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TOP 10 Most Read Articles in DCDSS, February 2015
1 
The De Giorgi method for regularity of solutions of elliptic equations and its applications to fluid dynamics
Volume 3, Number 3, Pages: 409  427, 2010
Luis A. Caffarelli
and Alexis F. Vasseur
Abstract
Full Text
Related Articles
This paper is dedicated to the application of the De GiorgiNashMoser
kind of techniques to regularity issues in fluid mechanics. In a first
section, we recall the original method introduced by De Giorgi to prove
$C^\alpha$regularity of solutions to elliptic problems with rough
coefficients. In a second part, we give the main ideas to apply those
techniques in the case of parabolic equations with fractional Laplacian.
This allows, in particular, to show the global regularity of the
Surface QuasiGeostrophic equation in the critical case. Finally, a last
section is dedicated to the application of this method to the 3D
NavierStokes equation.

2 
Turing instability in a coupled predatorprey
model with different Holling type functional responses
Volume 4, Number 6, Pages: 1621  1628, 2010
Zhifu Xie
Abstract
References
Full Text
Related Articles
In a reactiondiffusion system, diffusion can induce the instability
of a positive equilibrium which is stable with respect to a
constant perturbation, therefore, the diffusion may create new
patterns when the corresponding system without diffusion fails,
as shown by Turing in 1950s. In this paper we study a coupled
predatorprey model with different Holling type functional
responses, where crossdiffusions are included in such a way that
the prey runs away from predator and the predator chase preys. We
conduct the Turing instability analysis for each Holling functional response.
We prove that if a positive equilibrium solution is linearly stable with respect to the ODE system of
the predatorprey model, then it is
also linearly stable with respect to the model. So diffusion and
crossdiffusion in the predatorprey model with Holling type
functional responses given in this paper can not drive Turing
instability. However, diffusion and crossdiffusion can still create
nonconstant positive solutions for the model.

3 
Symmetries in an overdetermined problem for the Green's function
Volume 4, Number 4, Pages: 791  800, 2010
Virginia Agostiniani
and Rolando Magnanini
Abstract
References
Full Text
Related Articles
We consider in the plane the problem of reconstructing a domain from the
normal derivative of its Green's function with pole at a fixed point in the domain.
By means of the theory of conformal mappings, we obtain existence, uniqueness, (nonspherical) symmetry results,
and a formula relating the curvature of the boundary of the domain to the normal derivative of its Green's function.

4 
Anisotropic phase field equations of arbitrary order
Volume 4, Number 2, Pages: 311  350, 2010
G. Caginalp
and Emre Esenturk
Abstract
References
Full Text
Related Articles
We derive a set of higher order phase field equations using a microscopic
interaction Hamiltonian with detailed anisotropy in the interactions of the
form $a_{0}+\delta\sum_{n=1}^{N}{a_{n}\cos( 2n\theta) + b_{n}\sin( 2n\theta) }$ where $\theta$ is the angle with
respect to a fixed axis, and $\delta$ is a parameter. The Hamiltonian is
expanded using complex Fourier series, and leads to a free energy and phase
field equation with arbitrarily high order derivatives in the spatial
variable. Formal asymptotic analysis is performed on these phase field
equation in terms of the interface thickness in order to obtain the
interfacial conditions. One can capture $2N$fold anisotropy by retaining at
least $2N^{th}$ degree phase field equation. We derive, in the limit of small
$\delta,$ the classical result $( TT_{E} ) [s]_{E}=\kappa
{\sigma( \theta ) + \sigma^{''}(
\theta) }$ where $TT_{E}$ is the difference between the temperature
at the interface and the equilibrium temperature between phases, $[s]_{E}$ is
the entropy difference between phases, $\sigma$ is the surface tension and
$\kappa$ is the curvature. If there is only one mode in the anisotropy [i.e.,
the sum contains only one term: $A_{n}\cos( 2n\theta) $] then
this identity is exact (valid for any magnitude of $\delta$) if the surface
tension is interpreted as the sharp interface limit of excess free energy
obtained by the solution of the $2N^{th}$ degree differential equation. The
techniques rely on rewriting the sums of derivatives using complex variables
and combinatorial identities, and performing formal asymptotic analyses for
differential equations of arbitrary order.

5 
A mathematical model of a criminalprone society
Volume 4, Number 1, Pages: 193  207, 2010
Juan Carlos Nuño,
Miguel Angel Herrero
and Mario Primicerio
Abstract
References
Full Text
Related Articles
Criminals are common to all societies. To fight against them the
community takes different security measures as, for example, to
bring about a police. Thus, crime causes a depletion of the common
wealth not only by criminal acts but also because the cost of
hiring a police force. In this paper, we present a mathematical
model of a criminalprone selfprotected society that is divided
into socioeconomical classes. We study the effect of a nonnull
crime rate on a freeofcriminals society which is taken as a
reference system. As a consequence, we define a criminalprone
society as one whose freeofcriminals steady state is unstable
under small perturbations of a certain socioeconomical context.
Finally, we compare two alternative strategies to control crime:
(i) enhancing police efficiency, either by enlarging its size or
by updating its technology, against (ii) either reducing criminal
appealing or promoting social classes at risk.

6 
The degenerate driftdiffusion system with the Sobolev critical exponent
Volume 4, Number 4, Pages: 875  886, 2010
T. Ogawa
Abstract
References
Full Text
Related Articles
We consider the driftdiffusion system of degenerated type.
For $n\ge 3$,
$\partial_t \rho \Delta \rho^\alpha + \kappa\nabla\cdot
(\rho \nabla \psi ) =0, t>0, x \in R^n,$
$\Delta \psi = \rho, t>0, x \in R^n,$
$\rho(0,x) = \rho_0(x)\ge 0, x \in R^n,$
where $\alpha>1$ and $\kappa=1$.
There exists a critical exponent that classifies the global behavior of the weak solution. In particular, we consider the critical case $\alpha_*=\frac{2 n}{n+2}=(2^*)'$, where the Talenti function $U(x)$
solving $2^*\Delta U^{\frac{n2}{n+2}}=U$ in $R^n$
classifies the global existence of the weak solution and finite
blowup of the solution.

7 
A simple proof of wellposedness for the freesurface
incompressible Euler equations
Volume 3, Number 3, Pages: 429  449, 2010
Daniel Coutand
and Steve Shkoller
Abstract
Full Text
Related Articles
The purpose of this this paper is to present a new simple proof for the construction
of unique solutions to the moving freeboundary incompressible 3D Euler equations in vacuum.
Our method relies on the Lagrangian representation of the fluid, and the anisotropic smoothing
operation that we call horizontal convolutionbylayers. The method is general and can be applied
to a number of other moving freeboundary problems.

8 
The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity
Volume 4, Number 1, Pages: 209  222, 2010
Madalina Petcu
and Roger Temam
Abstract
References
Full Text
Related Articles
In the present article we consider the nonviscous Shallow Water Equations in space dimension one with
Dirichlet boundary conditions for the velocity and we
show the locally in time wellposedness of the model.

9 
On sharp interface limits of AllenCahn/CahnHilliard variational inequalities
Volume 1, Number 1, Pages: 1  14, 2007
John W. Barrett,
Harald Garcke
and Robert Nürnberg
Abstract
Full Text
Related Articles
Using formally matched asymptotic expansions we
identify the sharp interface asymptotic limit of an
AllenCahn/CahnHilliard system using a novel approach which enables
us to handle the case of variational inequalities.

10 
The periodic patch model for population dynamics with fractional diffusion
Volume 4, Number 1, Pages: 1  13, 2010
Henri Berestycki,
JeanMichel Roquejoffre
and Luca Rossi
Abstract
References
Full Text
Related Articles
Fractional diffusions arise in the study of models
from population dynamics. In this paper, we derive a class of integrodifferential reactiondiffusion equations
from simple principles. We then prove an approximation result for the first eigenvalue
of linear integrodifferential operators of the fractional diffusion type, and we study from that
the dynamics of a population in a fragmented environment with fractional diffusion.

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