In this talk we discuss the recent proof for the existence of $C^{1,1}$ isometric immersions of several classes of negatively curved surfaces into $\R^3$, including the Lobachevsky plane, metrics of helicoid type and a one-parameter family of perturbations of the Enneper surface. Our method, following Chen--Slemrod--Wang and Cao--Huang--Wang, is to transform the Gauss--Codazzi equations into a system of hyperbolic balance laws, and prove the existence of weak solutions by finding the invariant regions. In addition, we provide further characterisation of the $C^{1,1}$ isometrically immersed generalised helicoids/catenoids established in the literature.

# Past PDE CDT Lunchtime Seminar

Many problems arising in Physics can be posed as minimisation of energy functionals under linear partial differential constraints. For example, a prototypical example in the Calculus of Variations is given by functionals defined on curl-free fields, i.e., gradients. Most work done subject to more general constraints met significant difficulty due to the lack of associated potentials. We show that under the constant rank assumption, which holds true of almost all examples of constraints investigated in connection with lower-semicontinuity, linear constraints admit a potential in frequency space. As a consequence, the notion of A-quasiconvexity, which involves testing with periodic fields leading to difficulties in establishing sufficiency for weak sequential lower semi-continuity, can be tested against compactly supported fields. We will indicate how this can simplify the general framework.

We present some recent results on the regularity criteria for weak solutions to the incompressible Navier--Stokes equations (NSE) in 3 dimensions. By the work of Constantin--Fefferman, it is known that the alignment of vorticity directions is crucial to the regularity of NSE in $\R^3$. In this talk we show a boundary regularity theorem for NSE on curvilinear domains with oblique derivative boundary conditions. As an application, the boundary regularity of incompressible flows on balls, cylinders and half-spaces with Navier boundary condition is established, provided that the vorticity is coherently aligned up to the boundary. The effects of vorticity alignment on the $L^q$, $1<q<\infty$ norm of the vorticity will also be discussed.

When nematic liquid crystal droplets are produced in the form or tori (or such is the shapes of confining cavities), they may be called toroidal nematics, for short. When subject to degenerate planar anchoring on the boundary of a torus, the nematic director acquires a natural equilibrium configuration within the torus, irrespective of the values of Frank's elastic constants. That is the pure bend arrangement whose integral lines run along the parallels of all inner deflated tori. This lecture is concerned with the stability of such a universal equilibrium configuration. Whenever its stability is lost, new equilibrium configurations arise in pairs, the members of which are symmetric and exhibit opposite chirality. Previous work has shown that a rescaled saddle-splay constant may be held responsible for such a chiral symmetry breaking. We shall show that that is not the only possible instability mechanism and, perhaps more importantly, we shall attempt to describe the qualitative properties of the equilibrium nematic textures that prevail when the chiral symmetry is broken.

The basic mathematical models that describe the behavior of fluid flows date back to the eighteenth century, and yet many phenomena observed in experiments are far from being well understood from a theoretical viewpoint. For instance, especially challenging is the study of fundamental stability mechanisms when weak dissipative forces (generated, for example, by molecular friction) interact with advection processes, such as mixing and stirring. The goal of this talk is to have an overview on recent results on a variety of aspects related to hydrodynamic stability, such as the stability of vortices and laminar flows, the enhancement of dissipative force via mixing, and the statistical description of turbulent flows.

The Boltzmann equation is a well-studied PDE that describes the statistical evolution of a dilute gas of spherical particles. However, much less is known — both from the physical and mathematical viewpoints — about the Boltzmann equation for non-spherical particles. In this talk, we present some new results on the non-existence and non-uniqueness of weak solutions to the initial-boundary value problem for N non-spherical particles which have importance for the Boltzmann equation.

We present work which was done jointly with L. Saint-Raymond (ENS Lyon), and also with P. Palffy-Muhoray (Kent State), E. Virga (Pavia) and X. Zheng (Kent State).

In this talk, we propose a model describing the growth of tree stems and vine, taking into account also the presence of external obstacles. The system evolution is described by an integral differential equation which becomes discontinuous when the stem hits the obstacle. The stem feels the obstacle reaction not just at the tip, but along the whole stem. This fact represents one of the main challenges to overcome, since it produces a cone of possible reactions which is not normal with respect to the obstacle. However, using the geometric structure of the problem and optimal control tools, we are able to prove existence and uniqueness of the solution for the integral differential equation under natural assumptions on the initial data.

Recent results on viscous conservation laws with nonlocal flux will be presented. Such models contain, as a particular example, the celebrated parabolic-elliptic Keller-Segel model of chemotaxis. Here, global-in-time solutions are constructed under (nearly) optimal assumptions on the size of radial initial data. Moreover, criteria for blowup of solutions in terms of their local concerntariotions will be derived.

We will discuss nonlinear cross-diffusion models describing cell motility of two distinct populations. The continuum PDE model is derived systematically from a stochastic discrete model consisting of impenetrable diffusing spheres. In this talk, I will outline the derivation of the cross-diffusion model, discuss some of its features such as the gradient-flow structure, and show numerical results comparing the discrete stochastic system to the derived model.

I will introduce a McKean—Vlasov problem arising from a simple mean-field model of interacting neurons. The equation is nonlinear and captures the positive feedback effect of neurons spiking. This leads to a phase transition in the regularity of the solution: if the interaction is too strong, then the system exhibits blow-up. We will cover the mathematical challenges in defining, constructing and proving uniqueness of solutions, as well as explaining the connection to PDEs, integral equations and mathematical finance.