Past OxPDE Lunchtime Seminar

The question of local existence of a deformation of a simply connected body whose Left Cauchy Green Tensor matches a prescribed, symmetric, positive definite tensor field is considered. A sufficient condition is deduced after formulation as a problem in Riemannian Geometry. The compatibility condition ends up being surprisingly different from that of compatibility of a Right Cauchy Green Tensor field, a fact that becomes evident after the geometric formulation. The question involves determining conditions for the local existence of solutions to an overdetermined system of Pfaffian PDEs with algebraic constraints that is typically not completely integrable.

We present some recent results on singular solutions of the problem of travelling gravity water waves on flows with vorticity. We show that, for a certain class of vorticity functions, a sequence of regular waves converges to an extreme wave with stagnation points at its crests. We also show that, for any vorticity function, the profile of an extreme wave must have either a symmetric corner of 120 degrees or a horizontal tangent at any isolated stagnation point. Moreover, the profile necessarily has a symmetric corner of 120 degrees if the vorticity is nonnegative near the free surface.

This talk will review recent progress in understanding the effective

behavior of free boundaries in heterogeneous media.  Though motivated

by the pinning of martensitic phase boundaries, we shall explain

connections to other problems.  This talk is based on joint work with

Patrick Dondl.

In order to understand the nonlinear stability of many types of time-periodic travelling waves on unbounded domains, one must overcome two main difficulties: the presence of embedded neutral eigenvalues and the time-dependence of the associated linear operator. This problem is studied in the context of time-periodic Lax shocks in systems of viscous conservation laws. Using spatial dynamics and a decomposition into separate Floquet eigenmodes, it is shown that the linear evolution for the time-dependent operator can be represented using a contour integral similar to that of the standard time-independent case. By decomposing the resulting Green's distribution, the leading order behavior associated with the embedded eigenvalues is extracted. Sharp pointwise bounds are then obtained, which are used to prove that the time-periodic Lax shocks are linearly and nonlinearly stable under the necessary conditions of spectral stability and minimal multiplicity of the translational eigenvalues. The latter conditions hold, for example, for small-oscillation time-periodic waves that emerge through a supercritical Hopf bifurcation from a family of time-independent Lax shocks of possibly large amplitude.

The large-time behavior of solutions to Burgers equation with

small viscosity is described using invariant manifolds. In particular,

a geometric explanation is provided for a phenomenon known as

metastability,which in the present context means that

solutions spend a very long time near the family of solutions known as

diffusive N-waves before finally converging to a stable self-similar

diffusion wave. More precisely, it is shown that in terms of

similarity, or scaling, variables in an algebraically weighted $L^2$

space, the self-similar diffusion waves correspond to a one-dimensional

global center manifold of stationary solutions. Through each of these

fixed points there exists a one-dimensional, global, attractive,

invariant manifold corresponding to the diffusive N-waves. Thus,

metastability corresponds to a fast transient in which solutions

approach this ``metastable" manifold of diffusive N-waves, followed by

a slow decay along this manifold, and, finally, convergence to the

self-similar diffusion wave.

A function $u: \mathbb{R}^{n} \to \mathbb{R}^{m}$ is one-homogeneous if $u(ax)=au(x)$ for any positive real number $a$ and all $x$ in $\R^{n}$. Phillips(2002) showed that in two dimensions such a function cannot solve an elliptic system in divergence form, in contrast to the situation in higher dimensions where various authors have constructed one-homogeneous minimizers of regular variational problems. This talk will discuss an extension of Phillips's 2002 result to $x-$dependent systems. Some specific one-homogeneous solutions will be constructed in order to show that certain of the hypotheses of the extension of the Phillips result can't be dropped. The method used in the construction is related to nonlinear elasticity in that it depends crucially on polyconvex functions $f$ with the property that $f(A) \to \infty$ as $\det A \to 0$.

Travelling waves are highly symmetric solutions to the Hamiltonian lattice equation and are determined by nonlinear advance-delay differential equations. They provide much insight into the microscopic dynamics and are moreover fundamental building blocks for macroscopic

lattice theories.

In this talk we concentrate on travelling waves in convex FPU chains and study both periodic waves (wave trains) and homoclinic waves (solitons). We present a new existence proof which combines variational and dynamical concepts.

In particular, we improve the known results by showing that the profile functions are unimodal and even.

Finally, we study the complete localization of wave trains and address additional complications that arise for heteroclinic waves (fronts).(joint work with Jens D.M. Rademacher, CWI Amsterdam)

The proof of quasiconvexity based sufficient conditions for strong local minima in vectorial variational problems consists of three major parts: the Decomposition Theorem, the Orthogonality principle and the Localization principle. The first and the last are the most technical.
In these talks I will explain the technical difficulties and the ways in which they were overcome.

In the past several years it has come to be appreciated that in low Reynolds number flow the nonlinearities provided by non-Newtonian stresses of a complex fluid can provide a richness of dynamical behaviors more commonly associated with high Reynolds number Newtonian flow. For example, experiments by V. Steinberg and collaborators have shown that dilute polymer suspensions being sheared in simple flow geometries can exhibit highly time dependent dynamics and show efficient mixing. The corresponding experiments using Newtonian fluids do not, and indeed cannot, show such nontrivial dynamics. To better understand these phenomena we study the Oldroyd-B viscoelastic model. We first explain the derivation of this system and its relation to more familiar systems of Newtonian fluids and solids and give some analytical results for small data perturbations. Next we study this and related models numerically for low-Reynolds number flows in two dimensions. For low Weissenberg number (an elasticity parameter), flows are "slaved" to the four-roll mill geometry of the fluid forcing. For sufficiently large Weissenberg number, such slaved solutions are unstable and under perturbation transit in time to a structurally dissimilar flow state dominated by a single large vortex, rather than four vortices of the four-roll mill state. The transition to this new state also leads to regions of well-mixed fluid and can show persistent oscillatory behavior with continued destruction and generation of smaller-scale vortices.

This talk first introduces generalized Young measures (or DiPerna/Majda measures) in an $L^1$-setting. This extension to classical Young measures is able to quantitatively account for both oscillation and concentration phenomena in generating sequences.

We establish several fundamental properties like compactness and representation of nonlinear integral functionals and present some examples. Then, generalized Young measures generated by $W^{1,1}$- and $BV$-gradients are more closely examined and several tools to manipulate them (including averaging and approximation) are presented.

Finally, we address the question of characterizing the set of generalized Young measures generated by gradients in the spirit of the Kinderlehrer-Pedregal Theorem.

This is joint work with Jan Kristensen.

Γ-convergence is a variational convergence on functionals. The explicit characterization of the integrand of the Γ-limit of sequences of integral functionals with periodic integrands is by now well known. Here we focus on the explicit characterization of the limit energy density of a sequence of functionals with non-periodic integrands. Such characterization is achieved in terms of the Young measure associated with relevant sequences of functions. Interesting examples are considered.

Multiaxial mechanical proportional loadings on shape memory alloys undergoing phase transformation permit to determine the yield curve of phase transformation initiation in the stress space. We show how to transport this yield surface in the set of effective transformation strains of producted phase M. Two numerical applications are done concerning a Cu Al Be and a Ni Ti polycrystallines shape memory alloys. A special attention is devoted to establish a convexity criterium of these surfaces.

Moreover an application to the determination of the phase transformation surface around the crack tip for SMA fracture is performed.

At last some datas are given concerning the SMA damping behavior

AUTHORS

Christian Lexcellent, Rachid Laydi, Emmanuel Foltete, Manuel collet and Frédéric Thiebaud

FEMTO-ST Département de Mécanique Appliquée Université de Franche Comte Besançon France

We present a selection of recent results pertaining to Hessian

and Monge-Ampere equations, where the Hessian matrix is augmented by a

matrix valued lower order operator. Equations of this type arise in

conformal geometry, geometric optics and optimal transportation.In

particular we will discuss structure conditions, due to Ma,Wang and

myself, which imply the regularity of solutions.These conditions are a

refinement of a condition used originally by Pogorelev for general

equations of Monge-Ampere type in two variables and called strong

ellipticity by him.

I will begin by talking briefly about the Lavrentiev phenomenon and its implications for computations. In short, if a minimization problem exhibits a Lavrentiev gap then `naive' numerical methods cannot be used to solve it. In the past, several regularization techniques have been used to overcome this difficulty. I will briefly mention them and discuss their strengths and weaknesses.

The main part of the talk will be concerned with a class of convex problems, and I will show that for this class, relatively simple numerical methods, namely (i) the Crouzeix--Raviart FEM and (ii) the P2-FEM with under-integration, can successfully overcome the Lavrentiev gap.

We propose and analyze a fully discrete Fourier collocation scheme to

solve numerically a nonlinear equation in 2D space derived from a

pattern forming gradient flow. We prove existence and uniqueness of the

numerical solution and show that it converges to a solution of the

initial continuous problem. We also derive some error estimates and

perform numerical experiments to illustrate the theory.

In this talk we will start with various shock reflection-diffraction phenomena, their fundamental scientific issues, and their theoretical roles in the mathematical theory of multidimensional hyperbolic systems of conservation laws. Then we will describe how the global shock reflection-diffraction problems can be formulated as free boundary problems for nonlinear conservation laws of mixed-composite hyperbolic-elliptic type.

Finally we will discuss some recent developments in attacking the shock reflection-diffraction problems, including the existence, stability, and regularity of global regular configurations of shock reflection-diffraction by wedges. The approach includes techniques to handle free boundary problems, degenerate elliptic equations, and corner singularities, which is highly motivated by experimental, computational, and asymptotic results. Further trends and open problems in this direction will be also addressed. This talk will be mainly based on joint work with M. Feldman.

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.