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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 2015 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, May 2015
1 
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
Related Articles
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.

2 
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
Full Text
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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.

3 
$H^1$ 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.

4 
Renormalization group method: Application to NavierStokes equation
Volume 6, Number 1, Pages: 191  210, 1999
I. Moise
and Roger Temam
Abstract
Full Text
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The aim of this article is to present a rather unusual and partly heuristic
application of the renormalization group (RG) theory to the NavierStokes equations
with space periodic boundary conditions.
We obtain in this way a new nonlinear
renormalized equation with a nonlinear term which is invariant under the Stokes
operator.
Its relation to the NavierStokes equations is investigated for nonresonant
domains.

5 
On a limiting system in the LotkaVolterra competition with crossdiffusion
Volume 10, Number 1/2, Pages: 435  458, 2003
Yuan Lou,
WeiMing Ni
and Shoji Yotsutani
Abstract
Full Text
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In this paper we investigate a limiting system that arises from
the study of steadystates of the LotkaVolterra competition model with
crossdiffusion. The main purpose here is to understand
all possible solutions to this limiting system, which consists of a nonlinear
elliptic equation and an integral constraint. As far as existence and
nonexistence in one dimensional domain are concerned, our knowledge of the
limiting system is nearly complete. We also consider the qualitative
behavior of solutions to this limiting system as the remaining diffusion
rate varies. Our basic approach is to convert the problem of solving the
limiting system to a problem of solving its "representation" in a
different parameter space. This is first done without the
integral constraint, and then we use the integral constraint to find the
"solution curve" in the new parameter space as the diffusion rate varies.
This turns out to be a powerful method as it gives fairly precise
information about the solutions.

6 
Global existence results for nonlinear Schrödinger equations with quadratic potentials
Volume 13, Number 2, Pages: 385  398, 2005
Rémi Carles
Abstract
Full Text
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We prove that no finite time blow up can occur for nonlinear Schrödinger
equations with quadratic potentials, provided that the potential has a
sufficiently strong repulsive component. This is not obvious in
general, since the energy associated to the linear equation is not
positive. The proof relies essentially on two arguments: global
in time Strichartz estimates, and a refined analysis of the linear
equation, which makes it possible to
control the nonlinear effects.

7 
Lyapunov exponents for continuous transformations and dimension theory
Volume 13, Number 2, Pages: 469  490, 2005
Luis Barreira
and César Silva
Abstract
Full Text
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We generalize the concept of Lyapunov exponent to transformations
that are not necessarily differentiable. For fairly large classes of
repellers and of hyperbolic sets of differentiable maps, the new
exponents are shown to coincide with the classical ones. We also
discuss the relation of the new Lyapunov exponents with the
dimension theory of dynamical systems for invariant sets of
continuous transformations.

8 
Uniform exponential attractors for a singularly perturbed damped wave equation
Volume 10, Number 1/2, Pages: 211  238, 2003
Pierre Fabrie,
Cedric Galusinski,
A. Miranville
and Sergey Zelik
Abstract
Full Text
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Our aim in this article is to construct exponential attractors for
singularly perturbed damped wave equations that are continuous with
respect to the perturbation parameter. The main difficulty comes from
the fact that the phase spaces for the perturbed and unperturbed
equations are not the same; indeed, the limit equation is a
(parabolic) reactiondiffusion equation. Therefore, previous
constructions obtained for parabolic systems cannot be applied
and have to be adapted. In particular, this necessitates a
study of the time boundary layer in order to estimate the difference
of solutions between the perturbed and unperturbed equations. We note
that the continuity is obtained without time shifts that have been used
in previous results.

9 
Topological methods in the instability problem of Hamiltonian systems
Volume 14, Number 2, Pages: 295  328, 2005
Marian Gidea
and Rafael De La Llave
Abstract
Full Text
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We use topological methods to investigate some recently proposed
mechanisms of instability (Arnol'd diffusion) in Hamiltonian
systems.
In these mechanisms, chains of heteroclinic connections between
whiskered tori are constructed, based on the existence of a
normally hyperbolic manifold $\Lambda$, so that: (a) the manifold
$\Lambda$ is covered rather densely by transitive tori (possibly
of different topology), (b) the manifolds $W^\s_\Lambda$,
$W^\u_\Lambda$ intersect transversally, (c) the systems satisfies
some explicit nondegeneracy assumptions, which hold generically.
In this paper we use the method of correctly aligned windows to
show that, under the assumptions (a), (b), (c), there are orbits
that move a significant amount.
As a matter of fact, the method presented here does not require
that the tori are exactly invariant, only that they are
approximately invariant. Hence, compared with the previous
papers, we do not need to use KAM theory. This lowers the
assumptions on differentiability.
Also, the method presented here allows us to produce concrete
estimates on the time to move, which were not considered in the
previous papers.

10 
Nonlocal heat flows preserving the L^{2} energy
Volume 23, Number 1/2, Pages: 49  64, 2008
Luis Caffarelli
and Fanghua Lin
Abstract
Full Text
Related Articles
We shall study L^{2} energy conserved solutions to the heat equation.
We shall first establish the global existence, uniqueness and
regularity of solutions to such nonlocal heat flows. We then extend the
method to a family of singularly perturbed systems of nonlocal parabolic
equations. The main goal is to show that solutions to these perturbed
systems converges strongly to some suitable weaksolutions
of the limiting constrained nonlocal heat flows of maps into a singular
space. It is then possible to study further properties of such suitable
weak solutions and the corresponding free boundary problem, which will
be discussed in a forthcoming article.

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