Past OxPDE Lunchtime Seminar

It is well known that the three dimensional, incompressible Navier-Stokes equations have a unique, global solution provided the initial data is small enough in a scale invariant space (say L3 for instance). We are interested in finding examples for which no smallness condition is imposed, but nevertheless the associate solution is global and unique. The examples we will present are due to collaborations with Jean-Yves Chemin, and with Marius Paicu.

As all analysts know, solving a problem which contains some kind of nonlinearity is generally far from being obvious. This is of course the case when we deal with nonlinear PDEs, but also when we have linear systems with some nonlinear compatibility conditions. One example is constituted by the systems of first order linear partial differential equations. If the coefficients are regular, there is a classical way to solve this systems. However, if the coefficients are only of class L^p, the classical methods cannot be applied. We will show how this problem can be solved by using a method of regularization which allows to preserve the nonlinear compatibility conditions. Then we will present some possible applications to the theory of elasticity. In the end some open problems related to similar aspects will be discussed. Example of such a problem: the rigidity of deformations of class H¹.

We are concerned with the derivation of the Γ-limit to a three dimensional geometrically exact Cosserat model as the relative thickness h > 0 of a at domain tends to zero. The Cosserat bulk model involves already exact rotations as a second independent field and this model is meant to describe defective elastic crystals liable to fracture under shear.
It is shown that the Γ-limit based on a natural scaling assumption con- sists of a membrane like energy contribution and a homogenized transverse shear energy both scaling with h, augmented by an additional curvature stiffness due to the underlying Cosserat bulk formulation, also scaling with h. No specific bending term appears in the dimensional homogenization process. The formulation exhibits an internal length scale Lc which sur- vives the homogenization process. A major technical difficulty, which we encounter in applying the Γ-convergence arguments, is to establish equi- coercivity of the sequence of functionals as the relative thickness h tends to zero. Usually, equi-coercivity follows from a local coerciveness assump- tion. While the three-dimensional problem is well-posed for the Cosserat couple modulus μ_c ≥ 0, equi-coercivity forces us to assume a strictly pos- itive Cosserat couple modulus μ_c > 0. The Γ-limit model determines the midsurface deformation m ∈ H^1,2(ω;R³). For the case of zero Cosserat couple modulus μ_c= 0 we obtain an estimate of the Γ - lim inf and Γ - lim sup, without equi-coercivity which is then strenghtened to a Γ- convergence result for zero Cosserat couple modulus. The classical linear Reissner-Mindlin model is "almost" the linearization of the Γ-limit for μ_c = 0 apart from a stabilizing shear energy term.

We describe how to perform high resolution simulations of viscoelastic continuum mechanical models for martensitic transformations with diffuse interfaces. The computational methods described may also be of use in performing high resolution simulations of time dependent partial differential equations where solutions are sufficiently smooth.