On the Generalization of the Hébraud-Lequeux Model to Multidimensional Flows

In this article we build a model for multidimensional flows based on the idea of Hébraud and Lequeux for soft glassy materials. Care is taken to build a frame indifferent multi-dimensional model. The main goal of this article is to prove that the methodology we have developed to study the well-posedness and the glass transition for the original Hébraud-Lequeux model can be successfully generalized. Thus this work may be used as a starting point for more sophisticated studies in the modeling of general flows of glassy materials

Well-posedness of boundary layer equations for time-dependent flow of non-Newtonian fluids

We consider the flow of an upper convected Maxwell fluid in the limit of high Weissenberg and Reynolds number. In this limit, the no-slip condition cannot be imposed on the solutions. We derive equations for the resulting boundary layer and prove the well-posedness of these equations. A transformation to Lagrangian coordinates is crucial in the argument

Transition from rotating waves to modulated rotating waves on the sphere

We study non-resonant and resonant Hopf bifurcation of a rotating wave in SO(3)-equivariant reaction-diffusion systems on a sphere. We obtained reduced differential equations on so(3), the characterization of modulated rotating waves obtained by Hopf bifurcation of a rotating wave, as well as results regarding the resonant case. Our main tools are the equivariant center manifold reduction and the theory of Lie groups and Lie algebras, especially for the group SO(3) of all rigid rotations on a sphere

Symmetric factorization of the conformation tensor in viscoelastic fluid models

The positive definite symmetric polymer conformation tensor possesses a unique symmetric square root that satisfies a closed evolution equation in the Oldroyd-B and FENE-P models of viscoelastic fluid flow. When expressed in terms of the velocity field and the symmetric square root of the conformation tensor, these models' equations of motion formally constitute an evolution in a Hilbert space with a total energy functional that defines a norm. Moreover, this formulation is easily implemented in direct numerical simulations resulting in significant practical advantages in terms of both accuracy and stability.Comment: 7 pages, 5 figure

Bifurcation of singular solutions in reversible systems and applications to reaction - Diffusion equations

Dynamical systems that are reversible in the sense of Moser are investigated and bifurcation of trajectories connecting saddle points from stationary solutions is studied. As an application, reaction-diffusion models in one space dimension are considered. These equations are studied in the neighborhood of a point, where the set of spatially homogeneous solutions displays a Hopf bifurcation. It is shown that from such a point branches of solutions bifurcate, which can be described as waves travelling to or from a center. These waves may be exponentially damped at infinity or not. They can be regarded as one-dimensional analogues of “target patterns” or “spiral waves.

On a class of quasilinear partial integrodifferential equations with singular kernels

AbstractWe prove local and global existence theorems for a model equation in nonlinear viscoelasticity. In contrast to previous studies, we allow the memory function to have a singularity. We approximate the equation by equations with regular kernels and use energy estimates to prove convergence of the approximate solutions