Search

An algorithm for continuous data assimilation for the two-dimensional Bénard convection problem is introduced and analyzed. It is inspired by the data assimilation algorithm developed for the Navier–Stokes equations, which allows for the implementation of variety of observables: low Fourier modes, nodal values, finite volume averages, and finite elements. The novelty here is that the observed data is obtained for the velocity field alone; i.e. no temperature measurements are needed for this algorithm. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate, the unique exact unknown solution of the Bénard convection problem associated with the observed (finite dimensional projection of) velocity. [ABSTRACT FROM AUTHOR]

In this paper we show that the global attractor of the 1D damped, driven, nonlinear Schrödinger equation (NLS) is embedded in the long-time dynamics of a determining form. The determining form is an ordinary differential equation in a space of trajectories X = C b 1 ( R , P m H 2 ) where P m is the L 2 -projector onto the span of the first m Fourier modes. There is a one-to-one identification with the trajectories in the global attractor of the NLS and the steady states of the determining form. We also give an improved estimate for the number of the determining modes. [ABSTRACT FROM AUTHOR]

In this article we consider the Euler- α system as a regularization of the incompressible Euler equations in a smooth, two-dimensional, bounded domain. For the limiting Euler system we consider the usual non-penetration boundary condition, while, for the Euler- α regularization, we use velocity vanishing at the boundary. We also assume that the initial velocities for the Euler- α system approximate, in a suitable sense, as the regularization parameter α → 0 , the initial velocity for the limiting Euler system. For small values of α , this situation leads to a boundary layer, which is the main concern of this work. Our main result is that, under appropriate regularity assumptions, and despite the presence of this boundary layer, the solutions of the Euler- α system converge, as α → 0 , to the corresponding solution of the Euler equations, in L 2 in space, uniformly in time. We also present an example involving parallel flows, in order to illustrate the indifference to the boundary layer of the α → 0 limit, which underlies our work. [ABSTRACT FROM AUTHOR]

We study the effect of the rotation on the life-span of solutions to the 3D hydrostatic Euler equations with rotation and the inviscid Primitive equations (PEs) on the torus. The space of analytic functions appears to be the natural space to study the initial value problem for the inviscid PEs with general initial data, as they have been recently shown to exhibit Kelvin–Helmholtz type instability. First, for a short interval of time that is independent of the rate of rotation | Ω | , we establish the local well-posedness of the inviscid PEs in the space of analytic functions. In addition, thanks to a fine analysis of the barotropic and baroclinic modes decomposition, we establish two results about the long time existence of solutions. (i) Independently of | Ω | , we show that the life-span of the solution tends to infinity as the analytic norm of the initial baroclinic mode goes to zero. Moreover, we show in this case that the solution of the 3D inviscid PEs converges to the solution of the limit system, which is governed by the 2D Euler equations. (ii) We show that the life-span of the solution can be prolonged unboundedly with | Ω | → ∞ , which is the main result of this paper. This is established for "well-prepared" initial data, namely, when only the Sobolev norm (but not the analytic norm) of the baroclinic mode is small enough, depending on | Ω | . Furthermore, for large | Ω | and "well-prepared" initial data, we show that the solution to the 3D inviscid PEs is approximated by the solution to a simple limit resonant system with the same initial data. [ABSTRACT FROM AUTHOR]

This study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of Approximate Inertial Manifolds (AIMs); the particular motivation is the so-called post-processing Galerkin method of Garcia-Archilla, Novo and Titi. Its applicability can be extended: the need for accurate truncated Galerkin projections and for deriving closed-formed corrections can be circumvented using machine learning tools. When the right latent variables are not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a manifold learning scheme) can be used to discover good sets of latent variables and test their explainability. The proposed methodology can express the ROMs in terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both Black-Box and (theoretically-informed and data-corrected) Gray-Box models are described; the necessity for the latter arises when truncated Galerkin projections are so inaccurate as to not be amenable to post-processing. We use the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative partial differential equations to illustrate and successfully test the overall framework. • Machine learning motivates revisiting post-processing Galerkin ROMs. • Diffusion Maps & autoencoders discover relations between sets of latent variables. • Data-driven workflows produce ROM closures to enhance accuracy. • "Gray box" models improve first principles ROMs when they are inaccurate. [ABSTRACT FROM AUTHOR]

The purpose of this article is to give a complete proof of a [Formula: see text] regularity result for the pressure for weak solutions of the two-dimensional 'incompressible Euler equations' when the fluid velocity enjoys the same type of regularity in a compact simply connected domain with [Formula: see text] boundary. To accomplish our result, we realize that it is compulsory to introduce a new weak formulation for the boundary condition of the pressure, which is consistent with, and equivalent to, that of classical solutions. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation | Ω | , we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only L 2 in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large | Ω | , we show that the existence time of the solution can be prolonged, with "well-prepared" initial data. Finally, in the case of two spatial dimensions with Ω = 0 , we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables., Competing Interests: Conflict of interestOn behalf of all authors, the corresponding author states that there is no conflict of interest., (© The Author(s) 2022.)