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TOP 10 Most Read Articles in DCDSS, May 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 
Singular limit of a twophase flow problem in porous medium as the air viscosity tends to zero
Volume 5, Number 1, Pages: 93  113, 2011
Marie Henry,
Danielle Hilhorst
and Robert Eymard
Abstract
References
Full Text
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In this paper we consider a twophase flow problem in porous media and study its singular limit as the viscosity of
the air tends to zero; more precisely, we prove the convergence of subsequences to solutions of a generalized
Richards model.

3 
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
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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.

4 
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
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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.

5 
Birth of canard cycles
Volume 2, Number 4, Pages: 723  781, 2009
Freddy Dumortier
and Robert Roussarie
Abstract
Full Text
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In this paper we consider singular perturbation problems occuring in planar slowfast systems $(\dot x=yF(x,\lambda),\dot y=\varepsilon G(x,\lambda))$ where $F$ and $G$ are smooth or even real analytic for some results, $\lambda$ is a multiparameter and $\varepsilon$ is a small parameter. We deal with turning points that are limiting situations of (generalized) Hopf bifurcations and that we call slowfast Hopf points. We investigate the number of limit cycles that can appear near a slowfast Hopf point and this under very general conditions. One of the results states that for any analytic family of planar systems, depending on a finite number of parameters, there is a finite upperbound for the number of limit cycles that can bifurcate from a slowfast Hopf point.
The most difficult problem to deal with concerns the uniform treatment of the evolution that a limit cycle undergoes when it grows from a small limit cycle near the singular point to a canard cycle of detectable size. This explains the title of the paper. The treatment is based on blowup, good normal forms and appropriate Chebyshev systems. In the paper we also relate the slowdivergence integral as it is used in singular perturbation theory to Abelian integrals that have to be used in studying limit cycles close to the singular point.

6 
Periodic solutions of a model for tumor virotherapy
Volume 4, Number 6, Pages: 1587  1597, 2010
Daniel Vasiliu
and Jianjun Paul Tian
Abstract
References
Full Text
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In this article we study periodic solutions of a mathematical model
for brain tumor virotherapy by finding Hopf bifurcations with
respect to a biological significant parameter, the burst size of the
oncolytic virus. The model is derived from a PDE free boundary
problem. Our model is an ODE system with six variables, five of them
represent different cell or virus populations, and one represents
tumor radius. We prove the existence of Hopf bifurcations, and
periodic solutions in a certain interval of the value of the burst
size. The evolution of the tumor radius is much influenced by the
value of the burst size. We also provide a numerical confirmation.

7 
Largetime asymptotics of the generalized BenjaminOnoBurgers equation
Volume 4, Number 1, Pages: 15  50, 2010
Jerry L. Bona
and Laihan Luo
Abstract
References
Full Text
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In this paper, attention is given to pure initialvalue problems for the
generalized BenjaminOnoBurgers (BOB) equation
$
u_t + u_x +(P(u))_{x}\nu $u_{xx}$  H$u_{xx}=0,
where $H$ is the Hilbert transform, $\nu > 0$ and $P\ : R \to R$ is a smooth function. We study questions of global
existence and of the largetime asymptotics of solutions of the
initialvalue problem. If $\Lambda (s)$ is defined by $\Lambda '(s)
= P(s), \Lambda (0) = 0,$ then solutions of the initialvalue
problem corresponding to reasonable initial data maintain their
integrity for all $t \geq 0$ provided that $\Lambda$ and $P'$
satisfy certain growth restrictions. In case a solution
corresponding to initial data that is square integrable is global,
it is straightforward to conclude it must decay to zero when $t$
becomes unboundedly large. We investigate the detailed asymptotics
of this decay. For generic initial data and weak nonlinearity, it is
demonstrated that the final decay is that of the linearized equation
in which $P \equiv 0.$ However, if the initial data is drawn from
more restricted classes that involve something akin to a condition
of zero mean, then enhanced decay rates are established. These
results extend the earlier work of Dix who considered the case where
$P$ is a quadratic polynomial.

8 
Existence of global weak solutions to FokkerPlanck and NavierStokesFokkerPlanck equations in kinetic models of dilute polymers
Volume 3, Number 3, Pages: 371  408, 2010
John W. Barrett
and Endre Süli
Abstract
Full Text
Related Articles
This survey paper reviews recent developments concerning the existence of global
weak solutions to FokkerPlanck equations with unbounded drift terms, and coupled NavierStokesFokkerPlanck systems of partial differential equations, that arise in finitely extensible nonlinear elastic (FENE) type kinetic models of incompressible dilute polymeric fluids in the case of general noncorotational flow.

9 
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
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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.

10 
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.

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