Uniform in discretization error estimates for convection dominated convection-diffusion problems
Uniformly stable mixed hp-finite elements on multilevel adaptive grids with hanging nodes
We consider a family of quadrilateral or hexahedral mixed hp-finite elements for an incompressible flow problem with Qr-elements for the velocity and discontinuous -elements for the pressure where the order r can vary from element to element between 2 and an arbitrary bound. For multilevel adaptive grids with hanging nodes and a sufficiently small mesh size, we prove the inf-sup condition uniformly with respect to the mesh size and the polynomial degree.
Uniqueness of weak solutions of the Navier-Stokes equations
Consider the Navier-Stokes equation with the initial data . Let and be two weak solutions with the same initial value . If satisfies the usual energy inequality and if where is the multiplier space, then we have .
Uniqueness of weak solutions to a Keller-Segel-Navier-Stokes model with a logistic source
We prove a uniqueness result of weak solutions to the Cauchy problem of a Keller-Segel-Navier-Stokes system with a logistic term.
Uniqueness results for some PDEs
Existence of solutions to many kinds of PDEs can be proved by using a fixed point argument or an iterative argument in some Banach space. This usually yields uniqueness in the same Banach space where the fixed point is performed. We give here two methods to prove uniqueness in a more natural class. The first one is based on proving some estimates in a less regular space. The second one is based on a duality argument. In this paper, we present some results obtained in collaboration with Pierre-Louis...
Unsteady flow of a radiating gas between concentric cylinders
Unsteady incompressible boundary layer equation in full two and two-once localized parametric approximation.
Unsteady motion around a uniformly deforming rotating cylinder
Validation of numerical simulations of a simple immersed boundary solver for fluid flow in branching channels
This work deals with the flow of incompressible viscous fluids in a two-dimensional branching channel. Using the immersed boundary method, a new finite difference solver was developed to interpret the channel geometry. The numerical results obtained by this new solver are compared with the numerical simulations of the older finite volume method code and with the results obtained with OpenFOAM. The aim of this work is to verify whether the immersed boundary method is suitable for fluid flow in channels...
Vanishing viscosity limit for an expanding domain in space
Verallgemeinerung des Problems von dem ruhenden Ellipsoid in einer bewegten, unendlichen Flüssigkeit
Verallgemeinerung des Problems von den Bewegungen, welche in einer ruhenden, unelastischen Flüssigkeit die Bewegung eines Ellipsoids hervorbringt
Very weak solutions of the stationary Stokes equations in unbounded domains of half space type
We consider the theory of very weak solutions of the stationary Stokes system with nonhomogeneous boundary data and divergence in domains of half space type, such as , bent half spaces whose boundary can be written as the graph of a Lipschitz function, perturbed half spaces as local but possibly large perturbations of , and in aperture domains. The proofs are based on duality arguments and corresponding results for strong solutions in these domains, which have to be constructed in homogeneous...
Viscous damping of gravity waves over a permeable bed.
Viscous-inviscid coupled problem with interfacial data.
Vorticity dynamics and numerical resolution of Navier-Stokes equations
We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data...
Vorticity dynamics and numerical Resolution of Navier-Stokes Equations
We present a new methodology for the numerical resolution of the hydrodynamics of incompressible viscid newtonian fluids. It is based on the Navier-Stokes equations and we refer to it as the vorticity projection method. The method is robust enough to handle complex and convoluted configurations typical to the motion of biological structures in viscous fluids. Although the method is applicable to three dimensions, we address here in detail only the two dimensional case. We provide numerical data...
Wall laws for viscous fluids near rough surfaces
In this paper, we review recent results on wall laws for viscous fluids near rough surfaces, of small amplitude and wavelength ε. When the surface is “genuinely rough”, the wall law at first order is the Dirichlet wall law: the fluid satisfies a “no-slip” boundary condition on the homogenized surface. We compare the various mathematical characterizations of genuine roughness, and the corresponding homogenization results. At the next order, under...
Weak solutions for a fluid-elastic structure interaction model.
The purpose of this paper is to study a model coupling an incompressible viscous fiuid with an elastic structure in a bounded container. We prove the existence of weak solutions à la Leray as long as no collisions occur.
Weak solutions of a stochastic model for two-dimensional second grade fluids.