\epsilon$ close to the corresponding points of the orbit of $x$, i.e. $d(x,x_0) < \delta$ implies $d(f^i(x),f^i(x_0)) \leq \epsilon$ for $i = 1,2,\ldots$. If we suppose that the errors occur not only at the initial point but at each iterate we obtain not the orbit of $x_0$ but a $\delta$-chain, a sequence $\{x_0,x_1,x_2,\ldots\}$ such that $d(f(x_i),x_{i+1}) \leq \delta$ for $i = 0,1,\ldots$. The point $x$ is called a chain continuity point for $f$ if for every $\epsilon > 0$ there is a $\delta > 0$ so that all $\delta$ chains beginning $\delta$ close to $x$ remain $\epsilon$ close to the points of the orbit of $x$, i.e. $d(x,x_0) < \delta$ and $d(f(x_i),x_{i+1}) \leq \delta$ imply $d(f^i(x),x_i) \leq \epsilon$ for $i = 1,2,\ldots$. In this note we characterize this property of chain continuity. Despite the strength of this property, there is a class of systems $(X,f)$ for which the chain continuity points form a residual subset of the space $X$. For a manifold $X$ this class includes a residual subset of the space of homeomorphisms on $X$.
Lorenz equations part II: "randomly" rotated homoclinic orbits and chaotic trajectories
Xinfu Chen
1996, 2(1): 121-140 doi: 10.3934/dcds.1996.2.121 +[Abstract](320) +[PDF](3696.3KB)
Abstract:
The Lorenz equations are a system of ordinary differential equations

$x' =s(y-x), \quad y'= Rx -y-xz, \quad z'= xy -qz,$

where $s$, $R$, and $q$ are positive parameters. We show by a purely analytic proof that for each non-negative integer $N$, there are positive parameters $s, q, $ and $R$ such that the Lorenz system has homoclinic orbits associated with the origin (i.e., orbits that tend to the origin as $t\to \pm \infty$) which can rotate around the $z$-axis $N/2$ times; namely, the $x$-component changes sign exactly $N$ times, the $y$-component changes sign exactly $N+1$ times, and the zeros of $x$ and $y$ are simple and interlace.

On continuous dependence under approximation for groundwater flow models with distributed and pointwise observations
B.G. Fitzpatrick and M.A. Jeffris
1996, 2(1): 141-149 doi: 10.3934/dcds.1996.2.141 +[Abstract](293) +[PDF](177.3KB)
Abstract:
We present in this paper some results on continuous dependence for parameters in a groundwater flow model. These results are crucial for theoretical and computational aspects of least squares estimation of parameters. As is typically the case in field studies, the form of the data is pointwise observation of hydraulic head and hydraulic conductivity at a discrete collection of observation well sites. We prove continuous dependence results for the solution of the groundwater flow equation, with respect to conductivity and boundary values, under certain types of numerical approximation.

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