We prove higher-order and a Gevrey class (spatial analytic) regularity of solutions to the Euler-Voigt inviscid $\alpha$-regularization three-dimensional Euler equations ideal incompressible fluids. Moreover, we establish convergence strong model corresponding solution for flow on interval existence latter. Furthermore, derive criterion finite-time blow-up based this regularization. The coupling magnetic field is introduced form an regularization irresistive magneto-hydrodynamic (MHD)...

We study a continuous data assimilation algorithm proposed by Azouani, Olson, and Titi (AOT) in the context of an unknown viscosity. determine large-time error between true solution 2 dimensional Navier--Stokes equations assimilated due to discrepancy approximate viscosity physical Additionally, we develop that can be run tandem with AOT recover both using only spatially discrete velocity measurements.

<p style='text-indent:20px;'>Motivated by recent progress in data assimilation, we develop an algorithm to dynamically learn the parameters of a chaotic system from partial observations. Under reasonable assumptions, supply rigorous analytical proof that guarantees convergence this true parameter values when question is classic three-dimensional Lorenz system. Such result appears be first its kind for dynamical estimation nonlinear systems. Computationally, demonstrate efficacy on...

We introduce three new nonlinear continuous data assimilation algorithms. These models are compared with the linear algorithm introduced by Azouani, Olson, and Titi (AOT). As a proof-of-concept for these models, we computationally investigate algorithms in context of 1D Kuramoto-Sivashinsky equations. observe that experience super-exponential convergence time, converge to machine precision significantly faster than AOT our tests. For both simplicity completeness, provide key analysis...

Abstract We propose an approximate model for the 2D Kuramoto–Sivashinsky equations (KSE) of flame fronts and crystal growth. prove that this new ‘calmed’ version KSE is globally well-posed, moreover, its solutions converge to on time interval existence uniqueness at algebraic rate. In addition, we provide simulations calmed KSE, illuminating dynamics. These also indicate our analytical predictions convergence rates are sharp. discuss analogies with 3D Navier–Stokes fluid

<p style='text-indent:20px;'>We study a continuous data assimilation (CDA) algorithm for velocity-vorticity formulation of the 2D Navier-Stokes equations in two cases: nudging applied to velocity and vorticity, only. We prove that under typical finite element spatial discretization backward Euler temporal discretization, application CDA preserves unconditional long-time stability property method provides optimal accuracy. These properties hold if is only velocity, also vorticity then...

We establish global existence and uniqueness theorems for the two-dimensional non-diffusive Boussinesq system with viscosity only in horizontal direction, which arises Ocean dynamics. This work improves well-posedness results established recently by R. Danchin M. Paicu anisotropic zero diffusion. Although we follow some of their ideas, proving result, have used an alternative approach writing transported temperature (density) as $θ= Δξ$ adapting techniques V. Yudovich 2D incompressible Euler...

We introduce three new nonlinear continuous data assimilation algorithms. These models are compared with the linear algorithm introduced by Azouani, Olson, and Titi (AOT). As a proof-of-concept for these models, we computationally investigate algorithms in context of 1D Kuramoto-Sivashinsky equation. observe that experience super-exponential convergence time, converge to machine precision significantly faster than AOT our tests.

An algorithm is developed, rigorously justified, and numerically implemented that capable of determining the full body force used to generate chaotic, turbulent dynamics in two-dimensional Navier-Stokes fluid dynamics. The primary contribution this result accurate reconstruction requires only partial observation state, i.e. sparse observations state are sufficient recover not itself but unknown forcing function as well even fully developed setting.

We propose a data assimilation algorithm for the 2D Navier-Stokes equations, based on Azouani, Olson, and Titi (AOT) algorithm, but applied to Navier-Stokes-Voigt equations. Adapting AOT regularized versions of has been done before, innovation this work is drive equation with observational data, rather than from system. first prove that new system globally well-posed. Moreover, we any admissible initial $ L^2 H^1 norms error are bounded by constant times power Voigt-regularization parameter...