Roger Temam

Infinite-Dimensional Dynamical Systems in Mechanics and Physics

Contents: General results and concepts on invariant sets and attractors.- Elements of functional analysis.- Attractors of the dissipative evolution equation of the first order in time: reaction-diffusion equations.- Fluid mechanics and pattern formation equations.- Attractors of dissipative wave equations.- Lyapunov exponents and dimensions of attractors.- Explicit bounds on the number of degrees of freedom and the dimension of attractors of some physical systems.- Non-well-posed problems, unstable manifolds. lyapunov functions, and lower bounds on dimensions.- The cone and squeezing properties.- Inertial manifolds.- New chapters: Inertial manifolds and slow manifolds the nonselfadjoint case.

Navier–Stokes Equations

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Navier-Stokes equations : theory and numerical analysis

I. The Steady-State Stokes Equations . 1. Some Function Spaces. 2. Existence and Uniqueness for the Stokes Equations. 3. Discretization of the Stokes Equations (I). 4. Discretization of the Stokes Equations (II). 5. Numerical Algorithms. 6. The Penalty Method. II. The Steady-State Navier-Stokes Equations . 1. Existence and Uniqueness Theorems. 2. Discrete Inequalities and Compactness Theorems. 3. Approximation of the Stationary Navier-Stokes Equations. 4. Bifurcation Theory and Non-Uniqueness Results. III. The Evolution Navier-Stokes Equations . 1. The Linear Case. 2. Compactness Theorems. 3. Existence and Uniqueness Theorems. (n < 4). 4. Alternate Proof of Existence by Semi-Discretization. 5. Discretization of the Navier-Stokes Equations: General Stability and Convergence Theorems. 6. Discretization of the Navier-Stokes Equations: Application of the General Results. 7. Approximation of the Navier-Stokes Equations by the Projection Method. 8. Approximation of the Navier-Stokes Equations by the Artificial Compressibility Method. Appendix I: Properties of the Curl Operator and Application to the Steady-State Navier-Stokes Equations. Appendix II. (by F. Thomasset): Implementation of Non-Conforming Linear Finite Elements. Comments.

Inertial manifolds for nonlinear evolutionary equations

Abstract In this paper we introduce the concept of an inertial manifold for nonlinear evolutionary equations, in particular for ordinary and partial differential equations. These manifolds, which are finite dimensional invariant Lipschitz manifolds, seem to be an appropriate tool for the study of questions related to the long-time behavior of solutions of the evolutionary equations. The inertial manifolds contain the global attractor, they attract exponentially all solutions, and they are stable with respect to perturbations. Furthermore, in the infinite dimensional case they allow for the reduction of the dynamics to a finite dimensional ordinary differential equation.

On some control problems in fluid mechanics

The issue of minimizing turbulence in an evolutionary Navier-Stokes flow is addressed from the point of view of optimal control. We derive theoretical results for various physical situations: distributed control, Bénard-type problems with boundary control, and flow in a channel. For each case that we consider, our results include the formulation of the problem as an optimal control problem and proof of the existence of an optimal control (which is not expected to be unique). Finally, we describe a numerical algorithm based on the gradient method for the corresponding cost function. For readers who are not interested in the mathematical details and the mathematical justifications, a nontechnical description of our results is included in Section 5.

DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms

Direct numerical simulations (DNS) and optimal control theory are used in a predictive control setting to determine controls that effectively reduce the turbulent kinetic energy and drag of a turbulent flow in a plane channel at Re τ = 100 and Re τ = 180. Wall transpiration (unsteady blowing/suction) with zero net mass flux is used as the control. The algorithm used for the control optimization is based solely on the control objective and the nonlinear partial differential equation governing the flow, with no ad hoc assumptions other than the finite prediction horizon, T , over which the control is optimized. Flow relaminarization, accompanied by a drag reduction of over 50%, is obtained in some of the control cases with the predictive control approach in direct numerical simulations of subcritical turbulent channel flows. Such performance far exceeds what has been obtained to date in similar flows (using this type of actuation) via adaptive strategies such as neural networks, intuition-based strategies such as opposition control, and the so-called ‘suboptimal’ strategies, which involve optimizations over a vanishingly small prediction horizon T + → 0. To achieve flow relaminarization in the predictive control approach, it is shown that it is necessary to optimize the controls over a sufficiently long prediction horizon T + [gsim ] 25. Implications of this result are discussed. The predictive control algorithm requires full flow field information and is computationally expensive, involving iterative direct numerical simulations. It is, therefore, impossible to implement this algorithm directly in a practical setting. However, these calculations allow us to quantify the best possible system performance given a certain class of flow actuation and to qualify how optimized controls correlate with the near-wall coherent structures believed to dominate the process of turbulence production in wall-bounded flows. Further, various approaches have been proposed to distil practical feedback schemes from the predictive control approach without the suboptimal approximation, which is shown in the present work to restrict severely the effectiveness of the resulting control algorithm. The present work thus represents a further step towards the determination of optimally effective yet implementable control strategies for the mitigation or enhancement of the consequential effects of turbulence.

Nonlinear Galerkin methods

This article presents a new method of integrating evolution differential equations—the non-linear Galerkin method—that is well adapted to the long-term integration of such equations.While the usual Galerkin method can be interpreted as a projection of the considered equation on a linear space, the methods considered here are related to the projection of the equation on a nonlinear manifold. From the practical point of view some terms have been identified as small, and sometimes.(but not always) disregarded.

On the equations of the large-scale ocean

As a preliminary step towards understanding the dynamics of the ocean and the impact of the ocean on the global climate system and weather prediction, the authors study the mathematical formulations and attractors of three systems of equations of the ocean, i.e. the primitive equations (the PEs), the primitive equations with vertical viscosity (the PEV2s), and the Boussinesq equations (the BEs), of the ocean. These equations are fundamental equations of the ocean. The BEs are obtained from the general equations of a compressible fluid under the Boussinesq approximation, i.e. the density differences are neglected in the system except in the buoyancy term and in the equation of state. The PEs are derived from the BEs under the hydrostatic approximation for the vertical momentum equation. The PEV2s are the PEs with the viscosity for the vertical velocity retained. This retention is partially based on the important role played by the viscosity in studying the long time behaviour of the ocean, and the Earths climate.

New formulations of the primitive equations of atmosphere and applications

The primitive equations are the fundamental equations of atmospheric dynamics. With the purpose of understanding the mechanism of long-term weather prediction and climate changes, the authors study as a first step towards this long-range project what is widely considered as the basic equations of atmospheric dynamics in meteorology, namely the primitive equations of the atmosphere. The mathematical formulation and attractors of the primitive equations, with or without vertical viscosity, are studied. First of all, by integrating the diagnostic equations they present a mathematical setting, and obtain the existence and time analyticity of solutions to the equations. They then establish some physically relevant estimates for the Hausdorff and fractal dimensions of the attractors of the problems.

Some global dynamical properties of the Kuramoto-Sivashinsky equations: Nonlinear stability and attractors

Abstract The Kuramoto-Sivashinsky equations model pattern formations on unstable flame fronts and thin hydrodynamic films. They are characterized by the coexistence of coherent spatial structures with temporal chaos. We investigate some global dynamical properties, including nonlinear stability. We demonstrate their low modal behavior, in terms of determining modes; and that the fractal dimension of all attractors is bounded by a universal constant times ≈L 13 8 , where ≈L is a dimensionless pattern cell size (in the one-dimensional case). Such equations are indeed a paradigm of low-dimensional behavior for infinite-dimensional systems.