We construct maximal surfaces with Neumann boundary conditions in Minkowski space using mean curvature flow. In particular we find curvature conditions on a boundary manifold so that mean curvature flow may be shown to exist for all time, and give conditions under which the maximal hypersurfaces are stable under the flow.

# Past PDE CDT Lunchtime Seminar

We present a new model for multi-agent dynamics where each agent is described by its position and body attitude: agents travel at a constant speed in a given direction and their body can rotate around it adopting different configurations. Agents try to coordinate their body attitudes with the ones of their neighbours. This model is inspired by the Vicsek model. The goal of this talk will be to present this new flocking model, its relevance and the derivation of the macroscopic equations from the particle dynamics.

After recalling some backgrounds and motivations, we'll report some recent results on the optimal L^2 extensions and multiplier ideal sheaves, with emphasizing the close relations between SCV and PDE.

After a brief review on the classical Prandtl system, we introduce our recent work on the well-posedness and high Reynolds numbers limit for the MHD boundary layer that shows the tangential magnetic field stabilizes the boundary layer. And then we will discuss some instability phenomena of the shear flow for both the classical Prandtl and MHD boundary layer systems. The talk includes some recent joint works with Chengjie Liu, Yaguang Wang on the classical Prandtl equation, and with Chengjie Liu and Feng Xie on the magnetohydrodynamic boundary layer.

We will discuss models for vehicular traffic flow on networks. The models include both the Lighthill-Whitham-Richards (LWR) model and Follow-the-Leader (FtL) models.

The emphasis will be on the Braess paradox in which adding a road to a traffic network can make travel times worse for all drivers.

In addition we will present a novel proof how FtL models approximate the LWR model in case of heavy traffic.

I will talk about the fluid equations used to model pedestrian motion and traffic. I will present the compressible-incompressible Navier-Stokes two phase system describing the flow in the free and in the congested regimes, respectively. I will also show how to approximate such system by the compressible Navier-Stokes equations with singular pressure for the fixed barrier densities and also some recent developments for the barrier densities varying in the space and time.

This is a talk based on several papers in collaboration with: D. Bresch, C. Perrin, P. Degond, P. Minakowski, and L. Navoret.

Divergence-measure fields are $L^{p}$-summable vector fields on $\mathbb{R}^{n}$ whose divergence is a Radon measure. Such vector fields form a new family of function spaces, which in a sense generalize the $BV$ fields, and were introduced at first by Anzellotti, before being rediscovered in the early 2000s by many authors for different purposes.

Chen and Frid were interested in the applications to the theory of systems of conservation laws with the Lax entropy condition and achieved a Gauss-Green formula for divergence-measure fields, for any $1 \le p \le \infty$, on open bounded sets with Lipschitz deformable boundary. We show in this talk that any Lipschitz domain is deformable.

Later, Chen, Torres and Ziemer extended this result to the sets of finite perimeter in the case $p = \infty$, showing in addition that the interior and exterior normal traces of the vector field are essentially bounded functions.

The Gauss-Green formula for $1 \le p \le \infty$ has been also studied by Silhavý on general open sets, and by Schuricht on compact sets. In such cases, the normal trace is not in general a summable function: it may even not be a measure, but just a distribution of order 1. However, we can show that such a trace is the limit of the integral of classical normal traces on (smooth) approximations of the integration domain.

According to a classical result of Bertoin (1998), if the initial data for Burgers equation is a Levy Process with no positive jump, then the same is true at later times, and there is an explicit equation for the evolution of the associated Levy measures. In 2010, Menon and Srinivasan published a conjecture for the statistical structure of solutions to scalar conservation laws with certain Markov initial conditions, proposing a kinetic equation that should suffice to describe the solution as a stochastic process in x with t fixed (or in t with x fixed). In a joint work with Dave Kaspar, we have been able to establish this conjecture. Our argument uses a particle system representation of solutions.

The Constantin-Lax-Majda model is a 1d system which shares certain features (related to vortex stretching) with the 3d Euler equation. The model is explicitly solvable and exhibits finite-time blow-up for an open subset of smooth initial data. In 1990s De Gregorio suggested adding a transport term to the system, which is analogous to the transport term in the Euler equation. It turns out the transport term has some regularizing effects, which we will discuss in the lecture.

I will address the study of decay rates of solutions to dissipative equations. The characterization of these rates will first be given for a wide class of linear systems by the decay character, which is a number associated to the initial datum that describes the behavior of the datum near the origin in frequency space. The understanding of the behavior of the linear combined with the decay character and the Fourier Splitting method is then used to obtain some upper and lower bounds for decay of solutions to appropriate dissipative nonlinear equations, both in the incompressible and compressible case.