One-dimensional parametric determining form for the two-dimensional Navier-Stokes equations

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 initial trajectory and the fixed steady state, with a dynamical convexity parameter $\theta$, which will be called the characteristic determining parameter. That is, we show a remarkable separation of variables formula for the solution of the determining form. Moreover, for a given initial trajectory, the dynamics of the infinite-dimensional determining form are equivalent to those of the characteristic determining parameter $\theta$ which is governed by a one-dimensional ODE. %for the parameter specifying the position on the line segment. This one-dimensional ODE is used to show that if the solution to the determining form converges to the fixed state it does so no faster than $\mathcal{O}(\tau^{-1/2})$, otherwise it converges to a projection of some other trajectory in the global attractor of the NSE, but no faster than $\mathcal{O}(\tau^{-1})$, as $\tau \to \infty$, where $\tau$ is the evolutionary variable in determining form. The one-dimensional ODE also exploited in computations which suggest that the one-sided convergence rate estimates are in fact achieved. The ODE is then modified to accelerate the convergence to an exponential rate. Remarkably, it is shown that the zeros of the scalar function that governs the dynamics of $\theta$, which are called characteristic determining values, identify in a unique fashion the trajectories in the global attractor of the 2D NSE. Furthermore, the one-dimensional characteristic determining form enables us to find unanticipated geometric features of the global attractor, a subject of future research

Exact Solutions of a Remarkable Fin Equation

A model "remarkable" fin equation is singled out from a class of nonlinear (1+1)-dimensional fin equations. For this equation a number of exact solutions are constructed by means of using both classical Lie algorithm and different modern techniques (functional separation of variables, generalized conditional symmetries, hidden symmetries etc).Comment: 6 page

Acoustic Spectroscopy of the DNA in GHz range

We find a parametric resonance in the GHz range of the DNA dynamics, generated by pumping hypersound . There are localized phonon modes caused by the random structure of elastic modulii due to the sequence of base pairs

Stability of Filters for the Navier-Stokes Equation

Data assimilation methodologies are designed to incorporate noisy observations of a physical system into an underlying model in order to infer the properties of the state of the system. Filters refer to a class of data assimilation algorithms designed to update the estimation of the state in a on-line fashion, as data is acquired sequentially. For linear problems subject to Gaussian noise filtering can be performed exactly using the Kalman filter. For nonlinear systems it can be approximated in a systematic way by particle filters. However in high dimensions these particle filtering methods can break down. Hence, for the large nonlinear systems arising in applications such as weather forecasting, various ad hoc filters are used, mostly based on making Gaussian approximations. The purpose of this work is to study the properties of these ad hoc filters, working in the context of the 2D incompressible Navier-Stokes equation. By working in this infinite dimensional setting we provide an analysis which is useful for understanding high dimensional filtering, and is robust to mesh-refinement. We describe theoretical results showing that, in the small observational noise limit, the filters can be tuned to accurately track the signal itself (filter stability), provided the system is observed in a sufficiently large low dimensional space; roughly speaking this space should be large enough to contain the unstable modes of the linearized dynamics. Numerical results are given which illustrate the theory. In a simplified scenario we also derive, and study numerically, a stochastic PDE which determines filter stability in the limit of frequent observations, subject to large observational noise. The positive results herein concerning filter stability complement recent numerical studies which demonstrate that the ad hoc filters perform poorly in reproducing statistical variation about the true signal

Calculation of AGARD Wing 445.6 Flutter Using Navier-Stokes Aerodynamics

An unsteady, 3D, implicit upwind Euler/Navier-Stokes algorithm is here used to compute the flutter characteristics of Wing 445.6, the AGARD standard aeroelastic configuration for dynamic response, with a view to the discrepancy between Euler characteristics and experimental data. Attention is given to effects of fluid viscosity, structural damping, and number of structural model nodes. The flutter characteristics of the wing are determined using these unsteady generalized aerodynamic forces in a traditional V-g analysis. The V-g analysis indicates that fluid viscosity has a significant effect on the supersonic flutter boundary for this wing

Downscaling data assimilation algorithm with applications to statistical solutions of the Navier-Stokes equations

Based on a previously introduced downscaling data assimilation algorithm, which employs a nudging term to synchronize the coarse mesh spatial scales, we construct a determining map for recovering the full trajectories from their corresponding coarse mesh spatial trajectories, and investigate its properties. This map is then used to develop a downscaling data assimilation scheme for statistical solutions of the two-dimensional Navier-Stokes equations, where the coarse mesh spatial statistics of the system is obtained from discrete spatial measurements. As a corollary, we deduce that statistical solutions for the Navier-Stokes equations are determined by their coarse mesh spatial distributions. Notably, we present our results in the context of the Navier-Stokes equations; however, the tools are general enough to be implemented for other dissipative evolution equations