AIMS

Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

ISSN 1078-0947(print) ISSN 1553-5231(online)	Current volume Go	Journal archive Go	Advanced Search
	DCDS, series A includes peer-reviewed original papers and invited expository papers on the theory and methods of analysis, differential equations and dynamical systems. This journal is committed to recording important new results in its field and maintains the highest standards of innovation and quality. To be published in this journal, an original paper must be correct, new, nontrivial and of interest to a substantial number of readers. DCDS is published monthly in 2013 and is a publication of the American Institute of Mathematical Sciences. All rights reserved.

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

1	Pointwise asymptotic convergence of solutions for a phase separation model Volume 16, Number 1, Pages: 1 - 18, 2006 Pavel Krejčí and Songmu Zheng Abstract Full Text Related Articles A new technique, combining the global energy and entropy balance equations with the local stability theory for dynamical systems, is used for proving that every solution to a non-smooth temperature-driven phase separation model with conserved energy converges pointwise in space to an equilibrium as time tends to infinity. Three main features are observed: the limit temperature is uniform in space, there exists a partition of the physical body into at most three constant limit phases, and the phase separation process has a hysteresis-like character.
2	A biharmonic-modified forward time stepping method for fourth order nonlinear diffusion equations Volume 29, Number 4, Pages: 1367 - 1391, 2010 Andrea L. Bertozzi, Ning Ju and Hsiang-Wei Lu Abstract References Full Text Related Articles We consider a class of splitting schemes for fourth order nonlinear diffusion equations. Standard backward-time differencing requires the solution of a higher order elliptic problem, which can be both computationally expensive and work-intensive to code, in higher space dimensions. Recent papers in the literature provide computational evidence that a biharmonic-modified, forward time-stepping method, can provide good results for these problems. We provide a theoretical explanation of the results. For a basic nonlinear 'thin film' type equation we prove $H^1$ stability of the method given very simple boundedness constraints of the numerical solution. For a more general class of long-wave unstable problems, we prove stability and convergence, using only constraints on the smooth solution. Computational examples include both the model of 'thin film' type problems and a quantitative model for electrowetting in a Hele-Shaw cell (Lu et al J. Fluid Mech. 2007). The methods considered here are related to 'convexity splitting' methods for gradient flows with nonconvex energies.
3	$H^1$ Solutions of a class of fourth order nonlinear equations for image processing Volume 10, Number 1/2, Pages: 349 - 366, 2003 John B. Greer and Andrea L. Bertozzi Abstract Full Text Related Articles Recently fourth order equations of the form $u_t = -\nabla\cdot((\mathcal G(J_\sigma u)) \nabla \Delta u)$ have been proposed for noise reduction and simplification of two dimensional images. The operator $\mathcal G$ is a nonlinear functional involving the gradient or Hessian of its argument, with decay in the far field. The operator $J_\sigma$ is a standard mollifier. Using ODE methods on Sobolev spaces, we prove existence and uniqueness of solutions of this problem for $H^1$ initial data.
4	Renormalization group method: Application to Navier-Stokes equation Volume 6, Number 1, Pages: 191 - 210, 1999 I. Moise and Roger Temam Abstract Full Text Related Articles The aim of this article is to present a rather unusual and partly heuristic application of the renormalization group (RG) theory to the Navier-Stokes equations with space periodic boundary conditions. We obtain in this way a new nonlinear renormalized equation with a nonlinear term which is invariant under the Stokes operator. Its relation to the Navier-Stokes equations is investigated for non-resonant domains.
5	On a limiting system in the Lotka--Volterra competition with cross-diffusion Volume 10, Number 1/2, Pages: 435 - 458, 2003 Yuan Lou, Wei-Ming Ni and Shoji Yotsutani Abstract Full Text Related Articles In this paper we investigate a limiting system that arises from the study of steady-states of the Lotka-Volterra competition model with cross-diffusion. The main purpose here is to understand all possible solutions to this limiting system, which consists of a nonlinear elliptic equation and an integral constraint. As far as existence and non-existence in one dimensional domain are concerned, our knowledge of the limiting system is nearly complete. We also consider the qualitative behavior of solutions to this limiting system as the remaining diffusion rate varies. Our basic approach is to convert the problem of solving the limiting system to a problem of solving its "representation" in a different parameter space. This is first done without the integral constraint, and then we use the integral constraint to find the "solution curve" in the new parameter space as the diffusion rate varies. This turns out to be a powerful method as it gives fairly precise information about the solutions.
6	Global existence results for nonlinear Schrödinger equations with quadratic potentials Volume 13, Number 2, Pages: 385 - 398, 2005 Rémi Carles Abstract Full Text Related Articles We prove that no finite time blow up can occur for nonlinear Schrödinger equations with quadratic potentials, provided that the potential has a sufficiently strong repulsive component. This is not obvious in general, since the energy associated to the linear equation is not positive. The proof relies essentially on two arguments: global in time Strichartz estimates, and a refined analysis of the linear equation, which makes it possible to control the nonlinear effects.
7	Lyapunov exponents for continuous transformations and dimension theory Volume 13, Number 2, Pages: 469 - 490, 2005 Luís Barreira and César Silva Abstract Full Text Related Articles We generalize the concept of Lyapunov exponent to transformations that are not necessarily differentiable. For fairly large classes of repellers and of hyperbolic sets of differentiable maps, the new exponents are shown to coincide with the classical ones. We also discuss the relation of the new Lyapunov exponents with the dimension theory of dynamical systems for invariant sets of continuous transformations.
8	Uniform exponential attractors for a singularly perturbed damped wave equation Volume 10, Number 1/2, Pages: 211 - 238, 2003 Pierre Fabrie, Cedric Galusinski, A. Miranville and Sergey Zelik Abstract Full Text Related Articles Our aim in this article is to construct exponential attractors for singularly perturbed damped wave equations that are continuous with respect to the perturbation parameter. The main difficulty comes from the fact that the phase spaces for the perturbed and unperturbed equations are not the same; indeed, the limit equation is a (parabolic) reaction-diffusion equation. Therefore, previous constructions obtained for parabolic systems cannot be applied and have to be adapted. In particular, this necessitates a study of the time boundary layer in order to estimate the difference of solutions between the perturbed and unperturbed equations. We note that the continuity is obtained without time shifts that have been used in previous results.
9	Topological methods in the instability problem of Hamiltonian systems Volume 14, Number 2, Pages: 295 - 328, 2005 Marian Gidea and Rafael De La Llave Abstract Full Text Related Articles We use topological methods to investigate some recently proposed mechanisms of instability (Arnol'd diffusion) in Hamiltonian systems. In these mechanisms, chains of heteroclinic connections between whiskered tori are constructed, based on the existence of a normally hyperbolic manifold $\Lambda$, so that: (a) the manifold $\Lambda$ is covered rather densely by transitive tori (possibly of different topology), (b) the manifolds $W^\s_\Lambda$, $W^\u_\Lambda$ intersect transversally, (c) the systems satisfies some explicit non-degeneracy assumptions, which hold generically. In this paper we use the method of correctly aligned windows to show that, under the assumptions (a), (b), (c), there are orbits that move a significant amount. As a matter of fact, the method presented here does not require that the tori are exactly invariant, only that they are approximately invariant. Hence, compared with the previous papers, we do not need to use KAM theory. This lowers the assumptions on differentiability. Also, the method presented here allows us to produce concrete estimates on the time to move, which were not considered in the previous papers.
10	Nonlocal heat flows preserving the L² energy Volume 23, Number 1/2, Pages: 49 - 64, 2008 Luis Caffarelli and Fanghua Lin Abstract Full Text Related Articles We shall study L² energy conserved solutions to the heat equation. We shall first establish the global existence, uniqueness and regularity of solutions to such nonlocal heat flows. We then extend the method to a family of singularly perturbed systems of nonlocal parabolic equations. The main goal is to show that solutions to these perturbed systems converges strongly to some suitable weak-solutions of the limiting constrained nonlocal heat flows of maps into a singular space. It is then possible to study further properties of such suitable weak solutions and the corresponding free boundary problem, which will be discussed in a forthcoming article.

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