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TOP 10 Most Read Articles in DCDSS, October 2016
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
Full Text
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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 
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.

3 
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.

4 
On two phase free
boundary problems governed by elliptic equations with distributed sources
Volume 7, Number 4, Pages: 673  693, 2014
Daniela De Silva,
Fausto Ferrari
and Sandro Salsa
Abstract
References
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We present some recent progress on the analysis of twophase free boundary
problems governed by elliptic operators, with nonzero right hand side. We
also discuss on several open questions, object of future investigations.

5 
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.

6 
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.

7 
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.

8 
Global solutions for a nonlinear integral equation with a generalized heat kernel
Volume 7, Number 4, Pages: 767  783, 2014
Kazuhiro Ishige,
Tatsuki Kawakami
and Kanako Kobayashi
Abstract
References
Full Text
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We study
the existence and the large time behavior of globalintime solutions of a nonlinear integral equation with a generalized heat kernel
\begin{eqnarray*}
& & u(x,t)=\int_{{\mathbb R}^N}G(xy,t)\varphi(y)dy\\
& & \qquad\quad
+\int_0^t\int_{{\mathbb R}^N}G(xy,ts)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds,
\end{eqnarray*}
where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ and $\ell\in\{0,1,\dots\}$.
The arguments of this paper are applicable to
the Cauchy problem for various nonlinear parabolic equations
such as fractional semilinear parabolic equations, higher order semilinear parabolic equations
and viscous HamiltonJacobi equations.

9 
Asymptotic behavior of the Caginalp phasefield system with coupled dynamic boundary conditions
Volume 5, Number 3, Pages: 485  505, 2011
Monica Conti,
Stefania Gatti
and Alain Miranville
Abstract
References
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This paper deals with the longtime behavior of the Caginalp phasefield system
with coupled dynamic boundary conditions on both state variables.
We prove that the system generates a dissipative semigroup in a suitable phasespace
and possesses the finitedimensional smooth global attractor and an exponential attractor.

10 
Some degenerate parabolic problems: Existence and decay properties
Volume 7, Number 4, Pages: 617  629, 2014
Lucio Boccardo
and Maria Michaela Porzio
Abstract
References
Full Text
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We study the existence of solutions $u $ belonging to $L^1(0,T; W_0^{1,1}(\Omega)) \cap L^{\infty}(0,T;L^2(\Omega))$ of a class of nonlinear problems whose prototype is the following
\begin{equation} \label{prob1}
\left\{
\begin{array}{lll}
\displaystyle
u_t  {\rm div} \left( \frac{\nabla u}{(1+u)^2} \right) = 0, & \hbox{in} & \Omega_T; \\
u=0, & \hbox{on} & \partial\Omega \times (0,T); & & & & \hbox{(1)}\\
u(x,0)= u_0(x) \in L^2(\Omega), & \hbox{ in} & \Omega.
\end{array}
\right.
\end{equation}
We investigate also the asymptotic estimates satisfied by distributional solutions that we find and the uniqueness.

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