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Discrete & Continuous Dynamical Systems - A

2014 , Volume 34 , Issue 1

Special issue
on Infinite Dimensional Dynamics and Applications

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Preface
Yuncheng You
2014, 34(1): i-iii doi: 10.3934/dcds.2014.34.1i +[Abstract](505) +[PDF](139.2KB)
Abstract:
The theory of infinite dimensional and stochastic dynamical systems is a rapidly expanding and vibrant field of mathematics. In the recent three decades it has been highlighted as a core knowledge and an advancing thrust in the qualitative study of complex systems and processes described by evolutionary partial differential equations in many different settings, stochastic differential equations, functional differential equations and lattice differential equations. The central research topics include the invariant and attracting sets, stability and bifurcation of patterns and waves, asymptotic theory of dissipative systems and reduction of dimensions, and more and more problems of nonlocal systems, ill-posed systems, multicomponent and network dynamics, random dynamics and chaotic dynamics.

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Global dynamics of boundary droplets
Peter W. Bates and Jiayin Jin
2014, 34(1): 1-17 doi: 10.3934/dcds.2014.34.1 +[Abstract](560) +[PDF](333.2KB)
Abstract:
We establish the existence of a global invariant manifold of bubble states for the mass-conserving Allen-Cahn Equation in two space dimensions and give the dynamics for the center of the bubble.
On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes
Anne Bronzi and Ricardo Rosa
2014, 34(1): 19-49 doi: 10.3934/dcds.2014.34.19 +[Abstract](573) +[PDF](516.3KB)
Abstract:
In this paper statistical solutions of the 3D Navier-Stokes-$\alpha$ model with periodic boundary condition are considered. It is proved that under certain natural conditions statistical solutions of the 3D Navier-Stokes-$\alpha$ model converge to statistical solutions of the exact 3D Navier-Stokes equations as $\alpha$ goes to zero. The statistical solutions considered here arise as families of time-projections of measures on suitable trajectory spaces.
On differential equations with delay in Banach spaces and attractors for retarded lattice dynamical systems
Tomás Caraballo, Francisco Morillas and José Valero
2014, 34(1): 51-77 doi: 10.3934/dcds.2014.34.51 +[Abstract](754) +[PDF](489.0KB)
Abstract:
In this paper we first prove a rather general theorem about existence of solutions for an abstract differential equation in a Banach space by assuming that the nonlinear term is in some sense weakly continuous.
    We then apply this result to a lattice dynamical system with delay, proving also the existence of a global compact attractor for such system.
Pathwise solutions of SPDEs driven by Hölder-continuous integrators with exponent larger than $1/2$ and random dynamical systems
Yong Chen, Hongjun Gao, María J. Garrido–Atienza and Björn Schmalfuss
2014, 34(1): 79-98 doi: 10.3934/dcds.2014.34.79 +[Abstract](639) +[PDF](424.5KB)
Abstract:
This article is devoted to the existence and uniqueness of pathwise solutions to stochastic evolution equations, driven by a Hölder continuous function with Hölder exponent in $(1/2,1)$, and with nontrivial multiplicative noise. As a particular situation, we shall consider the case where the equation is driven by a fractional Brownian motion $B^H$ with Hurst parameter $H>1/2$. In contrast to the article by Maslowski and Nualart [17], we present here an existence and uniqueness result in the space of Hölder continuous functions with values in a Hilbert space $V$. If the initial condition is in the latter space this forces us to consider solutions in a different space, which is a generalization of the Hölder continuous functions. That space of functions is appropriate to introduce a non-autonomous dynamical system generated by the corresponding solution to the equation. In fact, when choosing $B^H$ as the driving process, we shall prove that the dynamical system will turn out to be a random dynamical system, defined over the ergodic metric dynamical system generated by the infinite dimensional fractional Brownian motion.
Analysis of a degenerate biofilm model with a nutrient taxis term
Hermann J. Eberl, Messoud A. Efendiev, Dariusz Wrzosek and Anna Zhigun
2014, 34(1): 99-119 doi: 10.3934/dcds.2014.34.99 +[Abstract](699) +[PDF](546.3KB)
Abstract:
We introduce and analyze a prototype model for chemotactic effects in biofilm formation. The model is a system of quasilinear parabolic equations into which two thresholds are built in. One occurs at zero cell density level, the second one is related to the maximal density which the cells cannot exceed. Accordingly, both diffusion and taxis terms have degenerate or singular parts. This model extends a previously introduced degenerate biofilm model by combining it with a chemotaxis equation. We give conditions for existence and uniqueness of weak solutions and illustrate the model behavior in numerical simulations.
Dissipative solutions and the incompressible inviscid limits of the compressible magnetohydrodynamic system in unbounded domains
Eduard Feireisl, Antonin Novotny and Yongzhong Sun
2014, 34(1): 121-143 doi: 10.3934/dcds.2014.34.121 +[Abstract](741) +[PDF](456.7KB)
Abstract:
We consider the compressible Navier-Stokes system coupled with the Maxwell equations governing the time evolution of the magnetic field. We introduce a relative entropy functional along with the related concept of dissipative solution. As an application of the theory, we show that for small values of the Mach number and large Reynolds number, the global in time weak (dissipative) solutions converge to the ideal MHD system describing the motion of an incompressible, inviscid, and electrically conducting fluid. The proof is based on frequency localized Strichartz estimates for the Neumann Laplacean on unbounded domains.
Longtime behavior of nonlocal Cahn-Hilliard equations
Ciprian G. Gal and Maurizio Grasselli
2014, 34(1): 145-179 doi: 10.3934/dcds.2014.34.145 +[Abstract](716) +[PDF](619.2KB)
Abstract:
Here we consider the nonlocal Cahn-Hilliard equation with constant mobility in a bounded domain. We prove that the associated dynamical system has an exponential attractor, provided that the potential is regular. In order to do that a crucial step is showing the eventual boundedness of the order parameter uniformly with respect to the initial datum. This is obtained through an Alikakos-Moser type argument. We establish a similar result for the viscous nonlocal Cahn-Hilliard equation with singular (e.g., logarithmic) potential. In this case the validity of the so-called separation property is crucial. We also discuss the convergence of a solution to a single stationary state. The separation property in the nonviscous case is known to hold when the mobility degenerates at the pure phases in a proper way and the potential is of logarithmic type. Thus, the existence of an exponential attractor can be proven in this case as well.
Regularity of pullback attractors and attraction in $H^1$ in arbitrarily large finite intervals for 2D Navier-Stokes equations with infinite delay
Julia García-Luengo, Pedro Marín-Rubio and José Real
2014, 34(1): 181-201 doi: 10.3934/dcds.2014.34.181 +[Abstract](486) +[PDF](458.6KB)
Abstract:
In this paper we strengthen some results on the existence and properties of pullback attractors for a non-autonomous 2D Navier-Stokes model with infinite delay. Actually we prove that under suitable assumptions, and thanks to regularity results, the attraction also happens in the $H^1$ norm for arbitrarily large finite intervals of time. Indeed, from comparison results of attractors we establish that all these families of attractors are in fact the same object. The tempered character of these families in $H^1$ is also analyzed.
Pullback attractors for the non-autonomous 2D Navier--Stokes equations for minimally regular forcing
Julia García-Luengo, Pedro Marín-Rubio, José Real and James C. Robinson
2014, 34(1): 203-227 doi: 10.3934/dcds.2014.34.203 +[Abstract](601) +[PDF](506.7KB)
Abstract:
This paper treats the existence of pullback attractors for the non-autonomous 2D Navier--Stokes equations in two different spaces, namely $L^2$ and $H^1$. The non-autonomous forcing term is taken in $L^2_{\rm loc}(\mathbb R;H^{-1})$ and $L^2_{\rm loc}(\mathbb R;L^2)$ respectively for these two results: even in the autonomous case it is not straightforward to show the required asymptotic compactness of the flow with this regularity of the forcing term. Here we prove the asymptotic compactness of the corresponding processes by verifying the flattening property -- also known as ``Condition (C)". We also show, using the semigroup method, that a little additional regularity -- $f\in L^p_{\rm loc}(\mathbb R;H^{-1})$ or $f\in L^p_{\rm loc}(\mathbb R;L^2)$ for some $p>2$ -- is enough to ensure the existence of a compact pullback absorbing family (not only asymptotic compactness). Even in the autonomous case the existence of a compact absorbing set for this model is new when $f$ has such limited regularity.
Uniform attractor of the non-autonomous discrete Selkov model
Xiaolin Jia, Caidi Zhao and Juan Cao
2014, 34(1): 229-248 doi: 10.3934/dcds.2014.34.229 +[Abstract](595) +[PDF](468.7KB)
Abstract:
This paper studies the asymptotic behavior of solutions for the non-autonomous lattice Selkov model. We prove the existence of a uniform attractor for the generated family of processes and obtain an upper bound of the Kolmogorov $\varepsilon$-entropy for it. Also we establish the upper semicontinuity of the uniform attractor when the infinite lattice systems are approximated by finite lattice systems.
Frequency domain conditions for finite-dimensional projectors and determining observations for the set of amenable solutions
Sergey Popov and Volker Reitmann
2014, 34(1): 249-267 doi: 10.3934/dcds.2014.34.249 +[Abstract](491) +[PDF](572.6KB)
Abstract:
Frequency domain conditions for the existence of finite-dimensional projectors and determining observations for the set of amenable solutions of semi-dynamical systems in Hilbert spaces are derived. Evolutionary variational equations are considered as control systems in a rigged Hilbert space structure. As an example we investigate a coupled system of Maxwell's equations and the heat equation in one-space dimension. We show the controllability of the linear part and the frequency domain conditions for this example.
Random attractors for non-autonomous stochastic wave equations with multiplicative noise
Bixiang Wang
2014, 34(1): 269-300 doi: 10.3934/dcds.2014.34.269 +[Abstract](1320) +[PDF](574.1KB)
Abstract:
This paper is concerned with the asymptotic behavior of solutions of the damped non-autonomous stochastic wave equations driven by multiplicative white noise. We prove the existence of pullback random attractors in $H^1(\mathbb{R}^n) \times L^2(\mathbb{R}^n)$ when the intensity of noise is sufficiently small. We demonstrate that these random attractors are periodic in time if so are the deterministic non-autonomous external terms. We also establish the upper semicontinuity of random attractors when the intensity of noise approaches zero. In addition, we prove the measurability of random attractors even if the underlying probability space is not complete.
Random attractors and robustness for stochastic reversible reaction-diffusion systems
Yuncheng You
2014, 34(1): 301-333 doi: 10.3934/dcds.2014.34.301 +[Abstract](819) +[PDF](609.3KB)
Abstract:
For a typical stochastic reversible reaction-diffusion system with multiplicative white noise, the trimolecular autocatalytic Gray-Scott system on a three-dimensional bounded domain with random noise perturbation proportional to the state of the system, the existence of a random attractor and its robustness with respect to the reverse reaction rates are proved through sharp and uniform estimates showing the pullback uniform dissipation and the pullback asymptotic compactness.

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