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Volume 1, 2002

Communications on Pure & Applied Analysis

2002 , Volume 1 , Issue 2

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Peierls instability with electron-electron interaction: the commensurate case
V. Mastropietro
2002, 1(2): 135-159 doi: 10.3934/cpaa.2002.1.135 +[Abstract](154) +[PDF](279.7KB)
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.
A center manifold technique for tracing viscous waves
Stefano Bianchini and Alberto Bressan
2002, 1(2): 161-190 doi: 10.3934/cpaa.2002.1.161 +[Abstract](182) +[PDF](290.0KB)
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.
Existence of a weak solution for a quasilinear wave equation with boundary condition
Alain Hertzog and Antoine Mondoloni
2002, 1(2): 191-219 doi: 10.3934/cpaa.2002.1.191 +[Abstract](154) +[PDF](320.9KB)
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.
The Lagrangian averaged Euler equations as the short-time inviscid limit of the Navier–Stokes equations with Besov class data in $\mathbb{R}^2$
Marcel Oliver
2002, 1(2): 221-235 doi: 10.3934/cpaa.2002.1.221 +[Abstract](219) +[PDF](186.0KB)
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.
Modified wave operators for the Hartree equation with data, image and convergence in the same space
Kenji Nakanishi
2002, 1(2): 237-252 doi: 10.3934/cpaa.2002.1.237 +[Abstract](239) +[PDF](214.6KB)
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.
Asymptotic behavior in a general diffusive three-species predator-prey model
E. J. Avila–Vales and T. Montañez–May
2002, 1(2): 253-267 doi: 10.3934/cpaa.2002.1.253 +[Abstract](169) +[PDF](183.0KB)
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.
Families of periodic orbits for some PDE’s in higher dimensions
Dario Bambusi and Simone Paleari
2002, 1(2): 269-279 doi: 10.3934/cpaa.2002.1.269 +[Abstract](202) +[PDF](171.5KB)
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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