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Optimal control of linearized compressible Navier–Stokes equations

Shirshendu Chowdhury, Mythily Ramaswamy (2013)

ESAIM: Control, Optimisation and Calculus of Variations

We study an optimal boundary control problem for the two dimensional unsteady linearized compressible Navier–Stokes equations in a rectangle. The control acts through the Dirichlet boundary condition. We first establish the existence and uniqueness of the solution for the two-dimensional unsteady linearized compressible Navier–Stokes equations in a rectangle with inhomogeneous Dirichlet boundary data, not necessarily smooth. Then, we prove the existence and uniqueness of the optimal solution over...

Optimal convergence results for the Brezzi-Pitkäranta approximation of the Stokes problem: Exterior domains

Serguei A. Nazarov, Maria Specovius-Neugebauer (2008)

Banach Center Publications

This paper deals with a strongly elliptic perturbation for the Stokes equation in exterior three-dimensional domains Ω with smooth boundary. The continuity equation is substituted by the equation -ε²Δp + div u = 0, and a Neumann boundary condition for the pressure is added. Using parameter dependent Sobolev norms, for bounded domains and for sufficiently smooth data we prove H 5 / 2 - δ convergence for the velocity part and H 3 / 2 - δ convergence for the pressure to the solution of the Stokes problem, with δ arbitrarily...

Optimal error Estimates for the Stokes and Navier–Stokes equations with slip–boundary condition

Eberhard Bänsch, Klaus Deckelnick (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

We consider a finite element discretization by the Taylor–Hood element for the stationary Stokes and Navier–Stokes equations with slip boundary condition. The slip boundary condition is enforced pointwise for nodal values of the velocity in boundary nodes. We prove optimal error estimates in the H1 and L2 norms for the velocity and pressure respectively.

Partial regularity of solution to generalized Navier-Stokes problem

Václav Mácha (2014)

Open Mathematics

In the presented work, we study the regularity of solutions to the generalized Navier-Stokes problem up to a C 2 boundary in dimensions two and three. The point of our generalization is an assumption that a deviatoric part of a stress tensor depends on a shear rate and on a pressure. We focus on estimates of the Hausdorff measure of a singular set which is defined as a complement of a set where a solution is Hölder continuous. We use so-called indirect approach to show partial regularity, for dimension...

Planar flows of incompressible heat-conducting shear-thinning fluids — existence analysis

Miroslav Bulíček, Oldřich Ulrych (2011)

Applications of Mathematics

We study the flow of an incompressible homogeneous fluid whose material coefficients depend on the temperature and the shear-rate. For large class of models we establish the existence of a suitable weak solution for two-dimensional flows of fluid in a bounded domain. The proof relies on the reconstruction of the globally integrable pressure, available due to considered Navier’s slip boundary conditions, and on the so-called L ∞ -truncation method, used to obtain the strong convergence of the velocity...

Poche de tourbillon pour Euler 2D incompressible dans un ouvert à bord

Nicolas Depauw (1998)

Journées équations aux dérivées partielles

Nous considérons l'équation d'Euler pour un fluide incompressible dans un domaine borné régulier du plan. Pour une donnée initiale avec un tourbillon de type poche, i.e valant 1 sur un ouvert lisse à bord höldérien et 0 en dehors, nous prouvons l'existence d'une solution de même type, pour tout temps si la poche initiale est décollée du bord du domaine et seulement localement en temps si la poche initiale est tangente au bord. Nous contrôlons l'influence du bord grâce à la théorie des problèmes...

Poches de tourbillon singulières dans un fluide faiblement visqueux.

Taoufik Hmidi (2006)

Revista Matemática Iberoamericana

In this paper, we study the singular vortex patches in the two-dimensional incompressible Navier-Stokes equations. We show, in particular, that if the initial vortex patch is C1+s outside a singular set Σ, so the velocity is, for all time, lipschitzian outside the image of Σ through the viscous flow. In addition, the correponding lipschitzian norm is independent of the viscosity. This allows us to prove some results related to the inviscid limit for the geometric structures of the vortex patch.

Point fixe d'une application non contractante.

Pierre Gilles Lemarié-Rieusset (2006)

Revista Matemática Iberoamericana

We study a multilinear fixed-point equation in a closed ball of a Banach space where the application is 1-Lipschitzian: existence, uniqueness, approximations, regularity.

Positivity theorem

Jozef Takács (1995)

Acta Mathematica et Informatica Universitatis Ostraviensis

Prediction-correction legendre spectral scheme for incompressible fluid flow

He Li-ping, Mao De-kang, Guo Ben-yu (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

The initial-boundary value problem of two-dimensional incompressible fluid flow in stream function form is considered. A prediction-correction Legendre spectral scheme is proposed, which is easy to be performed. The numerical solution possesses the accuracy of second-order in time and higher order in space. The numerical experiments show the high accuracy of this approach.

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