Well posedness of a linearized fractional derivative fluid model

The one-dimensional fractional derivative Maxwell model (e.g. Palade et al. Rheol. Acta 35, 265, 1996), of importance in modeling the linear viscoelastic response in the glass transition region, has been generalized in Palade et al. Int. J. Non-Linear Mech. 37, 315, 1999, to objective three-dimensional constitutive equations (CEs) for both fluids and solids. Regarding the rest state stability of the fluid CE, in Heibig and Palade J. Math. Phys. 49, 043101, 2008, we gave a proof for the existence of weak solutions to the corresponding boundary value problem. The aim of this work is to achieve the study of the existence and uniqueness of the aforementioned solutions and to present smoothness results

Existence and uniqueness of a density probability solution for the stationary Doi–Edwards equation

We prove the existence, uniqueness and non negativity of solutions for a nonlinear stationary Doi-Edwards equation. The existence is proved by a perturbation argument. We get the uniqueness and the non negativity by showing the convergence in time of the solution of the evolutionary Doi-Edwards equation towards any stationary solution

Some properties of the Doi–Edwards and K-BKZ equations and operators

International audienc

A generalized Cauchy-Lipschitz theorem in low regularity spaces.

We prove well-posedness for some abstract differential equations of the first order. Our result covers the usual case of Lipschitz composition operators. It also contains the case of some integro-differential operators acting on spaces with low regularity indexes. The loss of derivatives induced by such operators has to be lower than one, in order to be dominated by the first order derivative involved in the problem

Existence of solutions for a fractional derivative system of equations.

International audienceWe study a fractional derivative system of equations. A Newton polygonal associated with this system is partially described. Under some additional assumptions, this Newton polygonal is fully described and L² estimates are given, as well as an existence result. We finally discuss our assumptions

Well-posedness of a Debye type system endowed with a full wave equation

International audienceWe prove well-posedness for a transport-diffusion problem coupled with a wave equation for the potential. We assume that the initial data are small. A bilinear form in the spirit of Kato's proof for the Navier-Stokes equations is used, coupled with suitable estimates in Chemin-Lerner spaces. In the one dimensional case, we get well-posedness for arbitrarily large initial data by using Gagliardo-Nirenberg inequalities

Local existence result in time for a drift-diffusion system with Robin boundary conditions

International audienc

On the steady state solution of a Euler–Bernoulli beam under a moving load

This paper focuses on a steady state solution of a Euler-Bernoulli beam under a moving load, on a foundation composed of a continuous distribution of linear elastic springs associated in parallel with a continuous distribution of Coulomb frictional dampers. The motion of the beam is governed by a partial differential inclusion. Under appropriate regularity assumptions on the initial data, the problem possesses a weak solution which is obtained as the limit of a sequence of solutions of regularized problems

A dynamic Euler–Bernoulli beam equation frictionally damped on an elastic foundation

International audienc