# Seminars & Events for Analysis Seminar

##### Dynamics of the Ericksen-Leslie Model for Nematic Liquid Crystal Flow with General Leslie Stress

Consider the Ericksen-Leslie model for the flow of nematic liquid crystals in a bounded domain with general Leslie stress in the iso- and nonisothermal setting. We discuss recent local and global well-posedness results in the strong sense for this system and describe in addition the dynamical behaviour of its solutions. Our approach is based on the entropy principle and maximal $L^p$-estimates for the linearized system. It is remarkable that for these results no structural conditions on the Leslie coefficients are imposed and that in particular Parodi's relation is not being assumed. This is joint work with Jan Pruess.

##### On two dimensional gravity water waves with angled crests

In this talk, I will survey the recent understandings on the motion of water waves obtained via rigorous mathematical tools, this includes the evolution of smooth initial data and some typical singular behaviors. In particular, I will present our recent results on gravity water waves with angled crests.

##### On the motion of the free boundary of a self-gravitating incompressible fluid

**Please note special time and location of this talk. **The motion of the free boundary of an incompressible fluid body subject to its self gravitational force can be described by a free boundary problem of the Euler-Poisson system. This problem differs from the water wave problem in that the constant gravity in water waves is replaced by a nonlinear self-gravity. In this talk, we present some recent results on the well-posedness of this problem and give a lower bound on the lifespan of smooth solutions. In particular, we show that the Taylor sign condition always holds leading to local well-posedness, and for smooth data of size $\epsilon$ a unique smooth solution exists for time greater than or equal to $O(1/{\epsilon}^2)$.

##### Symplectic non-squeezing for Hamiltonian PDEs

In this talk, I will discuss symplectic non-squeezing for the nonlinear Klein-Gordon equation (NLKG) which can be (formally) regarded as an infinite dimensional Hamiltonian system. The symplectic phase space for this equation is at the critical regularity, and in this setting there is no global well-posedness nor any uniform control on the local time of existence for arbitrary initial data. We will present several non-squeezing results for the NLKG, including a conditional result which states that uniform bounds on the Strichartz norms of solutions for initial data in bounded subsets of the phase space imply global-in-time non-squeezing. The proofs rely on several approximation results for the flow, which we obtain using a combination of probabilistic and deterministic techniques.

##### Global nonlinear stability of Minkowski space for the massless Einstein--Vlasov system

Massless collisionless matter is described in general relativity by the massless Einstein--Vlasov system. I will present key steps in a proof that, for asymptotically flat Cauchy data for this system, sufficiently close to that of the trivial solution, Minkowski space, the resulting maximal development of the data exists globally in time and asymptotically decays appropriately. By appealing to the corresponding result for the vacuum Einstein equations, a monumental result first obtained by Christodoulou--Klainerman in the early '90s, the proof reduces to a semi-global problem. A key step is to gain a priori control over certain Jacobi fields on the mass shell, a submanifold of the tangent bundle of the spacetime endowed with the Sasaki metric.

##### A sharp counter example to local existence for Einstein equations in wave coordinates

We are concerned with how regular initial data have to be to ensure local existence for Einstein's equations in wave coordinates. Klainerman-Rodnianski and Smith-Tataru showed that there in general is local existence for data in H^s for s>2. We give example of data in H^2 for which there is no local solution in H^2. This is joint work with Boris Ettinger.

##### Lipschitz Metrics for Nonlinear Wave Equations

The talk is concerned with some classes of nonlinear wave equations: of first order, such as the Camassa-Holm equation, or of second order, as the variational wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$. In both cases, it is known that the equations determine a unique flow of conservative solutions within the natural ``energy" space $H^1(\mathbb{R})$. However, this flow is not continuous w.r.t.~the $H^1$ distance. Local well-posedness is usually recovered only on spaces with higher regularity. Our goal is to construct a new metric, which renders this flow uniformly Lipschitz continuous on bounded subsets of $H^1$. For this purpose, $H^1$ is given the structure of a Finsler manifold, where the norm of tangent vectors is defined in terms of an optimal transportation problem.

##### Stability and instability results for scalar waves on general asymptotically flat spacetimes

In the first part of this talk, we will prove a logarithmic decay result for solutions to the scalar wave equation $\square_{g}\psi=0$ on general asymptotically flat spacetimes $(\mathcal{M},g)$, possibly bounded by an event horizon with positive surface gravity and having a small ergosphere, provided a uniform energy boundedness estimate holds on $(\mathcal{M},g)$. This result generalises of a result of Burq for the wave equation on the complement of an arbitrary compact obstacle in flat space. The methods developed for the proof of this result will then be applied, in the second part of the talk, in obtaining a rigorous proof of Friedman's ergosphere instability for scalar waves.