Global Stabilization of the Navier-Stokes-Voight and the damped nonlinear wave equations by finite number of feedback controllers

In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of the Navier-Stokes-Voigt equations, the strongly damped nonlinear wave equations and the nonlinear wave equation with nonlinear damping term, the Benjamin-Bona-Mahony-Burgers equation and the KdV-Burgers equation. This algorithm capitalizes on the fact that such infinite-dimensional dissipative dynamical systems posses finite-dimensional long-time behavior which is represented by, for instance, the finitely many determining parameters of their long-time dynamics, such as determining Fourier modes, determining volume elements, determining nodes , etc..The algorithm utilizes these finite parameters in the form of feedback control to stabilize the relevant solutions. For the sake of clarity, and in order to fix ideas, we focus in this work on the case of low Fourier modes feedback controller, however, our results and tools are equally valid for using other feedback controllers employing other spatial coarse mesh interpolants

Nonlinear Evolution Problems

In this workshop geometric evolution equations of parabolic type, nonlinear hyperbolic equations, and dispersive equations and their interrelations were the subject of 21 talks and several shorter special presentations

Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains

We prove the existence of a compact random attractor for the stochastic Benjamin-Bona-Mahony Equation defined on an unbounded domain. This random attractor is invariant and attracts every pulled-back tempered random set under the forward flow. The asymptotic compactness of the random dynamical system is established by a tail-estimates method, which shows that the solutions are uniformly asymptotically small when space and time variables approach infinity.Comment: 37 page

Experimental study of integrable turbulence in shallow water

We analyze a set of bidirectional wave experiments in a linear wave flume of which some are conducive to integrable turbulence. In all experiments the wavemaker forcing is sinusoidal and the wave motion is recorded by seven high-resolution side-looking cameras. The periodic scattering transform is implemented and power spectral densities computed to discriminate linear wave motion states from integrable turbulence and soliton gas. Values of the wavemaker forcing Ursell number and relative amplitude are required to be above some threshold values for the integral turbulence to occur. Despite the unavoidable slow damping, soliton gases achieve stationary states because of the continuous energy input by the wavemaker. The statistical properties are given in terms of probability density distribution, skewness and kurtosis. The route to integrable turbulence, by the disorganization of the wavemaker induced sinusoidal wave motion, depends on the non-linearity of the waves but equally on the amplitude amplification and reduction due to the wavemaker feedback on the wave field

Beyond the KdV: post-explosion development

Several threads of the last 25 years’ developments in nonlinear wave theory that stem from the classical Korteweg–de Vries (KdV) equation are surveyed. The focus is on various generalizations of the KdV equation which include higher-order nonlinearity, large-scale dispersion, and a nonlocal integral dispersion. We also discuss how relatively simple models can capture strongly nonlinear dynamics and how various modifications of the KdV equation lead to qualitatively new, non-trivial solutions and regimes of evolution observable in the laboratory and in nature. As the main physical example, we choose internal gravity waves in the ocean for which all these models are applicable and have genuine importance. We also briefly outline the authors’ view of the future development of the chosen lines of nonlinear wave theory

Random attractors for stochastic 2D-Navier-Stokes equations in some unbounded domains

We show that the stochastic flow generated by the Stochastic Navier-Stokes equations in a 2-dimensional Poincar\'e domain has a unique random attractor. This result complements a recent result by Brze\'zniak and Li [10] who showed that the flow is asymptotically compact and generalizes a recent result by Caraballo et al. [12] who proved existence of a unique pullback attractor for the time-dependent deterministic Navier-Stokes equations in a 2-dimensional Poincar\'e domain

Smoothing properties of certain dispersive nonlinear partial differential equations

This thesis is primarily concerned with the smoothing properties of dispersive equations and systems. Smoothing in this context means that the nonlinear part of the solution flow is of higher regularity than the initial data. We establish this property, and some of its consequences, for several equations. The first part of the thesis studies a periodic coupled Korteweg-de Vries (KdV) system. This system, known as the Majda-Biello system, displays an interesting dependancy on the coupling coefficient α linking the two KdV equations. Our main result is that the nonlinear part of the evolution resides in a smoother space for almost every choice of α. The smoothing index depends on number-theoretic properties of α, which control the behavior of the resonant sets. We then consider the forced and damped version of the system and obtain similar smoothing estimates. These estimates are used to show the existence of a global attractor in the energy space. We also use a modified energy functional to show that when the damping is large, the attractor is trivial. The next chapter studies the Zakharov and related Klein-Gordon-Schrödinger (KGS) systems on Euclidean spaces. Again, the main result is that the nonlinear part of the solution is smoother than the initial data. The proof relies on a new bilinear Bourgain-space estimate, which is proved using delicate dyadic and angular decompositions of the frequency domain. As an application, we give a simplified proof of the existence of global attractors for the KGS flow in the energy space for dimensions two and three. We also use smoothing in conjunction with a high-low decomposition to show global well-posedness of the KGS evolution on R4 below the energy space for sufficiently small initial data. In the final portion of the thesis we consider well-posedness and regularity properties of the “good” Boussinesq equation on the half line. We obtain local existence, uniqueness and continuous dependence on initial data in low-regularity spaces. We also establish a smoothing result, obtaining up to half derivative smoothing of the nonlinear term. The results are sharp within the framework of the restricted norm method that we use and match known results on the full line

A Boussinesq-type Model for Waves Generated by Flow over a Bump

Abstract A uniform flow disturbed by a bump is studied. The effect of the disturbance is presented at the surface, generating wave. The wave propagation is modeled into a couple of equations, in terms of the surface elevation and the depth average velocity. The numerical solution of the equations is simulated to observe the propagation, especially for long run time, using a predictor-corrector method. A steady solitary surface profile is obtained for supercritical upstream flow, similarly for subcritical flow but for negative amplitude. In the transient process, more waves are generated but some of them propagates to the left or right, and only one wave remains above the bump. Mathematics Subject Classification: 35C20, 76B07, 76B2