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	Each issue is devoted to a specific area of the mathematical, physical and engineering sciences. This area will define a research frontier that is advancing rapidly, often bridging mathematics and sciences. DCDS-S is essential reading for mathematicians, physicists, engineers and other physical scientists. The journal is published bimonthly. DCDS-S publishes only Theme Issues: issues with a coherent topic proposed by guest editors. Click here to learn how to submit a theme issue proposal. Occasionally, proposals of an important current topic that is also the main theme of a high quality workshop/meeting are also considered, however, the same rigorous editorial process is applied.

TOP 10 Most Read Articles in DCDS-S, May 2012

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 Giorgi-Nash-Moser 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 Quasi-Geostrophic equation in the critical case. Finally, a last section is dedicated to the application of this method to the 3D Navier-Stokes equation.
2	Singular limit of a two-phase 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 Related Articles In this paper we consider a two-phase 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 predator-prey 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 reaction-diffusion 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 predator-prey model with different Holling type functional responses, where cross-diffusions 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 predator-prey model, then it is also linearly stable with respect to the model. So diffusion and cross-diffusion in the predator-prey model with Holling type functional responses given in this paper can not drive Turing instability. However, diffusion and cross-diffusion can still create non-constant positive solutions for the model.
4	A mathematical model of a criminal-prone 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 criminal-prone self-protected society that is divided into socio-economical classes. We study the effect of a non-null crime rate on a free-of-criminals society which is taken as a reference system. As a consequence, we define a criminal-prone society as one whose free-of-criminals steady state is unstable under small perturbations of a certain socio-economical 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 Related Articles In this paper we consider singular perturbation problems occuring in planar slow-fast systems $(\dot x=y-F(x,\lambda),\dot y=-\varepsilon G(x,\lambda))$ 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 slow-fast Hopf points. We investigate the number of limit cycles that can appear near a slow-fast 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 slow-fast 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 blow-up, good normal forms and appropriate Chebyshev systems. In the paper we also relate the slow-divergence 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 Related Articles 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	Large-time asymptotics of the generalized Benjamin-Ono-Burgers equation Volume 4, Number 1, Pages: 15 - 50, 2010 Jerry L. Bona and Laihan Luo Abstract References Full Text Related Articles In this paper, attention is given to pure initial-value problems for the generalized Benjamin-Ono-Burgers (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 large-time asymptotics of solutions of the initial-value problem. If $\Lambda (s)$ is defined by $\Lambda '(s) = P(s), \Lambda (0) = 0,$ then solutions of the initial-value 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 Fokker-Planck and Navier-Stokes-Fokker-Planck 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 Fokker-Planck equations with unbounded drift terms, and coupled Navier-Stokes-Fokker-Planck 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, Jean-Michel 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 integro-differential reaction-diffusion equations from simple principles. We then prove an approximation result for the first eigenvalue of linear integro-differential 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 well-posedness of the model.

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