ISSN:

1531-3492

eISSN:

1553-524X

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### Volume 10, 2008

## Discrete & Continuous Dynamical Systems - B

May 2003 , Volume 3 , Issue 2

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2003, 3(2): 145-162
doi: 10.3934/dcdsb.2003.3.145

*+*[Abstract](579)*+*[PDF](185.9KB)**Abstract:**

Due to their mathematical tractability, two-dimensional (2D) fluid equations are often used by mathematicians as a model for quasi-geostrophic (QG) turbulence in the atmosphere, using Charney's 1971 paper as justification. Superficially, 2D and QG turbulence both satisfy the twin conservation of energy and enstrophy and thus are unlike 3D flows, which do not conserve enstrophy. Yet QG turbulence differs from 2D turbulence in fundamental ways, which are not generally known. Here we discuss ingredients missing in 2D turbulence formulations of large-scale atmospheric turbulence. We argue that there is no proof that energy cannot cascade downscale in QG turbulence. Indeed, observational evidence supports a downscale flux of both energy and enstrophy in the mesoscales. It is suggested that the observed atmospheric energy spectrum is explainable if there is a downscale energy cascade of QG turbulence, but is inconsistent with 2D turbulence theories, which require an upscale energy flux. A simple solved example is used to illustrate some of the ideas discussed.

2003, 3(2): 163-177
doi: 10.3934/dcdsb.2003.3.163

*+*[Abstract](606)*+*[PDF](422.0KB)**Abstract:**

This work revisits a couple of well-known piecewise linear oscillators pointing out several unnoticed properties. In particular, for one of these oscillators we study under what conditions bounded motions are possible and investigate the effect of viscous damping on its trajectories. The article complements a relatively recent paper by Capecchi [10] and presents a non-trivial counterexample to the wide-spread belief according to which chaos is ubiquitous in piecewise linear systems.

2003, 3(2): 179-192
doi: 10.3934/dcdsb.2003.3.179

*+*[Abstract](592)*+*[PDF](257.6KB)**Abstract:**

We find the explicit solution of the so-called two-mode model for multicomponent Bose-Einstein condensates. We prove that all the solutions are constants or periodic functions and give explicit formulae for them.

2003, 3(2): 193-200
doi: 10.3934/dcdsb.2003.3.193

*+*[Abstract](608)*+*[PDF](126.0KB)**Abstract:**

Countable many weakly coupled reversible oscillators are investigated. Homoclinic structures are assumed for the anti-integrable limit equations. The existence of infinitely many homoclinic solutions is shown for the chains of perturbed oscillators and each of the homoclinic solutions is accumulated by continuum many breathers with periods tending to infinity. A similar result is shown for the case when heteroclinic loop structures are assumed for the anti-integrable limit equations. Applications are given to several models.

2003, 3(2): 201-228
doi: 10.3934/dcdsb.2003.3.201

*+*[Abstract](590)*+*[PDF](247.0KB)**Abstract:**

A class of upwind flux splitting methods in the Euler equations of compressible flow is considered in this paper. Using the property that Euler flux $F(U)$ is a homogeneous function of degree one in $U$, we reformulate the splitting fluxes with $F^{+}=A^{+} U$, $F^{-}=A^{-} U$, and the corresponding matrices are either symmetric or symmetrizable and keep only non-negative and non-positive eigenvalues. That leads to the conclusion that the first order schemes are positive in the sense of Lax-Liu [18], which implies that it is $L^2$-stable in some suitable sense. Moreover, the second order scheme is a stable perturbation of the first order scheme, so that the positivity of the second order schemes is also established, under a CFL-like condition. In addition, these splitting methods preserve the positivity of density and energy.

2003, 3(2): 229-253
doi: 10.3934/dcdsb.2003.3.229

*+*[Abstract](723)*+*[PDF](273.7KB)**Abstract:**

In this article, we describe the basic properties of the Wang Chang-Uhlenbeck equations. Then, we obtain the classical H-theorem, the Gibbs theorem and the convergence toward an unique maxwellian equilibrium in the spatially homogeneous case. And, by choosing a particular cross sections model, we formally deduce the fluid limit which is the hyperbolic multispecies Euler system closed with a non classical state equation.

2003, 3(2): 255-262
doi: 10.3934/dcdsb.2003.3.255

*+*[Abstract](694)*+*[PDF](131.4KB)**Abstract:**

We prove the existence of recurrent or Poisson stable motions in the Navier-Stokes fluid system under recurrent or Poisson stable forcing, respectively. We use an approach based on nonautonomous dynamical systems ideas.

2003, 3(2): 263-284
doi: 10.3934/dcdsb.2003.3.263

*+*[Abstract](502)*+*[PDF](592.2KB)**Abstract:**

We present an application of the transport theory developed for area preserving dynamical systems, to the problem of pollution and in particular patchiness in clouds of pollution in partially stratified estuaries. We model the flow in such estuaries using a $3+1$ dimensional uncoupled cartoon of the dominant underlying global circulation mechanisms present within the estuarine flow. We separate the cross section up into different regions, bounded by partial and complete barriers. Using these barriers we then provide predictions for the lower bound on the vertical local flux. We also present work on the relationship between the time taken for a particle to leave the estuary, (ie. the exit time), and the mixing within the estuary. This link is important as we show that to optimally discharge pollution into an estuary both concepts have to be considered. We finish by suggesting coordinates in space time for an optimal discharge site and a discharge policy to ensure the continually optimal discharge from such a site (or even a non optimal site).

2003, 3(2): 285-298
doi: 10.3934/dcdsb.2003.3.285

*+*[Abstract](713)*+*[PDF](180.7KB)**Abstract:**

In this paper we consider the coupled PDE system which describes a composite (sandwich) beam, as recently proposed in [H.1], [H-S.1]: it couples the transverse displacement $w$ and the effective rotation angle $\xi$ of the beam. We show that by introducing a suitable new variable $\theta$, the original model in the original variables $\{w,\xi\}$ of the sandwich beam is transformed into a

*canonical thermoelastic system*in the new variables $\{w,\theta\}$,

*modulo lower-order terms*. This reduction then allows us to re-obtain recently established results on the sandwich beam--which had been proved by a direct, ad hoc technical analysis [H-L.1]--simply as corollaries of previously established corresponding results [A-L.1], [A-L.2], [L-T.1]--[L-T.5] on thermoelastic systems. These include the following

*known*results [H-L.1] for sandwich beams: (i) well-posedness in the semigroup sense; (ii) analyticity of the semigroup when rotational forces are not accounted for; (iii) structural decomposition of the semigroup when rotational forces are accounted for; and (iv) uniform stability.

In addition, however, through the aforementioned reduction to thermoelastic problems, we here establish

*new*results for sandwich beams, when rotational forces are accounted for. They include: (i) a backward uniqueness property (Section 4), and (ii) a suitable singular estimate, critical in control theory (Section 5). Finally, we obtain a

*new*backward uniqueness property, this time for a structural acoustic chamber having a composite (sandwich) beam as its flexible wall (Section 6).

2003, 3(2): 299-309
doi: 10.3934/dcdsb.2003.3.299

*+*[Abstract](838)*+*[PDF](131.4KB)**Abstract:**

Periodic oscillations are proved for an SIRS disease transmission model in which the size of the population varies and the incidence rate is a nonlinear function. For this particular incidence function, analytical techniques are used to show that, for some parameter values, periodic solutions can arise through a Hopf bifurcation and disappear through a homoclinic loop bifurcation. The existence of periodicity is important as it may indicate different strategies for controlling disease.

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