Past PDE CDT Lunchtime Seminar

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. 

We determine the hydrodynamic limit of some kinetic equations with either stochastic Vlasov force term or stochastic collision kernel. We obtain stochastic second-order parabolic equations at the limit. In the regime we consider, we also observe (or do not observe) some phenomena of enhanced diffusion. Joint works with Nils Caillerie, Arnaud Debussche, Martina Hofmanová.
 

I will discuss the specific long-time behaviour of kinetic solutions to stochastic conservation laws with non-linear multiplicative noises.

We consider the Euler–Voigt equations in a bounded domain as an approximation for the 3D Euler equations. We adopt suitable physical conditions and show that the solutions of the Voigt equations are global, do not smooth out the solutions and converge to the solutions of the Euler equations, hence they represent a good model.

We study vortex sheets for the relativistic Euler equations in three-dimensional Minkowski spacetime. The problem is a nonlinear hyperbolic problem with a characteristic free boundary. The so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. A necessary condition for the weak stability is obtained by analyzing roots of the Lopatinskii determinant associated to the linearized problem. Under such stability condition,  we prove short-time existence and nonlinear stability of relativistic vortex sheets by the Nash-Moser iterative scheme.

We consider the free boundary problem for 2D current-vortex sheets in ideal incompressible magneto-hydrodynamics near the transition point between the linearized stability and instability. In order to study the dynamics of the discontinuity near the onset of the instability, Hunter and Thoo have introduced an asymptotic quadratically nonlinear integro-differential equation for the amplitude of small perturbations of the planar discontinuity. 
In this talk we present our results about the well-posedness of the problem in the sense of Hadamard, under a suitable stability condition, that is the 
local-in-time existence in Sobolev spaces and uniqueness of smooth solutions to the Cauchy problem, and the strong continuous dependence on the data in the same topology.
Joint works with: Alessandro Morando and Paola Trebeschi.
 

Consider a family of uniformly bounded $W^{2,p}$ isometric immersions of an $n$-dimensional (semi-) Riemannian manifold into (resp., semi-) Euclidean spaces. Are the weak limits still isometric immersions?

We answer the question in the affirmative for $p>n$ in the Riemannian case, by exploiting the div-curl structure of the Gauss-Codazzi-Ricci equations, which describe the curvature flatness of the isometric immersions. Along the way a generalised div-curl lemma in Banach spaces is established. Moreover, the endpoint case $p=n=2$ is settled. 

In the semi-Riemannian case we reduce the problem to the weak continuity of H. Cartan's structural equations in $W^{1,p}_{\rm loc}$, which is proved by a generalised compensated compactness theorem relating the weak continuity of quadratic forms to the principal symbols of differential constraints. Again for $p>n$ we obtain the weak rigidity. The case of degenerate hypersurfaces are also discussed, as well as connections to PDEs in fluid dynamics.

We report on recent developments in the study of nonlocal operators. The central object of the talk are quadratic forms similar to those that define Sobolev spaces of fractional order. These objects are naturally linked to Markov processes via the theory of Dirichlet forms. We provide regularity results for solutions to corresponding integrodifferential equations. Our emphasis is on forms with singularand anisotropic measures. Some of the objects under consideration are related to the Boltzmann equation, which leads to an interesting question of comparability of quadrativ forms. The talk is based on recent results joint with B. Dyda and with K.-U. Bux and T. Schulze.

The non-local Fisher KPP equation is used to model non-local interaction and competition in a population. I will discuss recent work on solutions of this equation with a compactly supported initial condition, which strengthens results on the spreading speed obtained by Hamel and Ryzhik in 2013. The proofs are probabilistic, using a Feynman-Kac formula and some ideas from Bramson's 1983 work on the (local) Fisher KPP equation.

Systems that have more than one conserved quantity (i.e. energy plus momentum, density etc.), can exhibit quite interesting temperature profiles in non-equilibrium stationary states. I will present some numerical experiment and mathematical result. I will also expose some other connected problems, always concerning thermal boundary conditions in hydrodynamic limits.
 

Patlak-Keller-Segel equations 
\[
\begin{aligned}
u_t - L u &= - \mathop{\text{div}\,} (u \nabla v) \\
v_t - \Delta v &= u,
\end{aligned}
\]
where L is a dissipative operator, stem from mathematical chemistry and mathematical biology.
Their variants describe, among others, behaviour of chemotactic populations, including feeding strategies of zooplankton or of certain insects. Analytically, Patlak-Keller-Segel equations reveal quite rich dynamics and a delicate global smoothness vs. blowup dichotomy. 
We will discuss smoothness/blowup results for popular variants of the equations, focusing on the critical cases, where dissipative and aggregative forces seem to be in a balance. A part of this talk is based on joint results with Rafael Granero-Belinchon (Lyon).

Tbd

We prove higher differentiability of bounded local minimizers to some degenerate functionals satisfying anisotropic growth conditions. In the two-dimensional case we also study the Lipschitz regularity of such minimizers without any limitation on the exponents of anisotropy.

Tbd