Mohammed Ziane

Some Mathematical Problems in Geophysical Fluid Dynamics

Abstract This chapter reviews the recently developed mathematical setting of the primitive equations (PEs) of the atmosphere, the ocean, and the coupled atmosphere and ocean. The mathematical issues that are considered here are the existence, uniqueness, and regularity of solutions for the time-dependent problems in space dimensions 2 and 3, the PEs being supplemented by a variety of natural boundary conditions. The emphasis is on the case of the ocean that encompasses most of the mathematical difficulties. This chapter is devoted to the PEs in the presence of viscosity, while the PEs without viscosity are considered in the chapter by Rousseau, Temam, and Tribbia in the same volume. Whereas the theory of PEs without viscosity is just starting, the theory of PEs with viscosity has developed since the early 1990s and has now reached a satisfactory level of completion. The theory of the PEs was initially developed by analogy with that of the incompressible Navier Stokes equations, but the most recent developments reported in this chapter have shown that unlike the incompressible Navier-Stokes equations and the celebrated Millenium Clay problem, the PEs with viscosity are well-posed in space dimensions 2 and 3, when supplemented with fairly general boundary conditions. This chapter is essentially self-contained, and all the mathematical issues related to these problems are developed. A guide and summary of results for the physics-oriented reader is provided at the end of the Introduction ( Section 1.4 ).

One component regularity for the Navier–Stokes equations

We establish sufficient conditions for the regularity of solutions of the Navier–Stokes system based on conditions on one component of the velocity. The first result states that if , where and 54/23 ≤ r ≤ 18/5, then the solution is regular. The second result is that if , where and 24/5 ≤ r ≤ ∞, then the solution is regular. These statements improve earlier results on one component regularity.

Navier-Stokes equations with regularity in one direction

We consider sufficient conditions for the regularity of Leray-Hopf solutions of the Navier-Stokes equations. We prove that if the third derivative of the velocity ∂u∕∂x3 belongs to the space Lts0Lxr0, where 2∕s0+3∕r0⩽2 and 9∕4⩽r0⩽3, then the solution is regular. This extends a result of Beirao da Veiga [Chin. Ann. Math., Ser. B 16, 407–412 (1995); C. R. Acad. Sci, Ser. I: Math. 321, 405–408 (1995)] by making a requirement only on one direction of the velocity instead of on the full gradient. The derivative ∂u∕∂x3 can be substituted with any directional derivative of u.

A general framework for robust control in fluid mechanics

Abstract The application of optimal control theory to complex problems in fluid mechanics has proven to be quite effective when complete state information from high-resolution numerical simulations is available [P. Moin, T.R. Bewley, Appl. Mech. Rev., Part 2 47 (6) (1994) S3–S13; T.R. Bewley, P. Moin, R. Temam, J. Fluid Mech. (1999), submitted for publication]. In this approach, an iterative optimization algorithm based on the repeated computation of an adjoint field is used to optimize the controls for finite-horizon nonlinear flow problems [F. Abergel, R. Temam, Theoret. Comput. Fluid Dyn. 1 (1990) 303–325]. In order to extend this infinite-dimensional optimization approach to control externally disturbed flows in which the controls must be determined based on limited noisy flow measurements alone, it is necessary that the controls computed be insensitive to both state disturbances and measurement noise. For this reason, robust control theory, a generalization of optimal control theory, has been examined as a technique by which effective control algorithms which are insensitive to a broad class of external disturbances may be developed for a wide variety of infinite-dimensional linear and nonlinear problems in fluid mechanics. An aim of the present paper is to put such algorithms into a rigorous mathematical framework, for it cannot be assumed at the outset that a solution to the infinite-dimensional robust control problem even exists. In this paper, conditions on the initial data, the parameters in the cost functional, and the regularity of the problem are established such that existence and uniqueness of the solution to the robust control problem can be proven. Both linear and nonlinear problems are treated, and the 2D and 3D nonlinear cases are treated separately in order to get the best possible estimates. Several generalizations are discussed and an appropriate numerical method is proposed.

On the Regularity of the Primitive Equations of the Ocean

We prove the existence of global strong solutions of the primitive equations of the ocean in the case of the Dirichlet boundary conditions on the side and the bottom boundaries including the varying bottom topography. Previously, the existence of global strong solutions was known in the case of the Neumann boundary conditions in a cylindrical domain (Cao and Titi 2007 Ann. Math. 166 245–67).

On a certain renormalization group method

In this paper, a mathematical study of the renormalization group method, recently introduced by Chen, Goldenfeld, and Oono [Phys. Rev. E 54, 376 (1997); Phys. Rev. Lett. 73, 1311 (1994), NATO Adv. Study Inst. Ser., Ser. B 284, 375 (1991)] is given for the case of autonomous nonlinear systems of differential equations. We also observe that the approximation results obtained by this method by Chen, Goldenfeld, and Ono [NATO Adv. Study Inst. Ser., Ser. B 284, 375 (1991)] are valid over long time intervals. Moreover, a connection between this method and the classical Poincare–Dulac normal forms and the averaging method is briefly discussed.

REGULARITY RESULTS FOR LINEAR ELLIPTIC PROBLEMS RELATED TO THE PRIMITIVE EQUATIONS

The authors study the regularity of solutions of the GFD-Stokes problem and of some second order linear elliptic partial differential equations related to the Primitive Equations of the ocean. The present work generalizes the regularity results in [18] by taking into consideration the non-homogeneous boundary conditions and the dependence of solutions on the thickness e of the domain occupied by the ocean and its varying bottom topography. These regularity results are important tools in the study of the PEs (see e.g. [6]), and they seem also to possess their own interest.

Regularity results for stokes type systems

The aim of thin work is to study the regularity of solutions of Stokes type systems related to the large scale equations of the ocean and the primitive equations of the coupled system atmosphere-ocean, which have appeared in the work of Lions et al12,13,144. We prove various regularity results for strongly elliptic boundary value problems in cylinder-type domains with nonhomogeneous boundary conditions, as well ss the H2-regularity for the Stokes-type system built into these equations.

Renormalization Group Method. Applications to Partial Differential Equations

Our aim in this article is to present a simplified form of the renormalization group (RG) method introduced by Chen, Goldenfeld, and Oono and to derive a rigorous study of the validity in time of the asymptotic solutions furnished by the RG method. We apply the renormalization group method to a slightly compressible fluid equation and to the Swift–Hohenberg equation.

Asymptotic analysis of the Navier-Stokes equations in the domains

We are interested in this article with the Navier–Stokes equations of viscous incompressible fluids in three dimensional thin domains. Let Ωe be the thin domain Ωe = ω × (0, e), where ω is a suitable domain in R and 0 < e < 1. Our aim is to derive an asymptotic expansion of the strong solution u of the Navier–Stokes equations in the thin domain Ωe when e is small, which is valid uniformly in time. This study should give a better understanding of the global existence results in thin domains obtained previously; see [15]–[17] and [23], [22]. We consider in this work two types of boundary conditions: the Dirichlet-periodic boundary condition and the purely periodic condition. For the first type of boundary condition we derive an asymptotic expansion of the solution u in terms of the solution of the associated Stokes problem. More precisely, we prove that the solution can be written, for e small, as