Abstract : Despite recent progress in the study of turbulent fluids, to date our mathematical understanding of it remains fundamentally incomplete. Furthermore, recent work on non-uniqueness of weak solutions and lack of global well-posedness of fluid equations, make their study even more pertinent and urgent. This mini-symposium will bring together researchers at all career stages to share their recent results on the interplay of topics such as uniqueness, regularity, boundary-layer theory, asymptotic dynamics and their connections to data assimilation, parameter estimation, machine and physics-informed deep learning algorithms, porous media flow simulations, and the study of statistical and stochastic solutions.
[01423] Parameter analysis in continuous data assimilation for three-dimensional Brinkman-Forchheimer-extended Darcy model
Format : Online Talk on Zoom
Author(s) :
Débora Aparecida Francisco Albanez (Universidade Tecnológica Federal do Parana)
Abstract : Analytical results of the long-time behavior of three-dimensional Brinkman-Forchheimer-extended Darcy model in the context that the parameters related to the damping nonlinear term are unknown is presented. We show estimates in $L^2$ and $H^1$ for large-time error between the true solution and the assimilated solution, which is constructed with the unknown damping parameters and observational measurements obtained continuously in time from a continuous data assimilation technique.
[04819] Boundary layers for a viscous fluid in a corner domain
Format : Online Talk on Zoom
Author(s) :
Anna Mazzucato (Penn State University)
Abstract : We study boundary layers for incompressible slightly viscous fluids in a rectangular domain when steady shears are applied to the top and bottom side. We establish the vanishing viscosity limit using various types of correctors. In particular, we introduce suitable corner layer correctors at the corners. This is joint work with Gung-Min Gie (U. Louisville) and James Kelliher (UC Riverside).
[03465] Numerical schemes for various stochastic models in hydrodynamic
Format : Online Talk on Zoom
Author(s) :
Hakima Bessaih (Florida International University)
Abstract : We will introduce space-time numerical schemes for some stochastic models in hydrodynamic. The models include, the stochastic Navier-Stokes equations, the Boussinesq equations and some other models in porous media. We will also discuss various rates of convergences in probability and in mean square.
[01351] Coupling of free flow and flow in porous media
Format : Online Talk on Zoom
Author(s) :
Xiaoming Wang (Missouri University of Science and Technology and Southern University of Science and Technology)
Abstract : We present some recent progress in the study of coupled free flow and porous media flow. In particular, we show that the several competing interface boundary conditions are asymptotically equivalent at the physically importance small Darcy number regime. We also offer a coarse-grained theory in predicting the deep vs shallow convections in the case when heat convection is involved. Effective numerical algorithms will be presented if time permits.
00869 (2/3) : 5C @D403 [Chair: Vincent R Martinez]
[01798] Wellposedness of stochastic PDEs arising in fluid dynamics
Format : Online Talk on Zoom
Author(s) :
Krutika Tawri (University of California Berkeley)
Abstract : Stochastic forcing terms are commonly added to the governing equations to account for numerical and physical uncertainties in applications. In this talk, we will discuss recent results and new techniques in the analysis of stochastic models, arising in fluid dynamics.
[03526] Error estimates for deep learning methods in fluid dynamics
Format : Talk at Waseda University
Author(s) :
Jing Tian (Towson university)
Animikh Biswas (University of Maryland, Baltimore County)
Suleyman Ulusoy (American University of Ras Al Khaimah)
Abstract : In this talk, we provide error estimates and stability analysis of deep learning techniques for certain partial differential equations including the incompressible Navier–Stokes equations. In particular, we obtain explicit error estimates for the solution computed by optimizing a loss function in a Deep Neural Network approximation of the solution, with a fixed complexity.
[01243] Reconstructing external driving forces in incompressible flow via low-mode observation
Format : Talk at Waseda University
Author(s) :
Vincent R Martinez (CUNY Hunter College & Graduate Center)
Abstract : In this talk, we describe a "spectral filtering" algorithm that reconstructs an apriori unknown external force in the 2D Navier-Stokes equations. This approach was developed by Celik, Olson, and Titi (2019) in order to recover the unobserved high-mode motion of the flow provided that sufficiently many low-modes are observed and that the external force is known. It is shown how this idea can be used to simultaneously recover both the unobserved motion and unknown forcing.
[04541] Analysis of a rotationally constrained convection model
Format : Talk at Waseda University
Author(s) :
Yanqiu Guo (Florida International University)
Abstract : This talk is about the analysis of an asymptotically reduced system for rotationally constrained convection. This reduced system was derived from the 3D Boussinesq equations using the asymptotic theory. On the one hand, the nonlinear convection term has a reduced complexity since it contains only the horizontal gradient. On the other hand, the regularizing viscosity acts in the horizontal direction only. I will present some of our results motivated by the global regularity problem.
[04953] Estimation of parameters on the fly via nudging methods
Format : Talk at Waseda University
Author(s) :
Jared P Whitehead (Brigham Young University)
Abstract : We demonstrate the utility of an algorithm that allows for the estimation and recovery of parameters in a dissipative dynamical system. Rigorous justification of the algorithm is established for specific settings, and numerical simulations are used to demonstrate that it works in various settings and for different circumstances including the estimation of a full forcing function and additive and multiplicative parameters.
[04093] Uniform Boundedness of Entropy to Compressible Navier-Stokes Equations with Vacuum
Format : Talk at Waseda University
Author(s) :
Jinkai Li (South China Normal University)
Abstract : In the presence of vacuum, the physical entropy for polytropic gases behave singularly and it is thus hard to study its dynamics. In this talk, we present some recent studies on the uniform boundedness of the entropy to the viscous compressible ideal gas in the presence of vacuum either at the far field or on the gas-vacuum interface. It will be shown in this talk that, in the case that the vacuum presents at the far fields only, the uniform boundedness of the entropy can be propagated locally or globally if the initial density decays slowly, while if the initial density decays sufficiently fast, the entropy becomes unbounded immediately after the initial time, in particular, the entropy tends to infinity at the far field.
[05089] A unified framework for the analysis of accuracy and stability of a class of data assimilation methods for the Navier-Stokes equations
Format : Talk at Waseda University
Author(s) :
Michal Branicki (University of Edinburgh)
Animikh Biswas (University of Maryland Baltimore County)
Abstract : Bayesian state estimation of a dynamical system utilising a stream of noisy measurements is important in many geophysical and engineering applications, where nonlinearities, high (or infinite) dimensionality of the state space, and sparse observations pose key challenges for deriving efficient and accurate data assimilation techniques. We develop a unified framework for the analysis of several well-known and empirically efficient data assimilation techniques derived from various Gaussian approximations of the Bayesian filtering problem for geophysical-type dissipative dynamics with quadratic nonlinearities. Our approach also elucidates the links between the approximate-Bayesian and control-theoretic approaches to data assimilation. We consider the `model' dynamics governed by the two-dimensional incompressible Navier–Stokes equations and observations given by noisy measurements of finite volume elements, modal or nodal points of the velocity field. In this setup the continuous-time data assimilation techniques, the so-called 3DVar and EnKF (Ensemble Kalman filter), are given by stochastically forced Navier–Stokes equations. We derive rigorous conditions for the (time-asymptotic) accuracy and stability of these algorithms and show the relevance of the so-called covariance inflation and localisation for assuring the necessary bounds. These conditions involve an interplay between the resolution of the observations associated with the covariance operator underlying the data assimilation algorithms and, for the first time, elucidate the properties of the EnKF as well as of the 3DVar for a general covariance operator which is common and relevant for volume and nodal observations.
[05669] Global well-posedness of 3D inhomogenous incompressible Navier-Stokes equations with variable viscosity
Author(s) :
Dongjuan Niu (Capital Normal University)
Abstract : In this talk, we investigate the global well-posedness in the critical spaces of 3D inhomogenous incompressible Navier-Stokes equations, which has variable kinematic viscosity under the smallness assumptions. In addition, the decay estimate of the velocity fields is also obtained.
