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A fairly general class of nonlinear evolution equations with a self-adjoint or non self-adjoint linear operator is considered, and a family of approximate inertial manifolds (AIMs) is constructed. Two cases are considered: when the spectral gap condition (SGC) is not satisfied and an exact inertial manifold is not known to exist the construction is such that the AIMs have exponential order, while when the SGC is satisfied (and hence there exists an exact inertial manifold) the construction is such that the AIMs converge exponentially to the exact inertial manifold.
On the convergence of statistical solutions of the 3D Navier-Stokes-$\alpha$ model as $\alpha$ vanishes
In this paper statistical solutions of the 3D Navier-Stokes-$\alpha$ model with periodic boundary condition are considered. It is proved that under certain natural conditions statistical solutions of the 3D Navier-Stokes-$\alpha$ model converge to statistical solutions of the exact 3D Navier-Stokes equations as $\alpha$ goes to zero. The statistical solutions considered here arise as families of time-projections of measures on suitable trajectory spaces.
A rigorous study of universal laws of 2-D turbulence is presented for time independent forcing at all length scales. Conditions for energy and enstrophy cascades are derived, both for a general force, and for one with a large gap in its spectrum. It is shown in the gap case that either a direct cascade of enstrophy or an inverse cascade of energy must hold, provided the gap modes of the velocity has a nonzero ensemble average. Partial rigorous support for 2-D analogs of Kolmogorov's 3-D dissipation law, as well as the power law for the distribution of energy are given.
Error in approximation of Lyapunov exponents on inertial manifolds: The Kuramoto-Sivashinsky equation
We provide an analysis of the error in approximating Lyapunov exponents of dissipative PDEs on inertial manifolds using QR techniques. The reduction in the number of modes needed for an inertial form facilitates the error analysis. Numerical computations on the Kuramoto-Sivashinsky equation illustrate the results.
Topological properties of the weak global attractor of the three-dimensional Navier-Stokes equations
The three-dimensional incompressible Navier-Stokes equations are considered along with its weak global attractor, which is the smallest weakly compact set which attracts all bounded sets in the weak topology of the phase space of the system (the space of square-integrable vector fields with divergence zero and appropriate periodic or no-slip boundary conditions). A number of topological properties are obtained for certain regular parts of the weak global attractor. Essentially two regular parts are considered, namely one made of points such that all weak solutions passing through it at a given initial time are strong solutions on a neighborhood of that initial time, and one made of points such that at least one weak solution passing through it at a given initial time is a strong solution on a neighborhood of that initial time. Similar topological results are obtained for the family of all trajectories in the weak global attractor.
The Bénard problem, a system with the Navier-Stokes equations for the velocity field coupled with a convection-diffusion equation for the temperature is considered. Non-homogeneous boundary conditions, external force and heat source in dual function spaces, and an arbitrary spatial domain (possibly nonsmooth and unbounded) as long as the Poincaré inequality holds on it (channel-like domain) are allowed. Moreover our approach, unlike in previous works, avoids the use of the maximum principle which would be problematic in this context. The mathematical formulation of the problem, the existence of global solution and the existence and finite dimensionality of the global attractor are proved.
An extension to the nonautonomous case of the energy equation method for proving the existence of attractors for noncompact systems is presented. A suitable generalization of the asymptotic compactness property to the nonautonomous case, termed uniform asymptotic compactness, is given, and conditions on the energy equation associated with an abstract class of equations that assure the uniform asymptotic compactness are obtained. This general formulation is then applied to a nonautonomous Navier-Stokes system on an infinite channel past an obstacle, with time-dependent forcing and boundary conditions, and to a nonautonomous, weakly damped, forced Korteweg-de Vries equation on the real line.
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