Ricardo Rosa

The global attractor for the 2D Navier-Stokes flow on some unbounded domains

We extend previous results on the existence of the global attractor for the 2D Navier-Stokes equations on some unbounded domains in the sense that the forcing term need not lie in any weighted space nor is the boundary of the domain required to be smooth. The existence of the global attractor is obtained on arbitrary open sets such that the Poincare inequality holds and for forces in the natural dual space V ′. The proof is based on the energy equation and the concept of asymptotic compactness. An estimate of the Hausdorff and fractal dimensions of the attractor is also given.

ATTRACTORS FOR NON-COMPACT SEMIGROUPS VIA ENERGY EQUATIONS

The energy-equation approach used to prove the existence of the global attractor by establishing the so-called asymptotic compactness property of the semigroup is considered, and a general formulation that can handle a number of weakly damped hyperbolic equations and parabolic equations on either bounded or unbounded spatial domains is presented. As examples, three specific and physically relevant problems are considered, namely the flows of a second-grade fluid, the flows of a Newtonian fluid in an infinite channel past an obstacle, and a weakly damped, forced Korteweg-de Vries equation on the whole line.

Statistical Estimates for the Navier–Stokes Equations and the Kraichnan Theory of 2-D Fully Developed Turbulence

A mathematical formulation of the Kraichnan theory for 2-D fully developed turbulence is given in terms of ensemble averages of solutions to the Navier–Stokes equations. A simple condition is given for the enstrophy cascade to hold for wavenumbers just beyond the highest wavenumber of the force up to a fixed fraction of the dissipation wavenumber, up to a logarithmic correction. This is followed by partial rigorous support for Kraichnans eddy breakup mechanism. A rigorous estimate for the total energy is found to be consistent with Kraichnans theory. Finally, it is shown that under our conditions for fully developed turbulence the fractal dimension of the attractor obeys a sharper upper bound than in the general case.

Estimates for the energy cascade in three-dimensional turbulent flows

Abstract The phenomenological theory of turbulence in three dimensions postulates that at large Reynolds numbers there exists an interval of wavenumbers within which the direct effects of the molecular viscosity are negligible. Within that interval, the so-called inertial range, an eddy characterized by a wavenumber given in that range decays principally by breaking down into smaller ones, with each of those smaller ones eventually breaking down into still smaller eddies, and so on, a process conventionally called a cascade in the wavenumber space. Such a cascade proceeds until the size of the descendant eddies is sufficiently small to enter the so-called dissipation range and disappear by the direct action of molecular viscosity. In this note, which is a continuation of [5], we prove the existence of the inertial range provided the Taylor wavenumber is sufficiently large. More precisely, we prove that the energy flux to higher modes is nearly equal to the energy dissipation rate throughout a certain range of wavenumbers much smaller than the Taylor wavenumber. These rigorous results show that the Taylor wavenumber is such that below it the conditions prevailing in the inertial range for the energy cascade are strictly satisfied. Moreover, we obtain several estimates concerning characteristic numbers and nondimensional numbers related to turbulent flows.

Accurate Computations on Inertial Manifolds

An algorithm is implemented for the computation of inertial manifolds to arbitrary accuracy. It is applied to the Kuramoto--Sivashinsky equation to recover the high wave number components of elements on the global attractor. The computational results demonstrate that this algorithm is significantly more accurate than previous methods, when compared after just a few iterations. An essential feature of the algorithm is that it does not require that the entire manifold be constructed. Hence the algorithm is particularly suited for computing trajectories on the manifold, and its computational complexity is independent of the dimension.

Inertial manifolds and normal hyperbolicity

Our aim in this article is to derive an existence theorem of inertial manifolds for fairly general equations with a self-adjoint or nonself-adjoint linear operator in a Banach space setting. A sharp form of the spectral gap condition is given. Many other properties are proven including an interesting characterization of the inertial manifold and the normal hyperbolicity of the inertial manifold.

Cascade of energy in turbulent flows

Abstract A starting point for the conventional theory of turbulence [12–14] is the notion that, on average, kinetic energy is transferred from low wave number modes to high wave number modes [19]. Such a transfer of energy occurs in a spectral range beyond that of injection of energy, and it underlies the so-called cascade of energy, a fundamental mechanism used to explain the Kolmogorov spectrum in three-dimensional turbulent flows. The aim of this Note is to prove this transfer of energy to higher modes in a mathematically rigorous manner, by working directly with the Navier–Stokes equations and stationary statistical solutions obtained through time averages. To the best of our knowledge, this result has not been proved previously; however, some discussions and partly intuitive proofs appear in the literature. See, e.g., [1,2,10,11,16,17,21], and [22]. It is noteworthy that a mathematical framework can be devised where this result can be completely proved, despite the well-known limitations of the mathematical theory of the three-dimensional Navier–Stokes equations. A similar result concerning the transfer of energy is valid in space dimension two. Here, however, due to vorticity constraints not present in the three-dimensional case, such energy transfer is accompanied by a similar transfer of enstrophy to higher modes. Moreover, at low wave numbers, in the spectral region below that of injection of energy, an inverse (from high to low modes) transfer of energy (as well as enstrophy) takes place. These results are directly related to the mechanisms of direct enstrophy cascade and inverse energy cascade which occur, respectively, in a certain spectral range above and below that of injection of energy [1,15]. In a forthcoming article [9] we will discuss conditions for the actual existence of the inertial range in dimension three.

Asymptotic regularity conditions for the strong convergence towards weak limit sets and weak attractors of the 3D Navier–Stokes equations

Abstract The asymptotic behavior of solutions of the three-dimensional Navier–Stokes equations is considered on bounded smooth domains with no-slip boundary conditions and on periodic domains. Asymptotic regularity conditions are presented to ensure that the convergence of a Leray–Hopf weak solution to its weak ω -limit set (weak in the sense of the weak topology of the space H of square-integrable divergence-free velocity fields with the appropriate boundary conditions) are achieved also in the strong topology. It is proved that the weak ω -limit set is strongly compact and strongly attracts the corresponding solution if and only if all the solutions in the weak ω -limit set are continuous in the strong topology of H . Corresponding results for the strong convergence towards the weak global attractor of Foias and Temam are also presented. In this case, it is proved that the weak global attractor is strongly compact and strongly attracts the weak solutions, uniformly with respect to uniformly bounded sets of weak solutions, if and only if all the global weak solutions in the weak global attractor are strongly continuous in H .

Some results on the Navier-Stokes equations in connection with the statistical theory of stationary turbulence

Some rigorous results connected with the conventional statistical theory of turbulence in both the two- and three-dimensional cases are discussed. Such results are based on the concept of stationary statistical solution, related to the notion of ensemble average for turbulence in statistical equilibrium, and concern, in particular, the mean kinetic energy and enstrophy fluxes and their corresponding cascades. Some of the results are developed here in the case of nonsmooth boundaries and a less regular forcing term and for arbitrary stationary statistical solutions.

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 1970s, 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.