Wall driven steady flow and heat transfer in a porous tube
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...
Water-wave problem for a vertical shell
The uniqueness theorem is proved for the linearized problem describing radiation and scattering of time-harmonic water waves by a vertical shell having an arbitrary horizontal cross-section. The uniqueness holds for all frequencies, and various locations of the shell are possible: surface-piercing, totally immersed and bottom-standing. A version of integral equation technique is outlined for finding a solution.
Wax deposition in crude oils: a new approach
The complex phenomenon of solid wax deposition in wax saturated crude oils subject to thermal gradients has been treated in a number of papers under very specific assumptions (e.g. thermodynamical equilibrium between dissolved wax and the wax suspended in the oil as a crystallized phase). Here we want to consider a more general framework in which thermodynamical equilibrium may not exist, the whole system may form a gel-like structure in which the segregated solid wax has no diffusivity, the thermal...
Weak and classical solutions of equations of motion for third grade fluids
Weak and classical solutions of equations of motion for third grade fluids
This paper shows that the decomposition method with special basis, introduced by Cioranescu and Ouazar, allows one to prove global existence in time of the weak solution for the third grade fluids, in three dimensions, with small data. Contrary to the special case where , studied by Amrouche and Cioranescu, the H1 norm of the velocity is not bounded for all data. This fact, which led others to think, in contradiction to this paper, that the method of decomposition could not apply to...
Weak solutions for a hyperbolic system with unilateral constraint and mass loss
Weak solutions for steady compressible Navier-Stokes-Fourier system in two space dimensions
We consider steady compressible Navier-Stokes-Fourier system in a bounded two-dimensional domain. We show the existence of a weak solution for arbitrarily large data for the pressure law if and if , , depending on the model for the heat flux.
Weak solutions of a stochastic model for two-dimensional second grade fluids.
Weak solutions of incompressible Euler equations with decreasing energy
Weak solutions to a nonlinear variational wave equation with general data
Weak solutions to the Navier-Stokes equations in a Y-shaped domain
We prove the existence of weak solutions to the Navier-Stokes equations describing the motion of a fluid in a Y-shaped domain.
Weak-strong uniqueness for a class of degenerate parabolic cross-diffusion systems
Bounded weak solutions to a particular class of degenerate parabolic cross-diffusion systems are shown to coincide with the unique strong solution determined by the same initial condition on the maximal existence interval of the latter. The proof relies on an estimate established for a relative entropy associated to the system.
Weighted L² and approaches to fluid flow past a rotating body
Consider the flow of a viscous, incompressible fluid past a rotating obstacle with velocity at infinity parallel to the axis of rotation. After a coordinate transform in order to reduce the problem to a Navier-Stokes system on a fixed exterior domain and a subsequent linearization we are led to a modified Oseen system with two additional terms one of which is not subordinate to the Laplacean. In this paper we describe two different approaches to this problem in the whole space case. One of them...
Well-posed solutions of the third order Benjamin-Ono equation in weighted Sobolev spaces.
Well-posedness for a class of non-Newtonian fluids with general growth conditions
The paper concerns uniqueness of weak solutions to non-Newtonian fluids with nonstandard growth conditions for the Cauchy stress tensor. We recall the results on existence of weak solutions and additionally provide the proof of existence of measure-valued solutions. Motivated by the fluids of strongly inhomogeneous behaviour and having the property of rapid shear thickening we observe that the described situation cannot be captured by power-law-type rheology. We describe the growth conditions with...
Well-posedness issues for the Prandtl boundary layer equations
These notes are an introduction to the recent paper [7], about the well-posedness of the Prandtl equation. The difficulties and main ideas of the paper are described on a simpler linearized model.