Stokes-type System Research Articles

Global Carleman inequalities for Stokes and penalized Stokes equations

In this note we use the result of [22] to prove a global Carleman inequality related to the null controllability of penalized Stokes kind systems. The constants of the obtained Carleman inequality are uniform in terms of the penalization parameter $\varepsilon$. It then provides a null control with a uniformly (in $\varepsilon$) bounded $L^2$ norm. With a limiting argument we also deduce a new Carleman inequality for Stokes type system. Thus, we apply theses results to obtain the null controllability of Oseen and Navier-Stokes system in the penalized and in the non penalized cases.

The Nonlinear Stokes Problem with General Potentials Having Superquadratic Growth

We discuss partial regularity results concerning local minimizers $${u : \mathbb{R}^3 \supset \Omega \rightarrow \mathbb{R}^3}$$ of variational integrals of the form $$\int\limits_{\Omega}\left\{h(|\varepsilon(w)|) - f \cdot w\right\}\,dx$$ defined on appropriate classes of solenoidal fields, where h is a N-function of rather general type. As a byproduct we obtain a theorem on partial C 1-regularity for weak solutions of certain non-uniformly elliptic Stokes-type systems modelling generalized Newtonian fluids.

A Note on Non-Uniformly Elliptic Stokes-Type Systems in Two Variables

We consider the regularity of weak solutions of a Stokes-type system of partial differential equations in 2D, which describes the stationary and also slow flow of an incompressible fluid. Here the nonlinear differential operator related to the stress tensor is generated by a potential H(ε) = h(|ε|) acting on symmetric (2 × 2)-matrices, where h is a N-function of rather general type leading to a non-uniformly elliptic problem.

A Small Eddy Correction Method for a 3D Navier–Stokes-Type System of Equations Related to the Primitive Equations of the Ocean

Considering the interaction between the baroclinic and barotropic flows and using the idea of the Newton iteration, a small eddy correction method is proposed for approximating and numerically solving the primitive equations of the ocean. We assume that the barotropic approximation to the solution is known. Formally applying the Newton iterative procedure to the baroclinic flow equation, we then generate approximate systems. It is shown that the first step leads to the well-known quasi-geostrophic equations. The convergence analysis is presented and the results show that the small eddy correction method can greatly improve the accuracy of the quasi-geostrophic approximate solution. More precisely, we prove that the approximate system derived from the procedure converges to the original primitive equations, and we estimate the rate of convergence as a function of the aspect ratio of the domain. Some numerical simulations are presented to illustrate the method.

A discrete de Rham complex with enhanced smoothness

Discrete de Rham complexes are fundamental tools in the con- struction of stable elements for some finite element methods. The purpose of this paper is to discuss a new discrete de Rham complex in three space dimensions, where the finite element spaces have extra smoothness com- pared to the standard requirements. The motivation for this construction is to produce discretizations which have uniform stability properties for cer- tain families of singular perturbation problem. In particular, we show how the spaces constructed here lead to discretizations of Stokes type systems which have uniform convergence properties as the Stokes flow approaches a Darcy flow.

Local strong solution for the incompressible Korteweg model

We consider a Navier–Stokes type system with a Korteweg stress tensor, coupled with a concentration equation without diffusion. We give a result concerning the local in time existence (and uniqueness) of strong solution for any data (and the global in time existence defined in a interval ( 0 , T ) for T < + ∞ fixed if data are small enough), in the case of a bounded domain Ω ⊂ R 3 . To cite this article: M. Sy et al., C. R. Acad. Sci. Paris, Ser. I 342 (2006).

New formulations of a Stokes type problem related to the primitive equations of the atmosphere and applications

The aim of this article is to propose new algorithms for a Stokes type system related to the primitive equations of atmosphere, which are the fundamental equations for the motion of the atmosphere [6]. We derive an equivalent formulation of these equations in which the natural constraint appearing in these equations is automatically satisfied without being explicitly imposed. Numerical algorithms based on the new formulation appeared to be very competitive compared to the Uzawa-Conjugate Gradient method.

Steady compressible Navier-Stokes equations with large potential forces via a method of decomposition

We investigate the steady compressible Navier–Stokes equations near the equilibrium state v = 0, ρ = ρ0 (v the velocity, ρ the density) corresponding to a large potential force. We introduce a method of decomposition for such equations: the velocity field v is split into a non-homogeneous incompressible part u (div (ρ0u) = (0) and a compressible (irrotational) part ∇ϕ. In such a way, the original complicated mixed elliptic–hyperbolic system is split into several ‘standard’ equations: a Stokes-type system for u, a Poisson-type equation for ϕ and a transport equation for the perturbation of the density σ = ρ − ρ0. For ρ0 = const. (zero potential forces), the method coincides with the decomposition of Novotny and Padula [21]. To underline the advantages of the present approach, we give, as an example, a ‘simple’ proof of the existence of isothermal flows in bounded domains with no-slip boundary conditions. The approach is applicable, with some modifications, to more complicated geometries and to more complicated boundary conditions as we will show in forthcoming papers. © 1998 B.G. Teubner Stuttgart–John Wiley & Sons Ltd.

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.

Regularity results for stokes type systems related to climatology

In this note, we prove the regularity of solutions of Stokes type systems related to the large scale equations of the ocean [1], the primitive equations of the atmosphere and the coupled system atmosphere-ocean [2,3]; the regularity of solutions of strongly elliptic nonhomogeneous problems in cylinder-type domains is proved with the same generality as in the homogeneous case and is used in the proof of our results.

The multigrid method applied to a finite-element scheme for a two-dimensional Stokes-type system

A two-dimensional system of the Stokes type, in which the coefficient of the highest order derivatives is a small parameter ε>0, is considered, and a version of the multigrid method is developed to handle a finite-element scheme for the system, obtained by using biquadratic elements for the velocities, linear elements for the pressure and an approximatively-solenoidal basis. Approximatively-solenoidal interpolations are defined and used to design the coarse-grid bases. Numerical experiments demonstrate a high rate of convergence even for very small ε values.