arXiv subjects

Ciprian Foias

Publications and source records attributed to Ciprian Foias.

At least 19 recordsLinked to original sources

arXiv2024

On Galerkin approximations of the Navier-Stokes equations in the limit of large Grashof numbers

We examine how stationary solutions to Galerkin approximations of the Navier--Stokes equations behave in the limit as the Grashof number $G$ tends to $\infty$. An appropriate scaling is used to place the Grashof number as a new coefficient of the nonlinear term, while the body force is fixed. A new type of asymptotic expansion, as $G\to\infty$, for a family of solutions is introduced. Relations among the terms in the expansion are obtained by following a procedure that compares and totally orders positive sequences generated by the expansion. The same methodology applies to the case of perturbed body forces and similar results are obtained. We demonstrate with a class of forces and solutions that have convergent asymptotic expansions in $G$. All the results hold in both two and three dimensions, as well as for both no-slip and periodic boundary conditions.

math.AP↗

arXiv2018

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.

math.AP↗

arXiv2017

Navier and Stokes meet Poincar\' e and Dulac

This paper surveys various precise (long-time) asymptotic results for the solutions of the Navier-Stokes equations with potential forces in bounded domains. It turns out that that the asymptotic expansion leads surprisingly to a Poincar\' e-Dulac normal form of the Navier-Stokes equations. We will also discuss some related results and a few open issues.

math.AP↗

arXiv2017

Contractions with Polynomial Characteristic Functions II. Analytic Approach

The simplest and most natural examples of completely nonunitary contractions on separable complex Hilbert spaces which have polynomial characteristic functions are the nilpotent operators. The main purpose of this paper is to prove the following theorem: Let $T$ be a completely nonunitary contraction on a Hilbert space $\mathcal{H}$. If the characteristic function $Θ_T$ of $T$ is a polynomial of degree $m$, then there exist a Hilbert space $\mathcal{M}$, a nilpotent operator $N$ of order $m$, a coisometry $V_1 \in \mathcal{L}(\overline{ran} (I - N N^*) \oplus \mathcal{M}, \overline{ran} (I - T T^*))$, and an isometry $V_2 \in \mathcal{L}(\overline{ran} (I - T^* T), \overline{ran} (I - N^* N) \oplus \mathcal{M})$, such that \[ Θ_T = V_1 \begin{bmatrix} Θ_N & 0 0 & I_{\mathcal{M}} \end{bmatrix} V_2. \]

math.FA↗

arXiv2016

Properties of stationary statistical solutions of the three-dimensional Navier-Stokes equations

The stationary version of a modified definition of statistical solution for the three-dimensional incompressible Navier-Stokes equations introduced in a previous work is investigated. Particular types of such stationary statistical solutions and their analytical properties are addressed. Results on the support and carriers of these stationary statistical solutions are also given, showing in particular that they are supported on the weak global attractor and are carried by a more regular part of the weak global attractor containing Leray-Hopf weak solutions which are locally strong solutions. Two recurrence-type results related to these measures are also proved.

math.AP↗

arXiv2016

A discrete data assimilation scheme for the solutions of the 2D Navier-Stokes equations and their statistics

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 systematic errors. Our algorithm is designed to work with a general class of observables, such as low Fourier modes and local spatial averages over finite volume elements. Under suitable conditions on the relaxation (nudging) parameter, the spatial mesh resolution and the time step between successive measurements, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of our discrete data assimilation algorithm is also provided.

math.AP↗

arXiv2016

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 $θ$, 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 $θ$ 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 $\mathcal{O}(τ^{-1/2})$, otherwise it converges to a projection of some other trajectory in the global attractor of the NSE, but no faster than $\mathcal{O}(τ^{-1})$, as $τ\to \infty$, where $τ$ 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 $θ$, 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.

math.DS↗

arXiv2015

On the emergence of the Navier-Stokes-$α$ model for turbulent channel flows

In a series of papers (see \cite{CDT02} and the pertinent references therein) the 3D Navier-Stokes-$α$ model were shown to be a useful complement to the 3D Navier-Stokes equations; and in particular, to be a good Reynolds version of the latter equations. In this work, we introduce a simple Reynolds averaging which, due to the wall roughness, transforms the Navier-Stokes equations into the Navier-Stokes-$α$ model.

math.AP↗

arXiv2015

On the Attractor for the Semi-Dissipative Boussinesq Equations

In this article, we study the long time behavior of solutions of a variant of the Boussinesq system in which the equation for the velocity is parabolic while the equation for the temperature is hyperbolic. We prove that the system has a global attractor which retains some of the properties of the global attractors for the 2D and 3D Navier-Stokes equations. Moreover, this attractor contains infinitely many invariant manifolds in which several universal properties of the Batchelor, Kraichnan, Leith theory of turbulence are potentially present.

math-ph↗

arXiv2015

Convergence of time averages of weak solutions of the three-dimensional Navier-Stokes equations

Using the concept of stationary statistical solution, which generalizes the notion of invariant measure, it is proved that, in a suitable sense, time averages of almost every Leray-Hopf weak solution of the three-dimensional incompressible Navier-Stokes equations converge as the averaging time goes to infinity. This system of equations is not known to be globally well-posed, and the above result answers a long-standing problem, extending to this system a classical result from ergodic theory. It is also showed that, from a measure-theoretic point of view, the stationary statistical solution obtained from a generalized limit of time averages is independent of the choice of the generalized limit. Finally, any Borel subset of the phase space with positive measure with respect to a stationary statistical solution is such that for almost all initial conditions in that Borel set and for at least one Leray-Hopf weak solution starting with that initial condition, the corresponding orbit is recurrent to that Borel subset and its mean sojourn time within that Borel subset is strictly positive.

math.AP↗

arXiv2013

On whether zero is in the global attractor of the 2D Navier-Stokes equations

The set of nonzero external forces for which the zero function is in the global attractor of the 2D Navier-Stokes equations is shown to be meagre in a Fréchet topology. A criterion in terms of a Taylor expansion in complex time is used to characterize the forces in this set. This leads to several relations between certain Gevrey subclasses of $C^{\infty}$ and a new upper bound for a Gevrey norm of solutions in the attractor, valid in the strip of analyticity in time.

math.AP↗

arXiv2013

Time analyticity with higher norm estimates for the 2D Navier-Stokes equations

This paper establishes bounds on norms of all orders for solutions on the global attractor of the 2D Navier-Stokes equations, complexified in time. Specifically, for periodic boundary conditions on $[0,L]^2$, and a force $g\in\calD(A^{\frac{α-1}{2}})$, we show there is a fixed strip about the real time axis on which a uniform bound $|A^αu|< m_ανκ_0^α$ holds for each $α\in \bN$. Here $ν$ is viscosity, $\k0=2π/L$, and $m_α$ is explicitly given in terms of $g$ and $α$. We show that if any element in $\calA$ is in $\D(A^α)$, then all of $\calA$ is in $\D(A^α)$, and likewise with $\D(A^α)$ replaced by $C^\infty(Ω)$. We demonstrate the universality of this "all for one, one for all" law on the union of a hierarchal set of function classes. Finally, we treat the question of whether the zero solution can be in the global attractor for a nonzero force by showing that if this is so, the force must be in a particular function class.

math.DS↗

arXiv2013

A unified approach to determining forms for the 2D Navier-Stokes equations - the general interpolants case

In this paper we show that the long time dynamics (the global attractor) of the 2D Navier-Stokes equation is embedded in the long time dynamics of an ordinary differential equation, named {\it determining form}, in a space of trajectories which is isomorphic to $C^1_b(\bR; \bR^N)$, for $N$ large enough depending on the physical parameters of the Navier-Stokes equations. We present a unified approach based on interpolant operators that are induced by any of the determining parameters for the Navier-Stokes equations, namely, determining nodal values, Fourier modes, finite volume elements, finite elements, etc... There are two immediate and interesting consequences of this unified approach. The first is that the constructed determining form has a Lyapunov function, thus its solutions converge, as time goes to infinity, to the set of steady states of the determining form. The second is that these steady states of the determining form are identified, one-to-one, with the trajectories on the global attractor of the Navier-Stokes equations. It is worth adding that this unified approach is general enough that it applies, in an almost straightforward manner, to a whole class of dissipative dynamical systems.

math.DS↗

arXiv2012

Properties of time-dependent statistical solutions of the three-dimensional Navier-Stokes equations

This work is devoted to the concept of statistical solution of the Navier-Stokes equations, proposed as a rigorous mathematical object to address the fundamental concept of ensemble average used in the study of the conventional theory of fully developed turbulence. Two types of statistical solutions have been proposed in the 1970's, one by Foias and Prodi and the other one by Vishik and Fursikov. In this article, a new, intermediate type of statistical solution is introduced and studied. This solution is a particular type of a statistical solution in the sense of Foias and Prodi which is constructed in a way akin to the definition given by Vishik and Fursikov, in such a way that it possesses a number of useful analytical properties.

math.AP↗

arXiv2012

Navier-Stokes equations, determining forms, determining modes, inertial manifolds, dissipative dynamical systems

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 variable $s\in\mathbb{R}$ with values in certain finite-dimensional linear space. This new evolution ODE, named {\it determining form}, induces an infinite-dimensional dynamical system in the space $X$ which is noteworthy for two reasons. One is that $F$ is globally Lipschitz from $X$ into itself. The other is that the long-term dynamics of the determining form contains that of the NSE; the traveling wave solutions of the determining form, i.e., those of the form $v(t,s)=v_0(t+s)$, correspond exactly to initial data $v_0$ that are projections of solutions of the global attractor of the NSE onto the determining modes. The determining form is also shown to be dissipative; an estimate for the radius of an absorbing ball is derived in terms of the number of determining modes and the Grashof number (a dimensionless physical parameter). Finally, a unified approach is outlined for an ODE satisfied by a variety of other determining parameters such as nodal values, finite volumes, and finite elements.

math.AP↗

arXiv2010

Canonical Models For Bi-Isometries

A canonical model, analogous to the one for contraction operators, is introduced for bi-isometries, two commuting isometries on a Hilbert space. This model involves a contractive analytic operator-valued function on the unit disk. Various pureness conditions are considered as well as bi-isometries for which both isometries are shifts. Several families of examples are introduced and classified.

math.FA↗

arXiv2010

Resolutions of Hilbert Modules and Similarity

Let H^2_m be the Drury-Arveson (DA) module which is the reproducing kernel Hilbert space with the kernel function (z, w) \in B^m \times B^m \raro (1 - )^{-1}. We investigate for which multipliers θ: \mathbb{B}^m \raro \cll(\cle, \cle_*) the quotient module \clh_θ is similar to H^2_m \otimes \clf for some Hilbert space \clf, where M_θ is the corresponding multiplication operator in \cll(H^2_m \otimes \cle, H^2_m \otimes \cle_*) for Hilbert spaces \cle and \cle_* and \clh_θ is the quotient module (H^2_m \otimes \cle_*)/ {clos} [M_θ(H^2_m \otimes \cle)]. We show that a necessary condition is the existence of a multiplier $ψ$ in \clm(\cle_*, \cle) such that θψθ= θ. Moreover, we show that the converse is equivalent to a structure theorem for complemented submodules of H^2_m \otimes \cle for a Hilbert space \cle, which is valid for the case of m=1. The latter result generalizes a known theorem on similarity to the unilateral shift, but the above statement is new. Further, we show that a finite resolution of DA-modules of arbitrary multiplicity using partially isometric module maps must be trivial. Finally, we discuss the analogous questions when the underlying operator tuple or algebra is not necessarily commuting. In this case the converse to the similarity result is always valid.

math.FA↗

arXiv2010

Contractions with Polynomial characteristic functions I. Geometric approach

In this note we study the completely non unitary contractions on separable complex Hilbert spaces which have polynomial characteristic functions. These operators are precisely those which admit a matrix representation of the form T = S & * & * 0 & N & * 0& 0& C, where $S$ and C^* are unilateral shifts of arbitrary multiplicities and $N$ is nilpotent. We prove that dimension of ker S^* and dimension of ker C are unitary invariants of $T$ and that N, up to a quasi-similarity is uniquely determined by T. Also, we give a complete classification of the subclass of those contractions for which their characteristic functions are monomials.

math.FA↗