# American Institute of Mathematical Sciences

ISSN:
1534-0392

eISSN:
1553-5258

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## Communications on Pure & Applied Analysis

June 2002 , Volume 1 , Issue 2

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2002, 1(2): 135-159 doi: 10.3934/cpaa.2002.1.135 +[Abstract](690) +[PDF](279.7KB)
Abstract:
We consider a quantum many-body model describing a system of electrons interacting with themselves and hopping from one ion to another of a one dimensional lattice. We show that the ground state energy of such system, as a functional of the ionic configurations, has local minima in correspondence of configurations described by smooth $\frac{\pi}{pF}$ periodic functions, if the interaction is repulsive and large enough and pF is the Fermi momentum of the electrons. This means physically that a $d=1$ metal develop a periodic distortion of its reticular structure (Peierls instability). The minima are found solving the Eulero-Lagrange equations of the energy by a contraction method.
2002, 1(2): 161-190 doi: 10.3934/cpaa.2002.1.161 +[Abstract](711) +[PDF](290.0KB)
Abstract:
In this paper we introduce a new technique for tracing viscous travlling profiles. To illustrate the method, we consider a special $2\times 2$ hyperbolic system of conversation laws with viscosity, and show that any solution can be locally decomposed as the sume of 2 viscous travlling proflies. This yields the global existence, stability and uniform BV bounds for every solution with suitably small BV data.
2002, 1(2): 191-219 doi: 10.3934/cpaa.2002.1.191 +[Abstract](812) +[PDF](320.9KB)
Abstract:
In this paper we get by the Glimm scheme the existence of a weak solution to the quasilinear wave equation $w_{t t}=( \sigma_n(w_x))_x$ where $\sigma_n(x)=ax+\gamma x^{2n+1}$, $\alpha$, $\gamma>0$ and $n$ is an integer $n\ge 1$ with $w_x(0,t)=0$ for initial data not necessarily small.
2002, 1(2): 221-235 doi: 10.3934/cpaa.2002.1.221 +[Abstract](944) +[PDF](186.0KB)
Abstract:
We compare the vorticity corresponding to a solution of the Lagrangian averaged Euler equations on the plane to a solution of the Navier–Stokes equation with the same initial data, assuming that the averaged Euler potential vorticity is in a certain Besov class of regularity. Then the averaged Euler vorticity stays close to the Navier–Stokes vorticity for a short interval of time as the respective smoothing parameters tend to zero with natural scaling.
2002, 1(2): 237-252 doi: 10.3934/cpaa.2002.1.237 +[Abstract](903) +[PDF](214.6KB)
Abstract:
We construct modified wave operators for the Hartree equation with the long-range potential $|x|^{-1}$ in the whole space of $(1+|x|)^{-s}L^2$ for $s>1/2$. We also have the image, strong continuity and strong asymptotic approximation in the same space. The lower bound of the weight is sharp from the scaling argument. Those maps are homeomorphic onto open subsets, which implies in particular asymptotic completeness for small data.
2002, 1(2): 253-267 doi: 10.3934/cpaa.2002.1.253 +[Abstract](643) +[PDF](183.0KB)
Abstract:
This paper considers a periodic diffusion system representing one predator-two preys and two predator-one prey. We construct via an iteration scheme, a box that closes in solutions imposing conditions in the spectra of linear differential operators associated with the original reaction-diffusion system.
2002, 1(2): 269-279 doi: 10.3934/cpaa.2002.1.269 +[Abstract](726) +[PDF](171.5KB)
Abstract:
In this paper we consider the following nonlinear plate equations:

$u_{t t} +\Delta \Delta u + m u = \psi(x, u) ,$

$\psi(x, u) = \pm u^3 + O(u^5),\quad \psi(-x,-u)=-\psi(x,u), \qquad\qquad\qquad$(1)

with Navier boundary conditions in a $n$–dimensional cube, here $\psi$ is a $C^\infty$ function, and $m$ is a positive parameter. For this equation we construct some Cantor families of periodic orbits. Our proof is very simple and is based on contraction mapping principle and on a suitable correspondence between Lyapunov Schmidt decomposition and averaging theory.

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