Universal approximation by translates of fundamental solutions of elliptic equations

In the present work, we investigate the approximability of solutions of elliptic partial differential equations in a bounded domain ω by universal series of translates of fundamental solutions of the underlying partial differential operator. The singularities of the fundamental solutions lie on a prescribed surface outside of, known as the pseudo-boundary. The domains under consideration satisfy a rather mild boundary regularity requirement, namely, the segment condition. We study approximations with respect to the norms of the spaces and we establish the existence of universal series. Analogous results are obtainable with respect to the norms of Holder spaces The sequence of coefficients of the universal series may be chosen in but it can not be chosen in. © 2011, by Oldenbourg Wissenschaftsverlag, Nicosia, Germany. All rights reserved

Predicting chaos for infinite dimensional dynamical systems: the Kuramoto-Sivashinsky equation, a case study.

The results of extensive computations are presented to accurately characterize transitions to chaos for the Kuramoto-Sivashinsky equation. In particular we follow the oscillatory dynamics in a window that supports a complete sequence of period doubling bifurcations preceding chaos. As many as 13 period doublings are followed and used to compute the Feigenbaum number for the cascade and so enable an accurate numerical evaluation of the theory of universal behavior of nonlinear systems, for an infinite dimensional dynamical system. Furthermore, the dynamics at the threshold of chaos exhibit a self-similar behavior that is demonstrated and used to compute a universal scaling factor, which arises also from the theory of nonlinear maps and can enable continuation of the solution into a chaotic regime. Aperiodic solutions alternate with periodic ones after chaos sets in, and we show the existence of a period six solution separated by chaotic regions

Surfactant destabilization and non-linear phenomena in two-fluid shear flows at small Reynolds numbers.

The flow of two superposed fluids in a channel in the presence of an insoluble surfactant is studied. Asymptotic analysis when one of the layers is thin yields a system of coupled weakly non-linear evolution equations for the film thickness and the local surface surfactant concentration. Film and main flow dynamics are coupled through a non-local term, and in the absence of surfactants the model is non-linearly stable with trivial large time solutions. Instability arises due to the presence of surfactants and the pseudo-differential non-linear system is solved numerically by implementing accurate linearly implicit methods. Extensive numerical experiments reveal that the dynamics are mostly organized into travelling or time-periodic travelling wave pulses, but spatiotemporal chaos is also supported when the length of the system is sufficiently large

Computational study of chaotic and ordered solutions of the Kuramoto-Sivashinsky equation

We report the results of extensive numerical experiments on the Kuramoto-Sivashinsky equation in the strongly chaotic regime as the viscosity parameter is decreased and increasingly more linearly unstable modes enter the dynamics. General initial conditions are used and evolving states do not assume odd-parity. A large numberofnumerical experiments are employed in order to obtain quantitative characteristics of the dynamics. We report on di erent routes to chaos and provide numerical evidence and construction of strange attractors with self-similar characteristics. As the \viscosity " parameter decreases the dynamics becomes increasingly more complicated and chaotic. In particular it is found that regular behavior in the form of steady state or steady state traveling waves is supported amidst the time-dependent and irregular motions. We show that multimodal steady states emerge and are supported on decreasing windows in parameter space. In addition we invoke a self-similarity property of the equation, to show that these pro les are obtainable from global xed point attractors of the Kuramoto-Sivashinsky equation at much larger values of the viscosity

Optimal analyticity estimates for non-linear active-dissipative evolution equations

Active–dissipative evolution equations emerge in a variety of physical and technological applications including liquid film flows, flame propagation, epitaxial film growth in materials manufacturing, to mention a few. They are characterized by three main ingredients: a term producing growth (active), a term providing damping at short length scales (dissipative) and a nonlinear term that transfers energy between modes and crucially produces a nonlinear saturation. The manifestation of these three mechanisms can produce large-time spatiotemporal chaos as evidenced by the Kuramoto-Sivashinsky equation (negative diffusion, fourth-order dissipation and a Burgers nonlinearity), which is arguably the simplest partial differential equation to produce chaos. The exact form of the terms (and in particular their Fourier symbol) determines the type of attractors that the equations possess. The present study considers the spatial analyticity of solutions under the assumption that the equations possess a global attractor. In particular, we investigate the spatial analyticity of solutions of a class of one-dimensional evolutionary pseudo-differential equations with Burgers nonlinearity, which are periodic in space, thus generalizing the Kuramoto-Sivashinsky equation motivated by both applications and their fundamental mathematical properties. Analyticity is examined by utilizing a criterion involving the rate of growth of suitable norms of the n th spatial derivative of the solution, with respect to the spatial variable, as n tends to infinity. An estimate of the rate of growth of the n th spatial derivative is obtained by fine-tuning the spectral method, developed elsewhere. We prove that the solutions are analytic if γ ⁠, the order of dissipation of the pseudo-differential operator, is higher than one. We also present numerical evidence suggesting that this is optimal, i.e. if γ is not larger that one, then the solution is not in general analytic. Extensive numerical experiments are undertaken to confirm the analysis and also to compute the band of analyticity of solutions for a wide range of active–dissipative terms and large spatial periods that support chaotic solutions. These ideas can be applied to a wide class of active–dissipative–dispersive pseudo-differential equations

Universal series in ∩p>1ℓp

In this paper an abstract condition is given yielding universal series defined by sequences a = {aj}∞j=1 in ∩p>1ℓp but not in ℓ1. We obtain a unification of some known results related to approximation by translates of specific functions including the Riemann ζ-function, or a fundamental solution of a given elliptic operator in ℝν with constant coefficients or an approximate identity as, for example, the normal distribution. Another application gives universal trigonometric series in simultaneously with respect to all σ-finite Borel measures in ℝν. Stronger results are obtained by using universal Dirichlet series. © 2009 London Mathematical Society

Linearly implicit schemes for multi-dimensional Kuramoto-Sivashinsky type equations arising in falling film flows.

This study introduces, analyses and implements space-time discretizations of two-dimensional active dissipative partial differential equations such as the Topper–Kawahara equation; this is the two-dimensional extension of the dispersively modified Kuramoto–Sivashinsky equation found in falling film hydro-dynamics. The spatially periodic initial value problem is considered as the size of the periodic box increases. The schemes utilized are implicit–explicit multistep (BDF) in time and spectral in space. Numerical analysis of these schemes is carried out and error estimates, in both time and space, are derived. Preliminary numerical experiments provided strong evidence of analyticity, thus yielding a practical rule-of-thumb that determines the size of the truncation in Fourier space. The accuracy of the BDF schemes (of order 1–6) is confirmed through computations. Extensive computations into the strongly chaotic regime (as the domain size increases), provided an optimal estimate of the size of the absorbing ball as a function of the size of the domain; this estimate is found to be proportional to the area of the periodic box. Numerical experiments were also carried out in the presence of dispersion. It is observed that sufficient amounts of dispersion reduce the complexity of the chaotic dynamics, and can organize solution into nonlinear travelling wave pulses of permanent form

An in-depth numerical study of the two-dimensional Kuramoto–Sivashinsky equation

The Kuramoto–Sivashinsky equation in one spatial dimension (1D KSE) is one of the most well-known and well-studied partial differential equations. It exhibits spatio-temporal chaos that emerges through various bifurcations as the domain length increases. There have been several notable analytical studies aimed at understanding how this property extends to the case of two spatial dimensions. In this study, we perform an extensive numerical study of the Kuramoto–Sivashinsky equation (2D KSE) to complement this analytical work. We explore in detail the statistics of chaotic solutions and classify the solutions that arise for domain sizes where the trivial solution is unstable and the long-time dynamics are completely two-dimensional. While we find that many of the features of the 1D KSE, including how the energy scales with system size, carry over to the 2D case, we also note several differences including the various paths to chaos that are not through period doubling