ON DEMAND International PDE Conference 2022 - DAY THREE
Prof. Alessio Figalli - ETH Zürich
Stable and finite Morse index solutions to semilinear elliptic equations
Stable and finite Morse index solutions to semilinear elliptic PDEs appear in several problems. It is known since the 1970's that, in dimension n > 9, there exist singularstable solutions. In this talk, I will describe a recent work with Cabré, Ros-Oton, and Serra, where we prove that stable solutions in dimension n ≤ 9 are smooth. This answers a famous open problem posed by Brezis, concerning the regularity of extremal solutions to the Gelfand problem. Also, I will discuss a recent analog result with Zhang for finite Morse index solutions.
Prof. Mikhail Feldman - University of Wisconsin-Madison
Shock reflection problems: existence, uniqueness and stability of global solutions
In this talk we will start with discussion of shock reflection phenomena, and von Neumann conjectures on transition between regular and Mach reflections. Then we describe the results on existence, uniqueness, stability and regularity of global solutions to shock reflection for potential flow, and discuss the techniques. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation in self-similar coordinates, with ellipticity degenerate near a part of the boundary (the sonic arc). We will also discuss the open problems in the area.
Further Research:
G.-Q. Chen, M. Feldman, The Mathematics of Shock reflection-diffraction and von Neumann's conjectures, Annals of Mathematics Studies, 197. Princeton University Press, Princeton, NJ, 2018. xiv+814 pp.
Prof. Kenneth H. Karlsen - University of Oslo
A singular limit problem for stochastic conservation laws
*No presentations slides published.
We investigate a singular limit problem for stochastic conservation laws with discontinuous flux, perturbed by vanishing diffusion—dynamic capillarity terms. Our convergence arguments use kinetic formulations, H-measures and velocity averaging for stochastic transport equations, and a.s. representations of random variables in quasi-Polish spaces. This talk is based on joint work M. Kunzinger and D. Mitrovic.
Prof. Michael Struwe - ETH Zürich
Plateau flow, alias the heat flow for half-harmonic maps
Using the Millot-Sire interpretation of the half-Laplacian on $S1$ as the Dirichlet-to-Neumann operator for the Laplace equation on the ball $B$, we devise a classical approach to the heat flow for half-harmonic maps from $S1$ to a closed target manifold $N\subset\R^n$, recently studied by Wettstein, and for arbitrary finite-energy data we obtain a result fully analogous to the classical results for the harmonic map heat flow of surfaces and in similar generality.
When $N$ is a smoothly embedded, oriented closed curve $\Gamma\subset\R^n$ the half-harmonic map heat flow may be viewed as an alternative gradient flow for the Plateau problem of disc-type minimal surfaces.
Prof. James Glimm - Stony Brook University
Entropy Admissibility and Turbulent Structure for Fluids
We show that a class of solutions defined by Lebesgue measure on phase space maximizes the rate of entropy production. Entropy is defined as the log volume of a surface of constant energy. Energy itself is dependent on details of the physics model, and for single fluid incompressible flow is the energy of velocity fluctuations and of vorticity fluctuations.
Purturbation theory suggested by quantum field theory is defined. It is believed to be asymptotic, not convergent. We show that resummation of the series converges. Infinite moments of turbulence require renormalization in this series, which is accomplished through use of Sobolev norms of negative index.
Insight into the nature of turbulence is derived from this series in the form of surfaces in 3-space (vortex spheres, tori of various genus, knotted and twised), describes the states of fully developed turbulent flow.
Prof. Irene Fonseca - Centre for Nonlinear Analysis / Carnegie Mellon University
Phase Separation in Heterogeneous Media
Modern technologies and biological systems, such as, temperature-responsive polymers and lipid rafts, respectively, take advantage of engineered inclusions, or natural heterogeneities of the medium, to obtain novel composite materials with specific physical properties. To model such situations by using a variational approach based on the gradient theory, the potential and the wells have to depend on the spatial position, even in a discontinuous way, and different regimes should be considered. In the case where the scale of the small heterogeneities is of the same order of the scale governing the phase transition, the interaction between homogenization and the phase transitions process leads to an anisotropic interfacial energy. In the case where the heterogeneities of the material are of a larger scale than that of the diffuse interface between different phases, it is noted that there is no macroscopic phase separation and that thermal fluctuations play a role in the formation of nanodomains.
In addition, a characterization of the large-scale limiting behavior of viscosity solutions to non-degenerate and periodic Eikonal equations in half-spaces is given.
This is joint work with Riccardo Cristoferi (Radboud University, The Netherlands), Likhit Ganedi (CMU, USA), Adrian Hagerty (USA), Cristina Popovici (USA), Rustum Choksi (McGill, Canada), Jessica Lin (McGill, Canada), and Raghavendra Venkatraman (NYU, USA).
Prof. Robert V. Kohn - Courant Institute of Mathematical Sciences / New York University
Mathematical analysis of some devices made using epsilon-near-zero materials.
A body of literature has emerged concerning the design of devices made using "epsilon-near-zero" (ENZ) materials. The underlying mathematics is linear -- it uses the time-harmonic Maxwell's equations (epsilon is the dielectric permittivity). In a transverse-magnetic setting, the governing PDE reduces to a divergence-form Helmholtz equation, - div( a(x) grad u) = k^2 u, in which the coefficient a(x) is 1/epsilon. One sees quickly what is special about ENZ materials: if epsilon is nearly 0 in some region, then a(x) is nearly infinite, and the function u is nearly constant. To understand the robustness of ENZ devices, it is important to understand the leading-order corrections to the ENZ limit. This leads to interesting questions of both analysis and optimal design. This is joint work with Raghavendra Venkatraman.