Past Partial Differential Equations Seminar

based on joint work with Charles Fefferman (Princeton) and Kevin Luli (Yale).

In 1932 Signorini formulated the first variational inequality as a model of an elastic body laying on a rigid surface. In this talk we will revisit this problem from the point of view of regularity theory.

We will sketch a proof of optimal regularity and regularity of the contact set. Similar result are known for scalar equations. The proofs for scalar equations are however based on maximum principles and thus not applicable to Signorini's problem which is modelled by a system of equations.

We consider the stationary flow of Prandtl-Eyring fluids in two

dimensions. This model is a good approximation of perfect plasticity.

The corresponding potential is only slightly super linear. Thus, many

severe problems arise in the existence theory of weak solutions. These

problems are overcome by use of a divergence free Lipschitz

truncation. As a second application of this technique, we generalize

the concept of almost harmonic functions to the Stokes system.

This talk will consist of a pure PDE part, and an applied part. The unifying topic is mean curvature flow (MCF), and particularly mean curvature flow starting at cones. This latter subject originates from the abstract consideration of uniqueness questions for flows in the presence of singularities. Recently, this theory has found applications in several quite different areas, and I will explain the connections with Harnack estimates (which I will explain from scratch) and also with the study of the dynamics of charged fluid droplets.

There are essentially no prerequisites. It would help to be familiar with basic submanifold geometry (e.g. second fundamental form) and intuition concerning the heat equation, but I will try to explain everything and give the talk at colloquium level.

Joint work with Sebastian Helmensdorfer.

In the first part, a variational model for composition of finitely many strongly elliptic

homogenous elastic materials in linear elasticity is considered. The notion of`universal coercivity' for the variational integrals is introduced which is independent of particular compositions of materials involved. Examples and counterexamples for universal coercivity are presented.

In the second part, some results of recent work with colleagues on image processing and feature extraction will be displayed.

We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time ``splash'' singularity, wherein the evolving 2-D hypersurface intersects itself at a point. Our approach is based on the Lagrangian description of the free-boundary problem, combined with novel approximation scheme. We do not assume the fluid is irrotational, and as such, our method can be used for a number of other fluid interface problems. This is joint work with Daniel Coutand.

I will describe a multiscale asymptotic framework for the analysis of the macroscopic behaviour of periodic

two-material composites with high contrast in a finite-strain setting. I will start by introducing the nonlinear

description of a composite consisting of a stiff material matrix and soft, periodically distributed inclusions. I shall then focus

on the loading regimes when the applied load is small or of order one in terms of the period of the composite structure.

I will show that this corresponds to the situation when the displacements on the stiff component are situated in the vicinity

of a rigid-body motion. This allows to replace, in the homogenisation limit, the nonlinear material law of the stiff component

by its linearised version. As a main result, I derive (rigorously in the spirit of $\Gamma$-convergence) a limit functional

that allows to establish a precise two-scale expansion for minimising sequences. This is joint work with M. Cherdantsev and

S. Neukamm.

We describe several bifurcation properties corresponding to various classes of nonlinear elliptic equations The purpose of this talk is two-fold. First, it points out different competition effects between the terms involved in the equations. Second, it provides several non standard phenomena that occur according to the structure of the differential operator.

This is a joint work with Craig Evans. We study the partial regularity of minimizers for certain functionals in the calculus of variations, namely the modified Landau-de Gennes energy functional in nematic liquid crystal theory introduced by Ball and Majumdar.

We prove existence of a global semigroup of conservative solutions of the nonlinear variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. The equation was derived by Saxton as a model for liquid crystals. This equation shares many of the peculiarities of the Hunter–Saxton and the Camassa–Holm equations. In particular, the equation possesses two distinct classes of solutions denoted conservative and dissipative. In order to solve the Cauchy problem uniquely it is necessary to augment the equation properly. In this talk we describe how this is done for conservative solutions. The talk is based on joint work with X. Raynaud.

In many applications it is of interest to compute minimizers of

a functional I(u) which is the of the form $J(u)=\Phi(u)+R(u)$,

with $R(u)$ quadratic. We describe a stochastic algorithm for

this problem which avoids explicit computation of gradients of $\Phi$;

it requires only the ability to sample from a Gaussian measure

with Cameron-Martin norm squared equal to $R(u)$, and the ability

to evaluate $\Phi$. We show that, in an appropriate parameter limit,

a piecewise linear interpolant of the algorithm converges weakly to a noisy

gradient flow. \\

Joint work with Natesh Pillai (Harvard) and Alex Thiery (Warwick).

The main mechanism for crystal plasticity is the formation and motion of a special class of defects, the dislocations. These are topological defects in the crystalline structure that can be identify with lines on which energy concentrates. In recent years there has been a considerable effort for the mathematical derivation of models that describe these objects at different scales (from an energetic and a dynamical point of view). The results obtained mainly concern special geometries, as one dimensional models, reduction to straight dislocations, the activation of only one slip system, etc.

The description of the problem is indeed extremely complex in its generality.

In the presentation will be given an overview of the variational models for dislocations that can be obtained through an asymptotic analysis of systems of discrete dislocations.

Under suitable scales we study the ``variational limit'' (by means of Gamma-convergence) of a three dimensional (static) discrete model and deduce a line tension anisotropic energy. The characterization of the line tension energy density requires a relaxation result for energies defined on curves.

The talk will address two recent results concerning the Doi-Smoluchowski equation and the Onsager model for nematic liquid crystals. The first result concerns the existence of inertial manifolds for the Smloluchowski equation both in the presence and in the absence of external flows. While the Doi-Smoluchowski equation as a PDE is an infinite-dimensional dynamical system, it reduces to a system of ODEs on a set coined inertial manifold, to which all other solutions converge exponentially fast.  The proof uses a non-standard method, which consists in circumventing the restrictive spectral-gap condition, which the original equation fails to satisfy by transforming the equation into a form that does. 

The second result concerns the isotropic-nematic phase transition for the Onsager model on the circle using more complicated potentials than the Maier-Saupe potential. Exact multiplicity of steady-states on the circle is proven for the two-mode truncation of the Onsager potential.    

Please note that this seminar has been cancelled due to unforeseen circumstances.

In this talk, we will present some recent mathematical features around two-fluid models. Such systems may be encountoured for instance to model internal waves, violent aerated flows, oil-and-gas mixtures. Depending on the context, the models used for simulation may greatly differ. However averaged models share the same structure. Here, we address the question whether available mathematical results in the case of a single fluid governed by the compressible barotropic equations for single flow may be extended to two phase model and discuss derivations of well-known multi-fluid models from single fluid systems by homogeneization (assuming for instance highly oscillating density). We focus on existence of local existence of strong solutions, loss of hyperbolicity, global existence of weak solutions, invariant regions, Young measure characterization.

Given a film of viscous heavy liquid with upper free boundary over an inclined plane, a steady laminar motion develops parallel to the flat bottom ofthe layer. We name this motion\emph{ Poiseuille Free Boundary} PFBflow because of its (half) parabolic velocity profile. In flowsover an inclined plane the free surface introduces additionalinteresting effects of surface tension and gravity. These effectschange the character of the instability in a parallel flow, see{Smith} [1]. \par\noindentBenjamin [2], and Yih [3], have solved the linear stabilityproblem of a uniform film on a inclined plane. Instability takesplace in the form of an infinitely long wave, however\emph{surface waves of finite wavelengths are observed}, see e.g.Yih [3]. Up to date direct nonlinear methods for the study ofstability seem to be still lacking.
Aim of this talk is the investigation of nonlinear stability ofPFB providing \emph{ a rigorous formulation of the problem by theclassical direct Lyapunov method assuming periodicity in theplane}, when above the liquid there is a uniform pressure due tothe air at rest, and the liquid is moving with respect to the air.Sufficient conditions on the non dimensional Reynolds, Webernumbers, on the periodicity along the line of maximum slope, onthe depth of the layer and on the inclination angle are computedensuring Kelvin-Helmholtz \emph{nonlinear stability}. We use\emph{a modified energy method, cf. [4],[5], which providesphysically meaningful sufficient conditions ensuring nonlinearexponential stability}. The result is achieved in the class ofregular solutions occurring in simply connected domains havingcone property.\par\noindentNotice that the linear equations, obtained by linearization of ourscheme around the basic Poiseuille flow, do coincide with theusual linear equations, cf. {Yih} [3]. \\ {\bf References}\\ [1]  M.K. Smith, \textit{The mechanism for the long-waveinstability in thin liquid films} J. Fluid Mech., \textbf{217},1990, pp.469-485.
\\ [2]  Benjamin T.B., \textit{Wave formation in laminar flow down aninclined plane}, J. Fluid Mech. \textbf{2}, 1957, 554-574.
\\ [3]  Yih Chia-Shun, \textit{Stability of liquid flow down aninclined plane}, Phys. Fluids, \textbf{6}, 1963, pp.321-334.
\\ [4] Padula M., {\it On nonlinear stability of MHD equilibriumfigures}, Advances in Math. Fluid Mech., 2009, 301-331.
\\ [5] Padula M., \textit{On nonlinear stability of linear pinch},Appl. Anal.  90 (1), 2011, pp. 159-192.

We give a characterization of divergence-free vector fields on the plane such that the Cauchy problem for the associated continuity (or transport) equation has a unique bounded solution (in the sense of distribution).

Unlike previous results in this directions (Di Perna-Lions, Ambrosio), the proof relies on a dimension-reduction argument, which can be regarded as a variant of the method of characteristics. Note that our characterization is not stated in terms of function spaces, but is based on a suitable weak formulation of the Sard property for the potential associated to the vector-field.

This is a joint work with S. Bianchini (SISSA, Trieste) and Gianluca Crippa (Parma).

Fluids that are not adequately described by a linear constitutive relation are usually referred to as   "non-Newtonian fluids". In the last 15 years we have seen a significant progress in the mathematical theory of generalized Newtonian fluids, which is an important subclass of non-Newtonian fluids. We present some recent results in the existence theory and in the error analysis for approximate solutions. We will also indicate how these techniques can be generalized to more general constitutive relations.