Past Partial Differential Equations Seminar

The purpose of this talk is to provide a large class of examples of large initial data which gives rise to a global smooth solution. We shall explain what we mean by large initial data. Then we shall explain the concept of slowly varying function and give some flavor of the proofs of global existence.

We consider the Relativistic Vlasov-Maxwell system of equations which

describes the evolution of a collisionless plasma. We show that under

rather general conditions, one can test for linear instability by

checking the spectral properties of Schrodinger-type operators that

act only on the spatial variable, not the full phase space. This

extends previous results that show linear and nonlinear stability and

instability in more restrictive settings.

We will describe some joint work with V. G. Maz’ya and I. E. Verbitsky, concerning homogeneous quasilinear differential operators. The model operator under consideration is:

\[ L(u) = - \Delta_p u - \sigma |u|^{p-2} u. \]

Here $\Delta_p$ is the p-Laplacian operator and $\sigma$ is a signed measure, or more generally a distribution. We will discuss an approach to studying the operator L under only necessary conditions on $\sigma$, along with applications to the characterisation of certain Sobolev inequalities with indefinite weight. Many of the results discussed are new in the classical case p = 2, when the operator L reduces to the time independent Schrödinger operator.

We investigate ground state configurations of atomic pair potential systems in two dimensions as the number of particles tends to infinity. Assuming crystallization (which has been proved for some cases such as the Radin potential, and is believed to hold more generally), we show that after suitable rescaling, the ground states converge to a unique macroscopic Wulff shape. Moreover, we derive a scaling law for the size of microscopic non-uniqueness which indicates larger fluctuations about the Wulff shape than intuitively expected.

Joint work with Yuen Au-Yeung and Bernd Schmidt (TU Munich),

to appear in Calc. Var. PDE

We show the existence of global-in-time weak solutions to a general class of bead-spring chain models that arise from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain in two or three space dimensions for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck-type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. We require no structural assumptions on the drag term in the Fokker-Planck equation; in particular, the drag term need not be corotational. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the model, we prove the existence of a global-in-time weak solution to the coupled Navier-Stokes-Fokker-Planck system. It is also shown that in the absence of a body force, the weak solution decays exponentially in time to the equilibrium solution, at a rate that is independent of the choice of the initial datum and of the centre-of-mass diffusion coefficient.

The talk is based on joint work with John W. Barrett [Imperial College London].

It will be shown how the critical mass classical Keller-Segel system and

the critical displacement convex fast-diffusion equation in two

dimensions are related. On one hand, the critical fast diffusion

entropy functional helps to show global existence around equilibrium

states of the critical mass Keller-Segel system. On the other hand, the

critical fast diffusion flow allows to show functional inequalities such

as the Logarithmic HLS inequality in simple terms who is essential in the

behavior of the subcritical mass Keller-Segel system. HLS inequalities can

also be recovered in several dimensions using this procedure. It is

crucial the relation to the GNS inequalities obtained by DelPino and

Dolbeault. This talk corresponds to two works in preparation together

with E. Carlen and A. Blanchet, and with E. Carlen and M. Loss.

Let $u \in W^{1,p}(\Omega,\R^N)$, $\Omega$ a bounded domain in

$\R^n$, be a minimizer of a convex variational integral or a weak solution to

an elliptic system in divergence form. In the vectorial case, various

counterexamples to full regularity have been constructed in dimensions $n

\geq 3$, and it is well known that only a partial regularity result can be

expected, in the sense that the solution (or its gradient) is locally

continuous outside of a negligible set. In this talk, we shall investigate

the role of the space dimension $n$ on regularity: In arbitrary dimensions,

the best known result is partial regularity of the gradient $Du$ (and hence

for $u$) outside of a set of Lebesgue measure zero. Restricting ourselves to

the partial regularity of $u$ and to dimensions $n \leq p+2$, we explain why

the Hausdorff dimension of the singular set cannot exceed $n-p$. Finally, we

address the possible existence of singularities in two dimensions.

We study the nonhomogeneous boundary value problem for the

Navier--Stokes equations

\[

\left\{ \begin{array}{rcl}

-\nu \Delta{\bf u}+\big({\bf u}\cdot \nabla\big){\bf u} +\nabla p&=&{0}\qquad \hbox{\rm in }\;\;\Omega,\\[4pt]

{\rm div}\,{\bf u}&=&0 \qquad \hbox{\rm in }\;\;\Omega,\\[4pt]

{\bf u}&=&{\bf a} \qquad \hbox{\rm on }\;\;\partial\Omega

\end{array}\right

\eqno(1)

\]

in a bounded multiply connected domain

$\Omega\subset\mathbb{R}^n$ with the boundary $\partial\Omega$,

consisting of $N$ disjoint components $\Gamma_j$.

Starting from the famous J. Leray's paper published in 1933,

problem (1) was a subject of investigation in many papers. The

continuity equation in (1) implies the necessary solvability

condition

$$

\int\limits_{\partial\Omega}{\bf a}\cdot{\bf

n}\,dS=\sum\limits_{j=1}^N\int\limits_{\Gamma_j}{\bf a}\cdot{\bf

n}\,dS=0,\eqno(2)

$$

where ${\bf n}$ is a unit vector of the outward (with respect to

$\Omega$) normal to $\partial\Omega$. However, for a long time

the existence of a weak solution ${\bf u}\in W^{1,2}(\Omega)$ to

problem (1) was proved only under the stronger condition

$$

{\cal F}_j=\int\limits_{\Gamma_j}{\bf a}\cdot{\bf n}\,dS=0,\qquad

j=1,2,\ldots,N. \eqno(3)

$$

During the last 30 years many partial results concerning the

solvability of problem (1) under condition (2) were obtained. A

short overview of these results and the detailed study of problem

(1) in a two--dimensional bounded multiply connected domain

$\Omega=\Omega_1\setminus\Omega_2, \;\overline\Omega_2\subset

\Omega_1$ will be presented in the talk. It will be proved that

this problem has a solution, if the flux ${\cal

F}=\int\limits_{\partial\Omega_2}{\bf a}\cdot{\bf n}\,dS$ of the

boundary datum through $\partial\Omega_2$ is nonnegative (outflow

condition).

The maps $u$ which are continuous in ${\mathbb R}^n$ and circle-valued are precisely the maps of the form $u=\exp (i\varphi)$, where the phase $\varphi$ is continuous and real-valued.

In the context of Sobolev spaces, this is not true anymore: a map $u$ in some Sobolev space $W^{s,p}$ need not have a phase in the same space. However, it is still possible to describe all the circle-valued Sobolev maps. The characterization relies on a factorization formula for Sobolev maps, involving three objects: good phases, bad phases, and topological singularities. This formula is the analog, in the circle-valued context, of Weierstrass' factorization theorem for holomorphic maps.

The purpose of the talk is to describe the factorization and to present a puzzling byproduct concerning sums of Dirac masses.

I will make a short review of some continous approximations to the Navier-Stokes equations, especially with the aim of introducing alpha models for the Large Eddy Simulation of turbulent flows.

Next, I will present some recent results about approximate deconvolution models, derived with ideas similar to image processing. Finally, I will show the rigorous convergence of solutions towards those of the averaged fluid equations.

We develop a theory based on relative entropy to show stabilityand uniqueness of extremal entropic Rankine-Hugoniot discontinuities forsystems of conservation laws (typically 1-shocks, n-shocks, 1-contactdiscontinuities and n-contact discontinuities of big amplitude), amongbounded entropic weak solutions having an additional strong traceproperty. The existence of a convex entropy is needed. No BV estimateis needed on the weak solutions considered. The theory holds withoutsmallness condition. The assumptions are quite general. For instance, thestrict hyperbolicity is not needed globally. For fluid mechanics, thetheory handles solutions with vacuum.

The fundamental models for lipid bilayers are curvature based and neglect the internal structure of the lipid layers. In this talk, we explore models with an additional order parameter which describes the orientation of the lipid molecules in the membrane and compare their predictions based on numerical simulations. This is joint work with Soeren Bartels (Bonn) and Ricardo Nochetto (College Park).

Hilbert Sixth Problem of Axiomatization of Physics is a problem of general nature and not of specific problem. We will concentrate on the kinetic theory; the relations between the Newtonian particle systems, the Boltzmann equation and the fluid dynamics. This is a rich area of applied mathematics and mathematical physics. We will illustrate the richness with some examples, survey recent progresses and raise open research directions.

In this talk we describe some recent work on shock

reflection problems for the potential flow equation. We will

start with discussion of shock reflection phenomena. Then we

will describe the results on existence, structure and

regularity of global solutions to regular shock reflection. The

approach is to reduce the shock reflection problem to a free

boundary problem for a nonlinear elliptic equation, with

ellipticity degenerate near a part of the boundary (the sonic

arc). We will discuss techniques to handle such free boundary

problems and degenerate elliptic equations. This talk is based

on joint works with Gui-Qiang Chen, and with Myoungjean Ba

• (a) nonconvergence for the unregularized mathematical problem or
• (b) nonuniqueness of the limit if it exists, or
• (c) limiting solutions only in the very weak form of a space time dependent probability distribution.

I will recount progress regarding the robustness of solitary waves in
nonintegrable Hamiltonian systems such as FPU lattices, and discuss 
a proof (with Shu-Ming Sun) of spectral stability of small
solitary waves for the 2D Euler equations for water of finite depth
without surface tension.

We investigate a dynamic model of two dimensional crystal growth

described by a forward-backward parabolic equation. The ill-posed

region of the equation describes the motion of corners on the surface.

We analyze a fourth order regularized version of this equation and

show that the dynamical behavior of the regularized corner can be

described by a traveling wave solution. The speed of the wave is found

by rigorous asymptotic analysis. The interaction between multiple

corners will also be presented together with numerical simulations.

This is joint work in progress with Fang Wan.

We consider the motion of a viscous incompressible fluid described by

the Navier-Stokes equations in a bounded cylinder with slip boundary

conditions. Assuming that $L_2$ norms of the derivative of the initial

velocity and the external force with respect to the variable along the

axis of the cylinder are sufficiently small we are able to prove long

time existence of regular solutions. By the regular solutions we mean

that velocity belongs to $W^{2,1}_2 (Dx(0,T))$ and gradient of pressure

to $L_2(Dx(0,T))$. To show global existence we prolong the local solution

with sufficiently large T step by step in time up to infinity. For this purpose

we need that $L_2(D)$ norms of the external force and derivative

of the external force in the direction along the axis of the cylinder

vanish with time exponentially.

Next we consider the inflow-outflow problem. We assume that the normal

component of velocity is nonvanishing on the parts of the boundary which

are perpendicular to the axis of the cylinder. We obtain the energy type

estimate by using the Hopf function. Next the existence of weak solutions is

proved.