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The evolution of a determining form for the 2D Navier-Stokes equations (NSE), which is an ODE on a space of trajectories is completely described. It is proved that at every stage of its evolution, the solution is a convex combination of the…

Dynamical Systems · Mathematics 2017-04-05 Ciprian Foias , Michael S. Jolly , Daniel Lithio , Edriss S. Titi

In this article, we consider the two-dimensional stochastic Navier-Stokes equation (SNSE) on a smooth bounded domain, driven by affine-linear multiplicative white noise and with random initial conditions and Dirichlet boundary conditions.…

Analysis of PDEs · Mathematics 2011-12-14 Salah Mohammed , Tusheng Zhang

We discuss a minimal generalization of the incompressible Navier-Stokes equations to describe the solvent flow in an active suspension. To account phenomenologically for the presence of an active component driving the ambient fluid flow, we…

Soft Condensed Matter · Physics 2015-04-10 Jonasz Słomka , Jörn Dunkel

We adapt a previously introduced continuous in time data assimilation (downscaling) algorithm for the 2D Navier-Stokes equations to the more realistic case when the measurements are obtained discretely in time and may be contaminated by…

Analysis of PDEs · Mathematics 2016-05-24 Ciprian Foias , Cecilia F. Mondaini , Edriss S. Titi

In view of the possibility that the 3D Navier-Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with…

Analysis of PDEs · Mathematics 2007-05-23 Alexey Cheskidov , Ciprian Foias

This paper deals with the problem of designing unknown input observers for a class of coupled semilinear wave partial differential equations (PDE) systems. A state observer is designed to estimate the uncertain coupled wave PDE systems.…

Optimization and Control · Mathematics 2026-01-13 Najmeh Ghaderi , Birgit Jacob

A nonlinear observer on the Special Euclidean group $\mathrm{SE(3)}$ for full pose estimation, that takes the system outputs on the real projective space directly as inputs, is proposed. The observer derivation is based on a recent advanced…

Optimization and Control · Mathematics 2015-09-11 Minh-Duc Hua , Tarek Hamel , Robert Mahony , Jochen Trumpf

We establish the global-in-time well-posedness of the two-dimensional incompressible Navier-Stokes equations with freely transported viscosity coefficient, under a scaling-invariant smallness condition on the initial data. The viscosity…

Analysis of PDEs · Mathematics 2024-09-11 Xian Liao , Rebekka Zimmermann

We revisit and sharpen a recent observable regularity criterion for the three-dimensional Navier-Stokes equations on the periodic cube by requiring only finitely many measurements of the flow on a given time interval. Two data models are…

Analysis of PDEs · Mathematics 2026-03-19 Abhishek Balakrishna , Animikh Biswas

The determining modes for the two-dimensional incompressible Navier-Stokes equations (NSE) are shown to satisfy an ordinary differential equation of the form $dv/dt=F(v)$, in the Banach space, $X$, of all bounded continuous functions of the…

Analysis of PDEs · Mathematics 2012-08-28 Ciprian Foias , Michael S. Jolly , Rostyslav Kravchenko , Edriss S. Titi

This paper addresses the three-dimensional Navier-Stokes equations for an incompressible fluid whose density is permitted to be inhomogeneous. We establish a theorem of global existence and uniqueness of strong solutions for initial data…

Analysis of PDEs · Mathematics 2013-04-23 Walter Craig , Xiangdi Huang , Yun Wang

In this paper, we introduce a new sliding mode observer for Lur'e set-valued dynamical systems, particularly addressing challenges posed by uncertainties not within the standard range of observation. Traditionally, most of Luenberger-like…

Optimization and Control · Mathematics 2024-04-26 Samir Adly , Jun Huang , Ba Khiet Le

We study the numerical performance of a continuous data assimilation (downscaling) algorithm, based on ideas from feedback control theory, in the context of the two-dimensional incompressible Navier--Stokes equations. Our model problem is…

Dynamical Systems · Mathematics 2016-05-04 Masakazu Gesho , Eric Olson , Edriss S. Titi

We propose a modification to the nonlinear term of the three-dimensional incompressible Navier-Stokes equations (NSE) in either advective or rotational form which "calms" the system in the sense that the algebraic degree of the nonlinearity…

Analysis of PDEs · Mathematics 2024-01-01 Matthew Enlow , Adam Larios , Jiahong Wu

For two dimensional inhomogeneous Navier-Stokes of incompressible flows, with the assumption that the viscosity depends on the density but with a positive lower bound, using a partial regularity approach, in particular some enhanced decay…

Analysis of PDEs · Mathematics 2016-10-11 Ning Jiang , Yilong Luo

We address the problem of attack detection and isolation for a class of discrete-time nonlinear systems under (potentially unbounded) sensor attacks and measurement noise. We consider the case when a subset of sensors is subject to additive…

Systems and Control · Computer Science 2019-01-04 Tianci Yang , Carlos Murguia , Margreta Kuijper , Dragan Nešić

We prove that a popular classical implicit-explicit scheme for the 2D incompressible Navier--Stokes equations that treats the viscous term implicitly while the nonlinear advection term explicitly is long time stable provided that the time…

Numerical Analysis · Mathematics 2011-05-25 Sigal Gottlieb , Florentina Tone , Cheng Wang , Xiaoming Wang , Djoko Wirosoetisno

In this paper we consider the numerical approximation of the incompressible surface Navier--Stokes equations on an evolving surface. For the discrete representation of the moving surface we use parametric finite elements of degree $\ell…

Numerical Analysis · Mathematics 2026-01-09 Harald Garcke , Robert Nürnberg

In various applications in the field of control engineering the estimation of the state variables of dynamic systems in the presence of unknown inputs plays an important role. Existing methods require the so-called observer matching…

Systems and Control · Electrical Eng. & Systems 2022-04-08 Helmut Niederwieser , Markus Tranninger , Richard Seeber , Markus Reichhartinger

Onboard visual sensing has been widely used in the unmanned ground vehicle (UGV) and/or unmanned aerial vehicle (UAV), which can be modeled as dynamic systems on SE(3). The onboard sensing outputs of the dynamic system can usually be…

Systems and Control · Electrical Eng. & Systems 2023-03-22 Tong Zhang , Ying Tan , Xiang Chen , Zike Lei