We discuss regularity results concerning local minimizers $u:{ℝ}^n\supset \Omega \to {ℝ}^n$ of variational integrals like $$\begin{array}{c}\hfill {\int }_{\Omega }\{F(·,\epsilon \left(w\right))-f·w\}dx\end{array}$$ defined on energy classes of solenoidal fields. For the potential $F$ we assume a $(p,q)$-elliptic growth condition. In the situation without $x$-dependence it is known that minimizers are of class $C^{1,\alpha }$ on an open subset ${\Omega }_0$ of $\Omega$ with full measure if $q<p\frac{n+2}{n}$ (for $n=2$ we have ${\Omega }_0=\Omega$). In this article we extend this to the case of nonautonomous integrands. Of course our result extends to weak solutions of the corresponding nonlinear...

We consider the stationary Stokes system with slip boundary conditions in a bounded domain. Assuming that data functions belong to weighted Sobolev spaces with weights equal to some power of the distance to some distinguished axis, we prove the existence of solutions to the problem in appropriate weighted Sobolev spaces.

This article addresses some theoretical questions related to the choice of boundary conditions, which are essential for modelling and numerical computing in mathematical fluids mechanics. Unlike the standard choice of the well known non slip boundary conditions, we emphasize three selected sets of slip conditions, and particularly stress on the interaction between the appropriate functional setting and the status of these conditions.

The instationary Stokes and Navier−Stokes equations are considered in a simultaneously space-time variational saddle point formulation, so involving both velocities u and pressure p. For the instationary Stokes problem, it is shown that the corresponding operator is a boundedly invertible linear mapping between H1 and H'2, both Hilbert spaces H1 and H2 being Cartesian products of (intersections of) Bochner spaces, or duals of those. Based on these results, the operator that corresponds to the Navier−Stokes...

We consider the Stokes problem provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. Next we propose a discretization by spectral element methods which relies on this formulation. A detailed numerical analysis leads to optimal error estimates for the three unknowns and numerical...

We consider the Stokes problem provided with non standard boundary conditions which involve the normal component of the velocity and the tangential components of the vorticity. We write a variational formulation of this problem with three independent unknowns: the vorticity, the velocity and the pressure. Next we propose a discretization by spectral element methods which relies on this formulation. A detailed numerical analysis leads to optimal error estimates for the three unknowns and numerical...

We study the generalized Stokes resolvent equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer $\Omega ₀={ℝ}^{n-1}\times (-1,1)$. Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. We discuss the results on unique solvability of the generalized Stokes resolvent equations as well as the existence of a bounded $H_{\infty }$-calculus for the associated Stokes operator and some...

Operator version of the Stokeslet method in the theory of creeping flow is suggested. The approach is analogous to the zero-range potential one in quantum mechanics and is based on the theory of self-adjoint operator extensions in the space L2 and in the Pontryagin?s space with an indefinite metric. The problem of Stokes flow in two channels connected through a small opening is considered in the framework of this approach. The case of a periodic system of small openings is studied too. The picture...

The development of velocity distribution in plane laminar flow is examined, neglecting inertial terms in respect to viscous ones. A solution is given, which satisfies all boundary conditions.

This paper presents a superconvergence result based on projection method for stabilized finite element approximation of the Stokes eigenvalue problem. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares method. The paper complements the work of Li et al. (2012), which establishes the superconvergence result of the Stokes equations by the stabilized finite element method. Moreover, numerical tests confirm the theoretical analysis.

On sait que toutes les solutions de l’équation de Navier-Stokes dans ${\mathbf{R}}^2$ dont le tourbillon est intégrable convergent lorsque $t\to \infty$ vers un écoulement autosimilaire appelé tourbillon d’Oseen. Dans cet article, nous donnons une estimation du temps nécessaire à la solution pour atteindre un voisinage du tourbillon d’Oseen à partir d’une donnée initiale arbitraire, mais bien localisée en espace. Nous obtenons ainsi une borne supérieure sur le temps de vie de la turbulence bidimensionnelle libre, en fonction...

We consider the second-order projection schemes for the time-dependent natural convection problem. By the projection method, the natural convection problem is decoupled into two linear subproblems, and each subproblem is solved more easily than the original one. The error analysis is accomplished by interpreting the second-order time discretization of a perturbed system which approximates the time-dependent natural convection problem, and the rigorous error analysis of the projection schemes is...