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TOP 10 Most Read Articles in DCDSS, November 2014
1 
Positivity for the Navier bilaplace, an antieigenvalue and an expected lifetime
Volume 7, Number 4, Pages: 839  855, 2014
Guido Sweers
Abstract
References
Full Text
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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.

2 
Discussion about traffic junction modelling: Conservation laws VS HamiltonJacobi equations
Volume 7, Number 3, Pages: 411  433, 2014
Guillaume Costeseque
and JeanPatrick Lebacque
Abstract
References
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In this paper, we consider a numerical scheme to solve first order HamiltonJacobi (HJ) equations posed on a junction. The main mathematical properties of the scheme are first recalled and then we give a traffic flow interpretation of the key elements. The scheme formulation is also adapted to compute the vehicles densities on a junction. The equivalent scheme for densities recovers the wellknown Godunov scheme outside the junction point. We give two numerical illustrations for a merge and a diverge which are the two main types of traffic junctions. Some extensions to the junction model are finally discussed.

3 
Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators
Volume 7, Number 4, Pages: 857  885, 2014
JuanLuis Vázquez
Abstract
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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 
Oscillations in suspension bridges, vertical and torsional
Volume 7, Number 4, Pages: 785  791, 2014
P. J. McKenna
Abstract
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We first review some history prior to the failure of the Tacoma Narrows suspension bridge.
Then we consider some popular accounts of this in the popular physics literature, and the scientific
and scholarly basis for these accounts and point out some failings. Later, we give a quick introduction
to three different models, one single particle, one a continuum model, and two systems with two degrees of freedom.

5 
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.

6 
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.

7 
Some degenerate parabolic problems: Existence and decay properties
Volume 7, Number 4, Pages: 617  629, 2014
Lucio Boccardo
and Maria Michaela Porzio
Abstract
References
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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.

8 
Special asymptotics for a critical fast diffusion equation
Volume 7, Number 4, Pages: 725  735, 2014
Marek Fila
and Hannes Stuke
Abstract
References
Full Text
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We find a continuum of extinction rates of solutions of the Cauchy problem for
the fast diffusion equation
$u_\tau=\nabla\cdot(u^{m1}\,\nabla u)$ with $m=m_*:=(n4)/(n2)$, here $n>2$ is the spacedimension.
The extinction rates
depend explicitly on the
spatial decay rates of initial data and contain a logarithmic term.

9 
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
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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.

10 
Hopf fibration and singularly perturbed elliptic equations
Volume 7, Number 4, Pages: 823  838, 2014
Bernhard Ruf
and P. N. Srikanth
Abstract
References
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In this article we show how the Hopf fibration can be used to generate special solutions of singularly perturbed elliptic equations on annuli. Indeed, by the Hopf fibration the equation can be reduced to a lower dimensional problem, to which known results on single (or multiple point) concentration can be applied. Reversing the reduction process, one obtains solutions concentrating on circles and spheres, which are given as the fibres of the Hopf fibration.

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