Past OxPDE Lunchtime Seminar

We prove the existence of weak solutions to steady Navier Stokes equations

$$\text{div}\, \sigma+f=\nabla\pi+(\nabla u)u.$$

Here $u:\mathbb{R}^2\supset \Omega\rightarrow \mathbb{R}^2$ denotes

the velocity field satisfying $\text{div}\, u=0$,

$f:\Omega\rightarrow\mathbb{R}^2$ and

$\pi:\Omega\rightarrow\mathbb{R}$ are external volume force and

pressure, respectively. In order to model the behavior of

Prandtl-Eyring fluids we assume

$$\sigma= DW(\varepsilon (u)),\quad W(\varepsilon)=|\varepsilon|\log

(1+|\varepsilon|).$$

A crucial tool in our approach is a modified Lipschitz truncation

preserving the divergence of a given function.

I will describe recent work with Charles Fefferman on a

construction of families of analytic almost-sharp fronts for SQG. These

are special solutions of SQG which have a very sharp transition in a

very thin layer. One of the main difficulties of the construction is the

fact that there is no formal limit for the family of equations. I will

show how to overcome this difficulty, linking the result to joint work

with C. Fefferman and Kevin Luli on the existence of a "spine" for

almost-sharp fronts. This is a curve, defined for every time slice by a

measure-theoretic construction, that describes the evolution of the

almost-sharp front.

Modern mathematical approaches to plasticity result in non-convex variational problems for which the standard methods of the calculus of variations are not applicable. In this contribution we consider geometrically nonlinear crystal elasto-plasticity in two dimensions with one active slip system. In order to derive information about macroscopic material behavior the relaxation of the corresponding incremental problems is studied. We focus on the question if realistic systems with an elastic energy leading to large penalization of small elastic strains can be well-approximated by models based on the assumption of rigid elasticity. The interesting finding is that there are qualitatively different answers depending on whether hardening is included or not. In presence of hardening we obtain a positive result, which is mathematically backed up by Γ-convergence, while the material shows very soft macroscopic behavior in case of no hardening. The latter is due to the vanishing relaxation for a large class of applied loads.

This is joint work with Sergio Conti and Georg Dolzmann.

We propose a general framework for the study of $L^1$ contractive semigroups of solutions to conservation laws with discontinuous flux. Developing the ideas of a number of preceding works we claim that the whole admissibility issue is reduced to the selection of a family of "elementary solutions", which are certain piecewise constant stationary weak solutions. We refer to such a family as a "germ". It is well known that (CL) admits many different $L^1$ contractive semigroups, some of which reflects different physical applications. We revisit a number of the existing admissibility (or entropy) conditions and identify the germs that underly these conditions. We devote specific attention to the anishing viscosity" germ, which is a way to express the "$\Gamma$-condition" of Diehl. For any given germ, we formulate "germ-based" admissibility conditions in the form of a trace condition on the flux discontinuity line $x=0$ (in the spirit of Vol'pert) and in the form of a family of global entropy inequalities (following Kruzhkov and Carrillo). We characterize those germs that lead to the $L^1$-contraction property for the associated admissible solutions. Our approach offers a streamlined and unifying perspective on many of the known entropy conditions, making it possible to recover earlier uniqueness results under weaker conditions than before, and to provide new results for other less studied problems. Several strategies for proving the existence of admissible solutions are discussed, and existence results are given for fluxes satisfying some additional conditions. These are based on convergence results either for the vanishing viscosity method (with standard viscosity or with specific viscosities "adapted" to the choice of a germ), or for specific germ-adapted finite volume schemes.

This is joint work with Boris Andreianov and Nils Henrik Risebro.

The reconstruction of the early universe amounts to recovering the tiny density fluctuations of the early universe (shortly after the "big bang") from the current observation of the matter distribution in the universe. Following Zeldovich, Peebles and, more recently Frisch and collaboratoirs, we use a newtonian gravitational model with time dependent coefficients taking into accont general relativity effects. Due to the (remarkable) convexity of the corresponding action, the reconstruction problem apparently reduces to a straightforward convex minimization problem. Unfortunately, this approach completely ignores the mass concentration effects due to gravitational instabilities.

In this lecture, we show a way of modifying the action in order to take concentrations into account. This is obtained through a (questionable) modification of the gravitation model,

by substituting the fully nonlinear Monge-Amp`ere equation for the linear Poisson equation. (This is a reasonable approximation in the sense that it makes exact some approximate solutions advocated by Zeldovich for the original gravitational model.) Then the action can be written as a perfect square in which we can input mass concentration effects in a canonical way, based on the theory of gradient flows with convex potentials and somewhat related to the concept of self-dual Lagrangians developped by Ghoussoub. A fully discrete algorithm is introduced for the EUR problem in one space dimension.

Considering kinetic equations (Boltzmann, BGK, say...) in the small mean free path regime lead to conservation laws (the Euler system, typically) When the problem is set in a domain, boundary layers might occur due to the fact that incoming fluxes could be far from equilibrium states. We consider the problem from a numerical perspective and we propose a definition of numerical fluxes for the Euler system which is intended to account for the formation of these boundary layers.

Free-discontinuity problems describe situations where the solution of

interest is defined by a function and a lower dimensional set consisting

of the discontinuities of the function. Hence, the derivative of the

solution is assumed to be a "small function" almost everywhere except on

sets where it concentrates as a singular measure.

This is the case, for instance, in certain digital image segmentation

problems and brittle fracture models.

In the first part of this talk we show new preliminary results on

the existence of minimizers for inverse free-discontinuity problems, by

restricting the solutions to a class of functions with piecewise Lipschitz

discontinuity set.

If we discretize such situations for numerical purposes, the inverse

free-discontinuity problem in the discrete setting can be re-formulated as

that of finding a derivative vector with small components at all but a few

entries that exceed a certain threshold. This problem is similar to those

encountered in the field of "sparse recovery", where vectors

with a small number of dominating components in absolute value are

recovered from a few given linear measurements via the minimization of

related energy functionals.

As a second result, we show that the computation of global minimizers in

the discrete setting is an NP-hard problem.

With the aim of formulating efficient computational approaches in such

a complicated situation, we address iterative thresholding algorithms that

intertwine gradient-type iterations with thresholding steps which were

designed to recover sparse solutions.

It is natural to wonder how such algorithms can be used towards solving

discrete free-discontinuity problems. This talk explores also this

connection, and, by establishing an iterative thresholding algorithm for

discrete inverse free-discontinuity problems, provides new insights on

properties of minimizing solutions thereof.

Tonelli gave the first rigorous treatment of one-dimensional variational problems, providing conditions for existence and regularity of minimizers over the space of absolutely continuous functions.  He also proved a partial regularity theorem, asserting that a minimizer is everywhere differentiable, possible with infinite derivative, and that this derivative is continuous as a map into the extended real line.  Some recent work has lowered the smoothness assumptions on the Lagrangian for this result to various Lispschitz and H\"older conditions.  In this talk we will discuss the partial regularity result, construct examples showing that mere continuity of the Lagrangian is an insufficient condition.

It has long been known that many materials are crystalline when in their energy-minimizing states. Two of the most common crystalline structures are the face-centred cubic (fcc) and hexagonal close-packed (hcp) crystal lattices. Here we introduce the problem of crystallization from a mathematical viewpoint and present an outline of a proof that the ground state of a large system of identical particles, interacting under a suitable potential, behaves asymptotically like fcc or hcp, as the number of particles tends to infinity. An interesting feature of this result is that it holds under no initial assumption on the particle positions. The talk is based upon a joint work in progress with Florian Theil.

It is well-known that Morrey's quasiconvexity is closely related to gradient Young measures,

i.e., Young measures generated by sequences of gradients in

$L^p(\Omega;\mathbb{R}^{m\times n})$. Concentration effects,

however, cannot be treated by Young measures. One way how to describe both oscillation and

concentration effects in a fair generality are the so-called DiPerna-Majda measures.

DiPerna and Majda showed that having a sequence $\{y_k\}$ bounded in $L^p(\Omega;\mathbb{R}^{m\times n})$,$1\le p$ $0$.

We consider thin films of a cholesteric liquid-crystal material subject to an applied electric field.  In such materials, the liquid-crystal "director" (local average orientation of the long axis of the molecules) has an intrinsic tendency to rotate in space; while the substrates that confine the film tend to coerce a uniform orientation.

The electric field encourages certain preferred orientations of the director as well, and these competing influences give rise to several different stable equilibrium states of the director field, including spatially uniform, translation invariant (functions only of position across the cell gap) and periodic (with 1-D or 2-D periodicity in the plane of the film).  These structures depend on two principal control parameters: the ratio of the cell gap to the intrinsic "pitch" (spatial period of rotation) of the cholesteric and the magnitude of the applied voltage.

We report on numerical work (not complete) on the bifurcation and phase behavior of this system.  The study was motivated by potential applications involving switchable gratings and eyewear with tunable transparency. We compare our results with experiments conducted in the Liquid Crystal Institute at Kent State University.

Numerical homogenization/upscaling for problems with multiple scales have attracted increasing attention in recent years. In particular, problems with non-separable scales pose a great challenge to mathematical analysis and simulation.

In this talk, we present some rigorous results on homogenization of divergence form scalar and vectorial elliptic equations with $L^\infty$ rough coefficients which allow for a continuum of scales. The first approach is based on a new type of compensation phenomena for scalar elliptic equations using the so-called ``harmonic coordinates''. The second approach, the so-called ``flux norm approach'' can be applied to finite dimensional homogenization approximations of both scalar and vectorial problems with non-separated scales. It can be shown that in the flux norm, the error associated with approximating the set of solutions of the PDEs with rough coefficients, in a properly defined finite-dimensional basis, is equal to the error associated with approximating the set of solutions of the same type of PDEs with smooth coefficients in a standard finite element space. We will also talk about the ongoing work on the localization of the basis in the flux norm approach.

{\bf Keble Workshop on Partial Differential Equations

in Science and Engineering}

\\

\\Place: Roy Griffiths Room in the ARCO Building, Keble College

\\Time: 1:00pm-5:10pm, Thursday, June 10.

\\

\\

Program:\\

\\ 1:00-1:20pm: Coffee and Tea

\\

\\ 1:20-2:10pm: Prof. Walter Craig (Joint with OxPDE Lunchtime Seminar)

\\

\\ 2:20-2:40pm Prof. Mikhail Feldman

\\

\\ 2:50-3:10pm Prof. Paul Taylor

\\

\\ 3:20-3:40pm Coffee and Biscuits

\\

\\ 3:40-4:00pm: Prof. Sir John Ball

\\

\\ 4:10-4:30pm: Dr. Apala Majumdar

\\

\\ 4:40-5:00pm: Prof. Robert Pego

\\

\\ 5:10-6:00pm: Free Discussion

\\

\\{\bf Titles and Abstracts:}

\\

1.{\bf Title: On the singular set of the Navier-Stokes equations

\\ Speaker: Prof. Walter Craig, McMaster University, Canada}

\\ Abstract:\\

The Navier-Stokes equations are important in

fluid dynamics, and a famous mathematics problem is the

question as to whether solutions can form singularities.

I will describe these equations and this problem, present

three inequalities that have some implications as to the

question of singularity formation, and finally, give a

new result which is a lower bound on the size of the

singular set, if indeed singularities exist.

\\

\\{\bf 2. Title: Shock Analysis and Nonlinear Partial Differential Equations of Mixed Type.

\\ Speaker: Prof. Mikhail Feldman, University of Wisconsin-Madison, USA}

\\

\\ Abstract:\\ Shocks in gas or compressible fluid arise in various physical

situations, and often exhibit complex structures. One example is reflection

of shock by a wedge. The complexity of reflection-diffraction configurations

was first described by Ernst Mach in 1878. In later works, experimental and

computational studies and asymptotic analysis have shown that various patterns

of reflected shocks may occur, including regular and Mach reflection. However,

many fundamental issues related to shock reflection are not understood,

including transition between different reflection patterns. For this reason

it is important to establish mathematical theory of shock reflection,

in particular existence and stability of regular reflection solutions for PDEs

of gas dynamics. Some results in this direction were obtained recently.

\\

In this talk we start by discussing examples of shocks in supersonic and

transonic flows of gas. Then we introduce the basic equations of gas dynamics:

steady and self-similar compressible Euler system and potential flow equation.

These equations are of mixed elliptic-hyperbolic type. Subsonic and supersonic

regions in the flow correspond to elliptic and hyperbolic regions of solutions.

Shocks correspond to certain discontinuities in the solutions. We discuss some

results on existence and stability of steady and self-similar shock solutions,

in particular the recent work (joint with G.-Q. Chen) on global existence of

regular reflection solutions for potential flow. We also discuss open problems

in the area.

\\

\\{\bf 3. Title: Shallow water waves - a rich source of interesting solitary wave

solutions to PDEs

\\ Speaker: Prof. Paul H. Taylor, Keble College and Department of Engineering Science, Oxford}

\\

\\Abstract:\\ In shallow water, solitary waves are ubiquitous: even the wave crests

in a train of regular waves can be modelled as a succession of solitary waves.

When successive crests are of different size, they interact when the large ones

catch up with the smaller. Then what happens? John Scott Russell knew by experiment

in 1844, but answering this question mathematically took 120 years!

This talk will examine solitary wave interactions in a range of PDEs, starting

with the earliest from Korteweg and De Vries, then moving onto Peregrine's

regularized long wave equation and finally the recently introduced Camassa-Holm

equation, where solitary waves can be cartoon-like with sharp corners at the crests.

For each case the interactions can be described using the conserved quantities,

in two cases remarkably accurately and in the third exactly, without actually

solving any of the PDEs.

The methodology can be extended to other equations such as the various versions

of the Boussinesq equations popular with coastal engineers, and perhaps even

the full Euler equations.

\\

{\bf 4. Title: Austenite-Martensite interfaces

\\ Speaker: Prof. Sir John Ball, Queen's College and Mathematical Institute, Oxford}

\\

\\Abstract:\\ Many alloys undergo martensitic phase transformations

in which the underlying crystal lattice undergoes a change of shape

at a critical temperature. Usually the high temperature phase (austenite)

has higher symmetry than the low temperature phase (martensite).

In order to nucleate the martensite it has to somehow fit geometrically

to the austenite. The talk will describe different ways in which this

occurs and how they may be studied using nonlinear elasticity and

Young measures.

\\

\\{\bf 5. Title: Partial Differential Equations in Liquid Crystal Science and

Industrial Applications

\\ Speaker: Dr. Apala Majumdar, Keble College and Mathematical Institute, Oxford}

\\

\\Abstract:\\

Recent years have seen a growing demand for liquid crystals in modern

science, industry and nanotechnology. Liquid crystals are mesophases or

intermediate phases of matter between the solid and liquid phases of

matter, with very interesting physical and optical properties.

We briefly review the main mathematical theories for liquid crystals and

discuss their analogies with mathematical theories for other soft-matter

phases such as the Ginzburg-Landau theory for superconductors. The

governing equations for the static and dynamic behaviour are typically

given by systems of coupled elliptic and parabolic partial differential

equations. We then use this mathematical framework to model liquid crystal

devices and demonstrate how mathematical modelling can be used to make

qualitative and quantitative predictions for practical applications in

industry.

\\

\\{\bf 6. Title: Bubble bath, shock waves, and random walks --- Mathematical

models of clustering

\\Speaker: Prof. Robert Pego, Carnegie Mellon University, USA}

\\Abstract:\\ Mathematics is often about abstracting complicated phenomena into

simple models. This talk is about equations that model aggregation

or clustering phenomena --- think of how aerosols form soot particles

in the atmosphere, or how interplanetary dust forms comets, planets

and stars. Often in such complex systems one observes universal trend

toward self-similar growth. I'll describe an explanation for this

phenomenon in two simple models describing: (a) ``one-dimensional

bubble bath,'' and (b) the clustering of random shock waves.

In this talk, we describe new profile decompositions for bounded sequences in Banach spaces of functions defined on $\mathbb{R}^d$. In particular, for "critical spaces" of initial data for the Navier-Stokes equations, we show how these can give rise to new proofs of recent regularity theorems such as those found in the works of Escauriaza-Seregin-Sverak and Rusin-Sverak. We give an update on the state of the former and a new proof plus new results in the spirit of the latter. The new profile decompositions are constructed using wavelet theory following a method of Jaffard.

In this talk we discuss the solution of the elastodynamic

equations in a bounded domain with hereditary-type linear

viscoelastic constitutive relation. Existence, uniqueness, and

regularity of solutions to this problem is demonstrated

for those viscoelastic relaxation tensors satisfying the condition

of being completely monotone. We then consider the non-self-adjoint

and non-linear eigenvalue problem associated with the

frequency-domain form of the elastodynamic equations, and show how

the time-domain solution of the equations can be expressed in

terms of an eigenfunction expansion.

We consider a 3-dimensional elastic continuum whose material points

can experience no displacements, only rotations. This framework is a

special case of the Cosserat theory of elasticity. Rotations of

material points of the continuum are described mathematically by

attaching to each geometric point an orthonormal basis which gives a

field of orthonormal bases called the coframe. As the dynamical

variables (unknowns) of our theory we choose the coframe and a

density.

In the first part of the talk we write down the general dynamic

variational functional of our problem. In doing this we follow the

logic of classical linear elasticity with displacements replaced by

rotations and strain replaced by torsion. The corresponding

Euler-Lagrange equations turn out to be nonlinear, with the source

of this nonlinearity being purely geometric: unlike displacements,

rotations in 3D do not commute.

In the second part of the talk we present a class of explicit

solutions of our Euler-Lagrange equations. We call these solutions

plane waves. We identify two types of plane waves and calculate

their velocities.

In the third part of the talk we consider a particular case of our

theory when only one of the three rotational elastic moduli, that

corresponding to axial torsion, is nonzero. We examine this case in

detail and seek solutions which oscillate harmonically in time but

depend on the space coordinates in an arbitrary manner (this is a

far more general setting than with plane waves). We show [1] that

our second order nonlinear Euler-Lagrange equations are equivalent

to a pair of linear first order massless Dirac equations. The

crucial element of the proof is the observation that our Lagrangian

admits a factorisation.

[1] Olga Chervova and Dmitri Vassiliev, "The stationary Weyl

equation and Cosserat elasticity", preprint http://arxiv.org/abs/1001.4726

In this talk we consider the Boltzmann equation arising in gas dynamics with long-range interactions. Mathematically, it involves bilinear singular integral operators known as collision operators with non-cutoff collision kernels. As for the associated Cauchy problem, we develop a theory of weak solutions and present some of its a priori estimates related with physical quantities including the energy and moments.

The aim of this talk is to analyze energy functionals concentrated on the jump set of 2D vector fields of unit length and of vanishing divergence.

The motivation of this study comes from thin-film micromagnetics where these functionals correspond to limiting wall-energies. The main issue consists in characterizing the wall-energy density (the cost function) so that the energy functional is lower semicontinuous (l.s.c.). The key point resides in the concept of entropies due to the scalar conservation law implied by our vector fields. Our main result identifies appropriate cost functions

associated to certain sets of entropies. In particular, certain power cost functions lead to l.s.c. energy functionals.

A second issue concerns the existence of minimizers of such energy functionals that we prove via a compactness result. A natural question is whether the viscosity solution is a minimizing configuration. We show that in general it is not the case for nonconvex domains.

However, the case of convex domains is still open. It is a joint work with Benoit Merlet, Ecole Polytechnique (Paris).

One of important subjects in the study of transonic flow is to understand a global structure of flow through a convergent-divergent nozzle so called a de Laval nozzle. Depending on the pressure at the exit of the de Laval nozzle, various patterns of flow may occur. As an attempt to understand such a phenomenon, we introduce a new potential flow model called 'non-isentropic potential flow system' which allows a jump of the entropy across a shock, and use this model to rigorously prove the unique existence and the stability of transonic shocks for a fixed exit pressure. This is joint work with Mikhail Feldman.

I will discuss recent results on dispersive estimates for linear PDE's with time dependent coefficients. Then I will discuss how such

estimates can be used to study stability of nonlinear solitary waves and resonance phenomena.

Given a bounded Lipschitz domain, we consider the Dirichlet problem with boundary data in Besov spaces

for divergence form strongly elliptic systems of arbitrary order with bounded complex-valued coefficients.

The main result gives a sharp condition on the local mean oscillation of the coefficients of the differential operator

and the unit normal to the boundary (automatically satisfied if these functions belong to the space VMO)

which guarantee that the solution operator associated with this problem is an isomorphism.

According to the Ernst-Geroch reduction, in an axially symmetric stationary electrovac spacetime, the Einstein-Maxwell equations reduce to a harmonic map problem with singular boundary data. I will discuss the “regularity” of the reduced harmonic maps near the boundary and its implication on the regularity of the corresponding spacetimes.

We consider the three dimensional Navier-Stokes equations with a large initial data and

we prove the existence of a global smooth solution. The main feature of the initial data

is that it varies slowly in the vertical direction and has a norm which blows up as the

small parameter goes to zero. In the language of geometrical optics, this type of

initial data can be seen as the ``ill prepared" case. Using analytical-type estimates

and the special structure of the nonlinear term of the equation we obtain the existence

of a global smooth solution generated by this large initial data. This talk is based on a

work in collaboration with J.-Y. Chemin and I. Gallagher and on a joint work with Z.

Zhang.