Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation

We consider the Kuramoto-Sivashinsky (KS) equation in one spatial dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam, and later improved by Collet, Eckmann, Epstein and Stubbe, and Goodman, to prove that ||u||_2 < C L^1.5. This result is slightly weaker than that recently announced by Giacomelli and Otto, but applies in the presence of an additional linear destabilizing term. We further show that for a large class of Lyapunov functions \phi the exponent 1.5 is the best possible from this line of argument. Further, this result together with a result of Molinet gives an improved estimate for L_2 boundedness of the Kuramoto-Sivashinsky equation in thin rectangular domains in two spatial dimensions.Comment: 17 pages, 1 figure; typos corrected, references added; figure modifie

Noise-induced Turbulence in Nonlocally Coupled Oscillators

We demonstrate that nonlocally coupled limit-cycle oscillators subject to spatiotemporally white Gaussian noise can exhibit a noise-induced transition to turbulent states. After illustrating noise-induced turbulent states with numerical simulations using two representative models of limit-cycle oscillators, we develop a theory that clarifies the effective dynamical instabilities leading to the turbulent behavior using a hierarchy of dynamical reduction methods. We determine the parameter region where the system can exhibit noise-induced turbulent states, which is successfully confirmed by extensive numerical simulations at each level of the reduction.Comment: 23 pages, 17 figures, to appear in Phys. Rev.

On the nonlinear interfacial instability of rotating core-annular flow

The interfacial stability of rotating core-annular flows is investigated. The linear and nonlinear effects are considered for the case when the annular region is very thin. Both asymptotic and numerical methods are used to solve the flow in the core and film regions which are coupled by a difference in viscosity and density. The long-term behavior of the fluid-fluid interface is determined by deriving its nonlinear evolution in the form of a modified Kuramoto-Sivashinsky equation. We obtain a generalization of this equation to three dimensions. The flows considered are applicable to a wide array of physical problems where liquid films are used to lubricate higher or lower viscosity core fluids, for which a concentric arrangement is desired. Linearized solutions show that the effects of density and viscosity stratification are crucial to the stability of the interface. Rotation generally destabilizes non-axisymmetric disturbances to the interface, whereas the centripetal forces tend to stabilize flows in which the film contains the heavier fluid. Nonlinear affects allow finite amplitude helically travelling waves to exist when the fluids have different viscosities

On a nonlocal analog of the Kuramoto-Sivashinsky equation

We study a nonlocal equation, analogous to the Kuramoto-Sivashinsky equation, in which short waves are stabilized by a possibly fractional diffusion of order less than or equal to two, and long waves are destabilized by a backward fractional diffusion of lower order. We prove the global existence, uniqueness, and analyticity of solutions of the nonlocal equation and the existence of a compact attractor. Numerical results show that the equation has chaotic solutions whose spatial structure consists of interacting traveling waves resembling viscous shock profiles

Computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model

We present a computer-assisted proof of heteroclinic connections in the one-dimensional Ohta-Kawasaki model of diblock copolymers. The model is a fourth-order parabolic partial differential equation subject to homogeneous Neumann boundary conditions, which contains as a special case the celebrated Cahn-Hilliard equation. While the attractor structure of the latter model is completely understood for one-dimensional domains, the diblock copolymer extension exhibits considerably richer long-term dynamical behavior, which includes a high level of multistability. In this paper, we establish the existence of certain heteroclinic connections between the homogeneous equilibrium state, which represents a perfect copolymer mixture, and all local and global energy minimizers. In this way, we show that not every solution originating near the homogeneous state will converge to the global energy minimizer, but rather is trapped by a stable state with higher energy. This phenomenon can not be observed in the one-dimensional Cahn-Hillard equation, where generic solutions are attracted by a global minimizer

Structure And Dynamics Of Modulated Traveling Waves In Cellular Flames

We describe spatial and temporal patterns in cylindrical premixed flames in the cellular regime, $Le < 1$, where the Lewis number $Le$ is the ratio of thermal to mass diffusivity of a deficient component of the combustible mixture. A transition from stationary, axisymmetric flames to stationary cellular flames is predicted analytically if $Le$ is decreased below a critical value. We present the results of numerical computations to show that as $Le$ is further decreased traveling waves (TWs) along the flame front arise via an infinite-period bifurcation which breaks the reflection symmetry of the cellular array. Upon further decreasing $Le$ different kinds of periodically modulated traveling waves (MTWs) as well as a branch of quasiperiodically modulated traveling waves (QPMTWs) arise. These transitions are accompanied by the development of different spatial and temporal symmetries including period doublings and period halvings. We also observe the apparently chaotic temporal behavior of a disordered cellular pattern involving creation and annihilation of cells. We analytically describe the stability of the TW solution near its onset+ using suitable phase-amplitude equations. Within this framework one of the MTW's can be identified as a localized wave traveling through an underlying stationary, spatially periodic structure. We study the Eckhaus instability of the TW and find that in general they are unstable at onset in infinite systems. They can, however, become stable for larger amplitudes.Comment: to appear in Physica D 28 pages (LaTeX), 11 figures (2MB postscript file

Learning a model is paramount for sample efficiency in reinforcement learning control of PDEs

The goal of this paper is to make a strong point for the usage of dynamical models when using reinforcement learning (RL) for feedback control of dynamical systems governed by partial differential equations (PDEs). To breach the gap between the immense promises we see in RL and the applicability in complex engineering systems, the main challenges are the massive requirements in terms of the training data, as well as the lack of performance guarantees. We present a solution for the first issue using a data-driven surrogate model in the form of a convolutional LSTM with actuation. We demonstrate that learning an actuated model in parallel to training the RL agent significantly reduces the total amount of required data sampled from the real system. Furthermore, we show that iteratively updating the model is of major importance to avoid biases in the RL training. Detailed ablation studies reveal the most important ingredients of the modeling process. We use the chaotic Kuramoto-Sivashinsky equation do demonstarte our findings

Time integration and steady-state continuation for 2d lubrication equations

Lubrication equations allow to describe many structurin processes of thin liquid films. We develop and apply numerical tools suitable for their analysis employing a dynamical systems approach. In particular, we present a time integration algorithm based on exponential propagation and an algorithm for steady-state continuation. In both algorithms a Cayley transform is employed to overcome numerical problems resulting from scale separation in space and time. An adaptive time-step allows to study the dynamics close to hetero- or homoclinic connections. The developed framework is employed on the one hand to analyse different phases of the dewetting of a liquid film on a horizontal homogeneous substrate. On the other hand, we consider the depinning of drops pinned by a wettability defect. Time-stepping and path-following are used in both cases to analyse steady-state solutions and their bifurcations as well as dynamic processes on short and long time-scales. Both examples are treated for two- and three-dimensional physical settings and prove that the developed algorithms are reliable and efficient for 1d and 2d lubrication equations, respectively.Comment: 33 pages, 16 figure

Roll-waves in bi-layer flows

In this paper, we derive consistent shallow water equations for bi-layer flows of Newtonian fluids flowing down a ramp. We carry out a complete spectral analysis of steady flows in the low frequency regime and show the occurence of hydrodynamic instabilities, so called roll-waves, when steady flows are unstable

Pulse dynamics in low-Reynolds-number interfacial hydrodynamics: Experiments and theory

a b s t r a c t We analyze interaction of nonlinear pulses in active-dispersive-dissipative nonlinear media. A particular example of such media is a viscous thin film coating a vertical fibre. Experiments for this system reveal that the interface evolves into a train of droplike solitary pulses in which numerous inelastic coalescence events take place. In such events, larger pulses catch up with smaller ones and annihilate them. However, for certain flow conditions and after a certain distance from the inlet, no more coalescence is observed and the flow is described by quasi-equilibrium solitary pulses interacting continuously with each other through attractions and repulsions, and, eventually they form bound states of groups of pulses in which the pulses travel with the same velocities as a whole. This experimental study represents the first evidence of formation of bound states in low-Reynolds-number interfacial hydrodynamics. To gain theoretical insight into the interaction of the pulses and formation of bound states, we derive a weakly nonlinear model for the flow, the generalized Kuramoto-Sivashinsky (gKS) equation, that retains the fundamental mechanisms of the wave evolution, namely, dominant nonlinearity, instability, stability and dispersion. Much like in the experiments, the spatio-temporal evolution of the gKS equation is dominated by quasistationary solitary pulses which continuously interact with each other through coalescence events or attractions/repulsions. To understand the latter case, we utilize a weak-interaction theory for the solitary pulses of the gKS equation. The theory is based on representing the solution of the equation as a superposition of the pulses and an overlap function and leads to a coupled system of ordinary differential equations describing the evolution of the locations of the pulses, or, alternatively, the evolution of the separation distances. By analyzing the fixed points of this system, we obtain bound states of interacting pulses. For two pulses, we provide a criterion for the existence of a countable infinite or finite number of bound states, depending on the strength of the dispersive term in the equation. The interaction theory and resulting bound states are corroborated by computations of the full equation. We also find qualitative agreement between the theory and the experiments