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DCDS, series A includes peerreviewed original papers and invited expository papers on the theory and methods of analysis, differential equations and dynamical systems. This journal is committed to recording important new results in its field and maintains the highest standards of innovation and quality. To be published in this journal, an original paper must be correct, new, nontrivial and of interest to a substantial number of readers.
DCDS is published monthly in 2016 and is a publication of the American Institute of Mathematical Sciences. All rights reserved.
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TOP 10 Most Read Articles in DCDSA, September 2016
1 
An interface problem: The twolayer shallow water equations
Volume 33, Number 11/12, Pages: 5327  5345, 2013
Madalina Petcu
and Roger Temam
Abstract
References
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The aim of this article is to study a model of two superposed layers of fluid governed by the shallow water equations in space dimension one. Under some suitable hypotheses the governing equations are hyperbolic. We introduce suitable boundary conditions and establish a result of existence and uniqueness of smooth solutions for a limited time for this model.

2 
$H^1$ Solutions of a class of fourth order nonlinear equations for image processing
Volume 10, Number 1/2, Pages: 349  366, 2003
John B. Greer
and Andrea L. Bertozzi
Abstract
Full Text
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Recently fourth order equations of the form
$u_t = \nabla\cdot((\mathcal G(J_\sigma u)) \nabla \Delta u)$ have been proposed
for noise reduction and simplification of two dimensional images.
The operator $\mathcal G$ is a nonlinear functional involving
the gradient or Hessian of its argument, with decay in the far field.
The operator $J_\sigma$ is a standard mollifier.
Using ODE methods on Sobolev spaces,
we prove existence and uniqueness of solutions of this problem
for $H^1$ initial data.

3 
Adaptation of an ecological territorial model to street gang spatial patterns in Los Angeles
Volume 32, Number 9, Pages: 3223  3244, 2012
Laura M. Smith,
Andrea L. Bertozzi,
P. Jeffrey Brantingham,
George E. Tita
and Matthew Valasik
Abstract
References
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Territorial behavior is often found in nature. Coyotes and wolves organize themselves around a den site and mark their territory to distinguish their claimed region.
Moorcroft et al. model the formation of territories and spatial distributions of coyote packs and their markings in [31].
We modify this ecological approach to simulate spatial gang dynamics in the Hollenbeck policing division of eastern Los Angeles. We incorporate important geographical features from the region that would inhibit movement, such as rivers and freeways. From the gang and marking densities created by this method, we create a rivalry network from overlapping territories and compare the graph to both the observed network and those constructed through other methods. Data on the locations of where gang members have been observed is then used to analyze the densities created by the model.

4 
The uniform attractor of a multivalued process generated by
reactiondiffusion delay equations on an unbounded domain
Volume 34, Number 10, Pages: 4343  4370, 2014
Yejuan Wang
and Peter E. Kloeden
Abstract
References
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The existence of a uniform attractor in a space of higher
regularity is proved for the multivalued process associated
with the nonautonomous reactiondiffusion equation on an unbounded
domain with delays for which the uniqueness of solutions need not hold. A new method for
checking the asymptotical uppersemicompactness of the solutions is used.

5 
A biharmonicmodified forward time stepping
method for fourth order nonlinear diffusion equations
Volume 29, Number 4, Pages: 1367  1391, 2010
Andrea L. Bertozzi,
Ning Ju
and HsiangWei Lu
Abstract
References
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We consider a class of splitting schemes for fourth order nonlinear
diffusion equations. Standard backwardtime differencing requires
the solution of a higher order elliptic problem, which can be both
computationally expensive and workintensive to code, in higher space
dimensions.
Recent papers in the literature provide computational evidence that
a biharmonicmodified, forward timestepping method, can provide good
results for these problems.
We provide a theoretical explanation of the results.
For a basic nonlinear 'thin film' type equation we prove $H^1$
stability of the method given very simple boundedness constraints
of the numerical solution. For a more general class of longwave
unstable problems, we prove stability and convergence, using
only constraints on the smooth solution.
Computational examples include both the model of 'thin film' type
problems and a quantitative model for electrowetting in a HeleShaw
cell (Lu et al J. Fluid Mech. 2007).
The methods considered here are related to 'convexity splitting'
methods for gradient flows with nonconvex energies.

6 
Understanding ThomasFermiLike approximations: Averaging over oscillating occupied orbitals
Volume 33, Number 11/12, Pages: 5319  5325, 2013
John P. Perdew
and Adrienn Ruzsinszky
Abstract
References
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The ThomasFermi equation arises from the earliest density functional approximation
for the groundstate energy of a manyelectron system. Its solutions have been
carefully studied by mathematicians, including J.A. Goldstein.
Here we will review the approximation and its validity conditions
from a physics perspective, explaining why the theory correctly describes
the core electrons of an atom but fails to bind atoms to form molecules and solids.
The valence electrons are poorly described in ThomasFermi theory, for two reasons:
(1) This theory neglects the exchangecorrelation energy, ``nature's glue".
(2) It also makes a local density approximation for the kinetic energy,
which neglects important shellstructure effects in the exact kinetic energy
that are responsible for the structure of the periodic table of the elements.
Finally, we present a tentative explanation for the fact that the shellstructure
effects are relatively unimportant for the exact exchange energy,
which can thus be more usefully described by a local density or semilocal approximation
(as in the popular KohnSham theory): The exact exchange energy
from the occupied KohnSham orbitals has an extra sum over orbital labels
and an extra integration over space, in comparison to the kinetic energy,
and thus averages out more of the atomic individuality of the orbital oscillations.

7 
Pointwise asymptotic convergence of solutions for a phase separation model
Volume 16, Number 1, Pages: 1  18, 2006
Pavel Krejčí
and Songmu Zheng
Abstract
Full Text
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A new technique, combining the global energy and entropy balance equations
with the local stability theory for dynamical systems, is used for proving
that every solution to a nonsmooth temperaturedriven phase separation model
with conserved energy converges pointwise in space to an equilibrium as time
tends to infinity. Three main features are observed: the limit
temperature is uniform in space, there exists a partition of the physical body
into at most three constant limit phases, and the phase separation process has
a hysteresislike character.

8 
Asymptotic behaviour of a nonautonomous Lorenz84 system
Volume 34, Number 10, Pages: 3901  3920, 2014
María Anguiano
and Tomás Caraballo
Abstract
References
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The so called Lorenz84 model has been used in climatological studies, for example by coupling it with a lowdimensional model for ocean dynamics. The behaviour of this model has been studied extensively since its introduction by Lorenz in 1984. In this paper we study the asymptotic behaviour of a nonautonomous Lorenz84 version with several types of nonautonomous features. We prove the existence of pullback and uniform attractors for the process associated to this model. In particular we consider that the nonautonomous forcing terms are more general than almost periodic. Finally, we estimate the Hausdorff dimension of the pullback attractor. We illustrate some examples of pullback attractors by numerical simulations.

9 
Wellposedness results for the NavierStokes equations in the rotational framework
Volume 33, Number 11/12, Pages: 5143  5151, 2013
Matthias Hieber
and Sylvie Monniaux
Abstract
References
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Consider the NavierStokes equations in the rotational framework either on $\mathbb{R}^3$ or on
open sets $\Omega \subset \mathbb{R}^3$ subject to Dirichlet boundary conditions. This paper
discusses recent wellposedness and illposedness results for both situations.

10 
Variational methods for nonlocal operators
of elliptic type
Volume 33, Number 5, Pages: 2105  2137, 2012
Raffaella Servadei
and Enrico Valdinoci
Abstract
References
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In this paper we study the existence of nontrivial solutions for
equations driven by a nonlocal integrodifferential
operator $\mathcal L_K$ with homogeneous Dirichlet boundary
conditions. More precisely, we consider the problem
$$ \left\{
\begin{array}{ll}
\mathcal L_K u+\lambda u+f(x,u)=0 in Ω \\
u=0 in \mathbb{R}^n \backslash Ω ,
\end{array} \right.
$$
where $\lambda$ is a real parameter and the nonlinear term $f$
satisfies superlinear and subcritical growth conditions at zero and
at infinity. This equation has a variational nature, and so its
solutions can be found as critical points of the energy functional
$\mathcal J_\lambda$ associated to the problem. Here we get such
critical points using both the Mountain Pass Theorem and the Linking
Theorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq
\lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of the
operator $\mathcal L_K$.
As a particular case, we derive an existence theorem for the
following equation driven by the fractional Laplacian
$$ \left\{
\begin{array}{ll}
(\Delta)^s u\lambda u=f(x,u) in Ω \\
u=0 in \mathbb{R}^n \backslash Ω.
\end{array} \right.
$$
Thus, the results presented here may be seen as the extension
of some classical nonlinear analysis theorems to the case of fractional
operators.

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