22 research outputs found
Existence of globally attracting fixed points of viscous Burgers equation with constant forcing. A computer assisted proof
We present a computer assisted method for proving the existence of globally
attracting fixed points of dissipative PDEs. An application to the viscous
Burgers equation with periodic boundary conditions and a forcing function
constant in time is presented as a case study. We establish the existence of a
locally attracting fixed point by using rigorous numerics techniques. To prove
that the fixed point is, in fact, globally attracting we introduce a technique
relying on a construction of an absorbing set, capturing any sufficiently
regular initial condition after a finite time. Then the absorbing set is
rigorously integrated forward in time to verify that any sufficiently regular
initial condition is in the basin of attraction of the fixed point.Comment: To appear in Topological Methods in Nonlinear Analysis, 201
Existence of globally attracting solutions for one-dimensional viscous Burgers equation with nonautonomous forcing - a computer assisted proof
We prove the existence of globally attracting solutions of the viscous
Burgers equation with periodic boundary conditions on the line for some
particular choices of viscosity and non-autonomous forcing. The attract- ing
solution is periodic if the forcing is periodic. The method is general and can
be applied to other similar partial differential equations. The proof is
computer assisted.Comment: 38 pages, 1 figur
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
Stabilizing effect of large average initial velocity in forced dissipative PDEs invariant with respect to Galilean transformations
We describe a topological method to study the dynamics of dissipative PDEs on
a torus with rapidly oscillating forcing terms. We show that a dissipative PDE,
which is invariant with respect to Galilean transformations, with a large
average initial velocity can be reduced to a problem with rapidly oscillating
forcing terms. We apply the technique to the Burgers equation, and the
incompressible 2D Navier-Stokes equations with a time-dependent forcing. We
prove that for a large initial average speed the equation admits a bounded
eternal solution, which attracts all other solutions forward in time. For the
incompressible 3D Navier-Stokes equations we establish existence of a locally
attracting solution
Efficient and Generic Algorithm for Rigorous Integration Forward in Time of dPDEs: Part I
We propose an efficient and generic algorithm for rigorous integration forward in time of partial differential equations written in the Fourier basis. By rigorous integration we mean a procedure which operates on sets and return sets which are guaranteed to contain the exact solution. The presented algorithm generates, in an efficient way, normalized derivatives which are used by the Lohner algorithm to produce a rigorous bound. The algorithm has been successfully tested on several partial differential equations (PDEs) including the Burgers equation, the Kuramoto-Sivashinsky equation, and the Swift-Hohenberg equation. The problem of rigorous integration in time of partial differential equations is a problem of large computational complexity and efficient algorithms are required to deal with PDEs on higher dimensional domains, like the Navier-Stokes equation. Technicalities regarding the various optimization techniques implemented in the software used in this paper will be reported elsewhere
Unified Long-Term Time-Series Forecasting Benchmark
In order to support the advancement of machine learning methods for
predicting time-series data, we present a comprehensive dataset designed
explicitly for long-term time-series forecasting. We incorporate a collection
of datasets obtained from diverse, dynamic systems and real-life records. Each
dataset is standardized by dividing it into training and test trajectories with
predetermined lookback lengths. We include trajectories of length up to
to ensure a reliable evaluation of long-term forecasting capabilities. To
determine the most effective model in diverse scenarios, we conduct an
extensive benchmarking analysis using classical and state-of-the-art models,
namely LSTM, DeepAR, NLinear, N-Hits, PatchTST, and LatentODE. Our findings
reveal intriguing performance comparisons among these models, highlighting the
dataset-dependent nature of model effectiveness. Notably, we introduce a custom
latent NLinear model and enhance DeepAR with a curriculum learning phase. Both
consistently outperform their vanilla counterparts
Stabilizing the Long-time Behavior of the Navier-Stokes Equations and Damped Euler Systems by Fast Oscillating Forces
The paper studies the issue of stability of solutions to the Navier-Stokes
and damped Euler systems in periodic boxes. We show that under action of fast
oscillating-in- time external forces all two dimensional regular solutions
converge to a time periodic flow. Unexpectedly, effects of stabilization can be
also obtained for systems with stationary forces with large total momentum
(average of the velocity). Thanks to the Galilean transformation and space
boundary conditions, the stationary force changes into one with time
oscillations. In the three dimensional case we show an analogical result for
weak solutions to the Navier- Stokes equations