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TOP 10 Most Read Articles in DCDSS, November 2015
1 
A framework for the development of implicit solvers for incompressible flow problems
Volume 5, Number 6, Pages: 1195  1221, 2012
David J. Silvester,
Alex Bespalov
and Catherine E. Powell
Abstract
References
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This survey paper reviews some recent developments in the design of
robust solution methods for the NavierStokes equations
modelling incompressible fluid flow. There are two
building blocks in our solution strategy. First, an implicit time integrator that uses
a stabilized trapezoid rule with an explicit
AdamsBashforth method for error control, and second, a
robust Krylov subspace solver for the spatially discretized system.
Numerical experiments are presented that illustrate the effectiveness
of our generic approach. It is further shown that the basic solution strategy can be
readily extended to more complicated models, including
unsteady flow problems with coupled physics and steady flow problems that
are nondeterministic in the sense that they have uncertain input data.

2 
Annihilation of two interfaces in a hybrid system
Volume 8, Number 5, Pages: 857  869, 2015
ShinIchiro Ei,
Kei Nishi,
Yasumasa Nishiura
and Takashi Teramoto
Abstract
References
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We consider the mixed ODEPDE system called a hybrid system,
in which the two interfaces interact with each other through a
continuous medium and their equations of motion are derived
in a weak interaction framework.
We study the bifurcation property of the resulting hybrid system
and construct an unstable standing pulse solution, which plays
the role of a separator
for dynamic transition from standing breather to annihilation behavior
between two interfaces.

3 
Dynamics of two phytoplankton species
competing for light and nutrient with internal storage
Volume 7, Number 6, Pages: 1259  1285, 2014
SzeBi Hsu
and ChiuJu Lin
Abstract
References
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We analyze a competition model of two phytoplankton species for a
single nutrient with internal storage and light in a well mixed
aquatic environment. We apply the theory of monotone dynamical
system to determine the outcomes of competition: extinction of two
species, competitive exclusion, stable coexistence and bistability
of two species. We also present the graphical presentation to
classify the competition outcomes and to compare outcome of models with
and without internal storage.

4 
Positivity for the Navier bilaplace, an antieigenvalue and an expected lifetime
Volume 7, Number 4, Pages: 839  855, 2014
Guido Sweers
Abstract
References
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We address the question, for which $\lambda \in \mathbb{R}$ is the boundary
value problem
\begin{equation*}
\left\{
\begin{array}{cc}
\Delta ^{2}u+\lambda u=f & \text{in }\Omega , \\
u=\Delta u=0 & \text{on }\partial \Omega ,
\end{array}
\right.
\end{equation*}
positivity preserving, that is, $f\geq 0$ implies $u\geq 0$. Moreover, we consider what
happens, when $\lambda $ passes the maximal value for which positivity is
preserved.

5 
Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators
Volume 7, Number 4, Pages: 857  885, 2014
JuanLuis Vázquez
Abstract
References
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We report on recent progress in the study of nonlinear diffusion equations involving nonlocal, longrange diffusion effects. Our main concern is the socalled fractional porous medium equation, $\partial_t u +(\Delta)^{s}(u^m)=0$, and some of its generalizations. Contrary to usual porous medium flows, the fractional version has infinite speed of propagation for all exponents $0 < s < 1$ and $m > 0$; on the other hand, it also generates an $L^1$contraction semigroup which depends continuously on the exponent of fractional differentiation and the exponent of the nonlinearity.
After establishing the general existence and uniqueness theory, the main properties are described: positivity, regularity, continuous dependence, a priori estimates, Schwarz symmetrization, among others. Selfsimilar solutions are constructed (fractional Barenblatt solutions) and they explain the asymptotic behaviour of a large class of solutions. In the fast diffusion range we study extinction in finite time and we find suitable special solutions. We discuss KPP type propagation. We also examine some related equations that extend the model and briefly comment on current work.

6 
On the motion of incompressible inhomogeneous EulerKorteweg
fluids
Volume 3, Number 3, Pages: 497  515, 2010
Miroslav Bulíček,
Eduard Feireisl,
Josef Málek
and Roman Shvydkoy
Abstract
Full Text
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We study a system of equations governing evolution of incompressible inhomogeneous EulerKorteweg fluids that describe a class of incompressible elastic materials. A local wellposedness theory is developed on a bounded smooth domain with noslip boundary condition on velocity and vanishing gradient of density. The cases of open space and periodic box are also considered, where the local existence and uniqueness of solutions is shown in Sobolev spaces up to the critical smoothness $\frac{n}{2}+1$.

7 
Continuation and bifurcations of breathers in a finite discrete
NLS equation
Volume 4, Number 5, Pages: 1227  1245, 2010
Panayotis Panayotaros
Abstract
References
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We present results on the continuation of
breathers in the discrete cubic nonlinear Schrödinger
equation in a finite onedimensional lattice with
Dirichlet boundary conditions. In the limit of small intersite
coupling the equation has a finite number of breather
solutions and as we increase the coupling we see numerically
that all breather branches
undergo either fold or pitchfork bifurcations.
We also see branches that persist
for arbitrarily large coupling and converge to the
linear normal modes of the system.
The stability of the breathers
that persist generally changes as
the coupling is varied, although there are at
least two branches that
preserve their linear and nonlinear stability
properties throughout the continuation.

8 
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.

9 
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
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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.

10 
Existence and uniqueness of timeperiodic solutions to the NavierStokes equations in the whole plane
Volume 6, Number 5, Pages: 1237  1257, 2013
Giovanni P. Galdi
Abstract
References
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We consider the twodimensional motion of a NavierStokes liquid in the whole plane, under the action of a timeperiodic body force $F$ of period $T$, and tending to a prescribed nonzero constant velocity at infinity. We show that if the magnitude of $F$, in suitable norm, is sufficiently small, there exists one and only one corresponding timeperiodic flow of period $T$ in an appropriate function class.

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