Past Partial Differential Equations Seminar

 When given an explicit solution to an evolutionary partial differential equation, it is natural to ask whether the solution is stable, and if yes, what is the mechanism for stability and whether this mechanism survives under perturbations of the equation itself. Many familiar linear equations enjoy some notion of stability for the zero solution: solutions of the heat equation dissipate and decay uniformly and exponentially to zero, solutions of the Schrödinger equations disperse at a polynomial rate in time depending on spatial dimension, while solutions of the wave equation enjoy radiative decay (in the presence of at least two spatial dimensions) also at polynomial rates.

For this set of short course sessions, we will focus on the wave equation and its nonlinear perturbations. As mentioned above, the stability mechanism for the linear wave equation is that of radiative decay. Radiative decay depends on the number of spatial dimensions, and hence so does the stability of the zero solution for nonlinear wave equations. By the mid-1980s it was well understood that the stability mechanism survives generally (for “smooth nonlinearities”) when the spatial dimension is at least four, but for lower dimensions (two and three specifically; in dimension one there is no linear stability mechanism to start with) obstructions can arise when the nonlinearities are “stronger” than can be controlled by radiative decay. This led to the discovery of the null condition as a structural condition on the nonlinearities preventing the aforementioned obstructions. But what happens when the null condition is violated? This development spanning a quarter of a century, from F. John’s qualitative analysis of the spherically symmetric case, though S. Alinhac’s sharp control of the asymptotic lifespan, and culminating in D. Christodoulou’s full description of the null geometry, is the subject of this short course.

(1) We will start by reviewing the radiative decay mechanism for wave equations, and indicate the nonlinear stability results for high spatial dimensions. We then turn our attention to the case of three spatial dimensions: after a quick discussion of the null condition for quasilinear wave equations, we sketch, at the semilinear level, what happens when the null condition fails (in particular the asymptotic approximation of the solution by a Riccati equation).

(2) The semilinear picture is built up using a version of the method of characteristics associated with the standard wave operator. Turning to the quasilinear problem we will hence need to understand the characteristic geometry for a variable coefficient wave operator. This leads us to introduce the optical/acoustical function and its associated null structure equations.

(3) From this modern geometric perspective we next discuss, in some detail, the blow-up results obtained in the mid-1980s by F. John for quasilinear wave equations assuming radial symmetry.

(4) Finally, we indicate the main difficulties in extending the analysis to the non-radially-symmetric case, and how they can be resolved à la the recent tour de force of D. Christodoulou. While some knowledge of Lorentzian geometry and dynamics of wave equations will be helpful, this short course should be accessible to also graduate students with training in partial differential equations.

Imperial College London, United Kingdom E-mail address: @email

École Polytechnique Fédérale de Lausanne, Switzerland E-mail address: @email

We consider a class of CR manifold which are defined as asymptotically

Heisenberg,

and for these we give a notion of mass. From the solvability of the

$\Box_b$ equation

in a certain functional class ([Hsiao-Yung]), we prove positivity of the

mass under the

condition that the Webster curvature is positive and that the manifold

is embeddable.

We apply this result to the Yamabe problem for compact CR manifolds,

assuming positivity

of the Webster class and non-negativity of the Paneitz operator. This is

joint work with

J.H.Cheng and P.Yang.

A new nematic phase has recently been discovered and characterized experimentally. It embodies a theoretical prediction made by Robert B. Meyer in 1973 on the basis of mere symmetry considerations to the effect that a nematic phase might also exist which in its ground state would acquire a 'heliconical' configuration, similar to the chiral molecular arrangement of cholesterics, but with the nematic director precessing around a cone about the optic axis. Experiments with newly synthetized materials have shown chiral heliconical equilibrium structures with characteristic pitch in the range of 1o nanometres and cone semi-amplitude of about 20 degrees. In 2001, Ivan Dozov proposed an elastic theory for such (then still speculative) phase which features a negative bend elastic constant along with a quartic correction to the nematic energy density that makes it positive definite. This lecture will present some thoughts about the possibility of describing the elastic response of twist-bend nematics within a purely quadratic gradient theory.

We show that the spatial norm in any critical homogeneous Besov

space in which local existence of strong solutions to the 3-d

Navier-Stokes equations is known must become unbounded near a singularity.

In particular, the regularity of these spaces can be arbitrarily close to

-1, which is the lowest regularity of any Navier-Stokes critical space.

This extends a well-known result of Escauriaza-Seregin-Sverak (2003)

concerning the Lebesgue space $L^3$, a critical space with regularity 0

which is continuously embedded into the spaces we consider. We follow the

``critical element'' reductio ad absurdum method of Kenig-Merle based on

profile decompositions, but due to the low regularity of the spaces

considered we rely on an iterative algorithm to improve low-regularity

bounds on solutions to bounds on a part of the solution in spaces with

positive regularity. This is joint work with I. Gallagher (Paris 7) and

F. Planchon (Nice).

The vanishing mean curvature flow in Minkowski space is the

natural evolutionary generalisation of the minimal surface equation,

and has applications in cosmology as a model equation for cosmic

strings and membranes. The equation clearly admits initial data which

leads to singularity formation in finite time; Nguyen and Tian have

even shown stability of the singularity formation in low dimension. On

the other hand, Brendle and Lindblad separately have shown that all

"nearly flat" initial data leads to global existence of solutions. In

this talk, I describe an intermediate regime where global existence

of solutions can be proven on a codimension 1 set of initial data; and

where the codimension 1 condition is optimal --- The

catenoid, being a minimal surface in R^3, is a static solution to the

vanishing mean curvature flow. Its variational instability as a

minimal surface leads to a linear instability under the flow. By

appropriately "modding out" this unstable mode we can show the

existence of a stable manifold of initial data that gives rise to

solutions which scatters toward to the

catenoid. This is joint work with Roland Donninger, Joachim Krieger,

and Jeremy Szeftel. The preprint is available at http://arxiv.org/abs/1310.5606v1

We show that the positive mass theorem holds for

asymptotically flat, $n$-dimensional Riemannian manifolds with a metric

that is continuous, lies in the Sobolev space $W^{2, n/2}_{loc}$, and

has non-negative scalar curvature in the distributional sense. Our

approach requires an analysis of smooth approximations to the metric,

and a careful control of elliptic estimates for a related conformal

transformation problem. If the metric lies in $W^{2, p}_{loc}$ for

$p>n/2$, then we show that our metrics may be approximated locally

uniformly by smooth metrics with non-negative scalar curvature.

This talk is based on joint work with N. Tassotti and conversations with

J.J. Bevan.

The description of the behaviour of local minima or evolution problems for families of energies cannot in general be deduced from their Gamma-limit, which is a concept designed to treat static global minimum problems. Nevertheless this can be taken as a starting point. Various issues that have been addressed are:

Find criteria that ensure the convergence of local minimizers and critical points. In case this does not occur then modify the Gamma-limit in order to match this requirement. We note that in this way we `correct' some limit theories, finding (or `validating') other ones present in the literature;

Modify the concept of local minimizer, so that it may be more `compatible' with the process of Gamma-limit;

Treat evolution problems for energies with many local minima obtained by a time-discrete scheme introducing the notion of `minimizing movements along a sequence of functionals'. In this case the minimizing movement of the Gamma-limit can always be obtained by a choice of the space- and time-scale, but more interesting behaviors can be obtained at a critical ratio between them. In many cases a `critical scale' can be computed and an effective motion, from which all other minimizing movements are obtained by scaling.

Relate minimizing movements to general variational evolution results, in particular recent theories of quasistatic motion and gradient flow in metric spaces.

I will illustrate some of these points.

Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions under which this projected measure can have atoms, and give an example in which the projected measure is singular with respect to the Lebesgue measure and diffuse. We discuss applications of our results to the classical limit of the Schrödinger equation. Finally we present various examples and counterexamples showing that our results are sharp.

The Einstein equation from general relativity is a

quasilinear hyperbolic, geometric PDE (when viewed in an appropriate

coordinate system) for a manifold. A particularly interesting set of

known, exact solutions describe black holes. The wave and Maxwell

equations on these manifolds are models for perturbations of the known

solutions and have attracted a significant amount of attention in the

last decade. Key estimates are conservation of energy and Morawetz (or

integrated local energy) estimates. These can be proved using both

Fourier analytic methods and more geometric methods. The main focus of

the talk will be on decay estimates for solutions of the Maxwell

equation outside a slowly rotating Kerr black hole.

This talk is regarding PDE systems from geophysical applications with multiple time scales, in which linear skew-self-adjoint operators of size 1/epsilon gives rise to highly oscillatory solutions. Analysis is performed in justifying the limiting dynamics as epsilon goes to zero; furthermore, the analysis yields estimates on the difference between the multiscale solution and the limiting solution. We will introduce a simple yet effective time-averaging technique which is especially useful in general domains where Fourier analysis is not applicable.

The study of wave equations on black hole backgrounds provides important insights for the non-linear stability problem for black holes. I will illustrate this in the context of asymptotically anti de Sitter black holes and present both stability and instability results. In particular, I will outline the main ideas of recent work with J. Smulevici (Paris) establishing a logarithmic decay in time for solutions of the massive wave equation on Kerr-AdS black holes and proving that this slow decay rate is in fact sharp.

We discuss several singular limits for a scaled system of equations

(barotropic Navier-Stokes system), where the characteristic numbers become

small or ``infinite''. In particular, we focus on the situations relevant

in certain geophysical models with low Mach, large Rossby and large

Reynolds numbers. The limit system is rigorously identified in the

framework of weak solutions. The relative entropy inequality and careful

analysis of certain oscillatory integrals play crucial role.

The talk is based on the joint papers [{\it Bourgain J., Korobkov

M.V. and Kristensen~J.}: Journal fur die reine und angewandte Mathematik

(Crelles

Journal).

DOI: 10.1515/crelle-2013-0002] \ and \

[{\it Korobkov~M.V., Pileckas~K. and Russo~R.}:

arXiv:1302.0731, 4 Feb 2013]

We establish Luzin $N$ and Morse--Sard

properties for functions from the Sobolev space

$W^{n,1}(\mathbb R^n)$. Using these results we prove

that almost all level sets are finite disjoint unions of

$C^1$-smooth compact manifolds of dimension

$n-1$. These results remain valid also within

the larger space of functions of bounded variation

$BV_n(\mathbb R^n)$.

As an application, we study the nonhomogeneous boundary value problem

for the Navier--Stokes equations of steady motion of a viscous

incompressible fluid in arbitrary bounded multiply connected

plane or axially-symmetric spatial domains. We prove that this

problem has a solution under the sole necessary condition of zero total

flux through the boundary.

The problem was formulated by Jean Leray 80 years ago.

The proof of the main result uses Bernoulli's law

for a weak solution to the Euler equations based on the above-mentioned

Morse-Sard property for Sobolev functions.

Modelling reactive flows, diffusion, transport and mechanical interactions in media consisting of multiple phases, e.g. of a fluid and a solid phase in a porous medium, is giving rise to many open problems for multi-scale analysis and simulation. In this lecture, the following processes are studied:

diffusion, transport, and reaction of substances in the fluid and the solid phase,

mechanical interactions of the fluid and solid phase,

change of the mechanical properties of the solid phase by chemical reactions,

volume changes (“growth”) of the solid phase.

These processes occur for instance in soil and in porous materials, but also in biological membranes, tissues and in bones. The model equations consist of systems of nonlinear partial differential equations, with initial-boundary conditions and transmission conditions on fixed or free boundaries, mainly in complex domains. The coupling of processes on different scales is posing challenges to the mathematical analysis as well as to computing. In order to reduce the complexity, effective macroscopic equations have to be derived, including the relevant information from the micro scale.

In case of processes in tissues, a homogenization limit leads to an effective, mechanical system, containing a pressure gradient, which satisfies a generalized, time-dependent Darcy law, a Biot-law, where the chemical substances satisfy diffusion-transport-reaction equations and are influencing the mechanical parameters.

The interaction of the fluid and the material transported in a vessel with its flexible wall, incorporating material and changing its structure and mechanical behavior, is a process important e.g. in the vascular system (plague-formation) or in porous media.

The lecture is based on recent results obtained in cooperation with A. Mikelic, M. Neuss-Radu, F. Weller and Y. Yang.

A time-invariant level surface is a (codimension one)

spatial surface on which, for every fixed time, the solution of an

evolution equation equals a constant (depending on the time). A

relevant and motivating case is that of the heat equation. The

occurrence of one or more time-invariant surfaces forces the solution

to have a certain degree of symmetry. In my talk, I shall present a

set of results on this theme and sketch the main ideas involved, that

intertwine a wide variety of old and new analytical and geometrical

techniques.

Self-gravitating elastic bodies provide models for extended

objects in general relativity. I will discuss constructions of static

and rotating self-gravitating bodies, as well as recent results on the

initial value problem for self-gravitating elastic bodies.