Free Boundary Problems: Lecture Series and Recent Advances in Theory and Applications

Conference Schedule

Titles and Abstracts

Long-range capillarity theory
Enrico Valdinoci (Mini-course, 05/30 9:30 am - 10:30 am and 11:00 am - 12:00 pm)
Capillarity theory aims at understanding the displacement of a liquid droplet in a container in view of surface tension. In principle, however, even the modelization of surface tension itself is a rather delicate business, being the average outcome of the attractive forces between molecules of the same type and reflecting the interfaces between the droplet, the air, and the container. It would be therefore desirable to understand surface tension directly from the possibly long-range cohesive forces of molecules. In this spirit, we describe some recent results motivated by a nonlocal theory of capillarity, as related to the formation of droplets due to long-range interaction potentials, in connection with the theory of nonlocal minimal surfaces. We will discuss the notion of contact angle in this setting, considering a nonlocal version of the classical Young's Law, together with some regularity and asymptotic properties.

Free-surface hydrodynamic models and boundary interactions
Roberto Camassa (Mini-course, 05/31 9:30 am - 10:30 am and 11:00 am - 12:00 pm, 06/01 9:30 am - 10:30 am)
Time and again, models of free surface flows have played a pivotal role in identifying phenomena that would be hard to see and analyze from full general theories, particularly when interactions with rigid boundaries play a significant role in the dynamics.  This mini-course will provide selected examples of derivation  and applications of such "minimal" models, and illustrate their effectiveness at both qualitative and quantitative levels (as well their shortcomings) with "simple" laboratory experiments.

The geometry of maximal development for multi-D compressible Euler
Vlad Vicol (Mini-course, 06/01 11:00 am - 12:00 pm, 06/02 9:00 am - 10:00 am and 11:30 am - 12:30 pm)
Solutions to the compressible Euler equations of gas dynamics exhibit shock waves. These are spacetime-codimension-2 surfaces of discontinuity that can emerge in finite time from smooth initial data, and dynamically evolve according to the Rankine-Hugoniot (RH) jump conditions. In addition to the physical variable unknowns – velocity, density, and energy – the location of the shock surface is also an unknown.  Christodoulou laid out a program for constructing unique shock wave solutions to the Euler equations in multiple space dimensions. Starting from smooth initial data, the first step is called “shock formation” in which smooth initial data is evolved up to a spacetime-codimension-2 surface of “first singularities”. If we think of a multi-dimensional wave steeping in a direction n, then this surface of first singularities can be parameterized by time, and the transverse spatial directions to n. We call this surface the surface of “pre-shocks”. The objective of shock formation is to fully classify this spacetime-codimension-2 surface of pre-shocks and give a complete description of the physical variables in an open neighborhood of this preshock. In standard Eulerian coordinates, this pre-shock is the valley of a singular hypersurface and the physical variables form Holder-1/3 cusp singularities. These lectures are based on joint work with Steve Shkoller in which besides “shock formation” we also establish the maximal Cauchy development for smooth initial data.

(Non)local logistic equations with Neumann conditions
Serena Dipierro (05/30 2:00 pm - 2:50 pm)
We consider a problem of population dynamics modelled on a logistic equation with both classical and nonlocal diffusion, possibly in combination with a pollination term. The environment considered is a niche with zero-flux, according to a new type of Neumann condition. We discuss the situations that are more favourable for the survival of the species, in terms of the first positive eigenvalue and we also analyze the role played by the optimization strategy in the distribution of the resources.

Generic global existence for the modified SQG equation
Javier Gomez-Serrano (05/30 4:00 pm - 4:50 pm)
In this talk we will present a construction of global existence of small solutions of the modified SQG equations, close to the disk. The proof uses KAM theory and a Nash-Moser argument, and does not involve any external parameters. We moreover prove that this phenomenon is generic: most solutions satisfy it. Joint work with Alex Ionescu and Jaemin Park.

Conformal mappings, dynamics on various Riemann surface sheets and integrability of surface dynamics
Pavel Lushnikov (05/31 2:00 pm - 2:50 pm)
A fully nonlinear dynamics for potential flow of ideal incompressible fluid with a free surface is considered in two dimensional geometry. Arbitrary large surface waves and motions can be efficiently characterized through a time-dependent conformal mapping of a fluid domain into the lower complex half-plane. We reformulate the exact Eulerian dynamics through a non-canonical nonlocal Hamiltonian system for the pair of new conformal variables. The corresponding non-canonical Poisson bracket is non-degenerate, i.e. it does not have any Casimir invariant. Any two functionals of the conformal mapping commute with respect to the Poisson bracket. We also consider a generalized hydrodynamics for two components of superfluid Helium with a free surface which has the same non-canonical Hamiltonian structure. In both cases the fluid dynamics is fully characterized by the complex singularities in the upper complex half-plane of the conformal map and the complex velocity. Analytical continuation through the branch cuts generically results in the Riemann surface with infinite number of sheets including Stokes wave, An infinite family of solutions with moving poles are found on the Riemann surface. Residues of poles are the constants of motion. These constants commute with each other in the sense of underlying non-canonical Hamiltonian dynamics which provides an argument in support of the conjecture of complete Hamiltonian integrability of surface dynamics. If we consider initial conditions with short branch cuts then fluid dynamics is reduced to the complex Hopf equation for the complex velocity coupled with the complex transport equation for the conformal mapping. These equations are fully integrable by characteristics producing the infinite family of solutions, including the pairs of moving square root branch points. The solutions are compared with the simulations of the full Eulerian dynamics giving excellent agreement. Beyond short branch cut approximation we obtain a formation of a pair of new branch points in 2nd sheet of Riemann surface during infinitely small time. A crossing a branch cut connecting these two new branch points into  3rd Sheet of Riemann reveals even more branch points there etc resulting in infinite number of branch points in infinite number of sheets. An analysis of these branch points suggests a way to address a fully nonlinear dynamics.

Classical wave-particle duality
Pedro Saenz (05/31 3:00 pm - 3:50 pm)
A millimetric liquid droplet may walk across the surface of a vibrating fluid bath, self-propelled through a resonant interaction with its own guiding wave field. By virtue of the coupling with their wave fields, these walking droplets, or `walkers', extend the range of classical mechanics to include certain features previously thought to be exclusive to the quantum realm. In this talk, we will combine experiments, simulations and theory to discuss a number of hydrodynamic quantum analogs, including orbital quantization in a rotating frame, hydrodynamic spin lattices, and the absence of diffusion over random topographies.

Derivation of the 1-D Groma-Balogh equations from the Peierls-Nabarro model
Stefania Patrizi (05/31 4:00 pm - 4:50 pm)
We consider a semi-linear integro-differential equation in dimension one associated to the half Laplacian whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls-Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a fully nonlinear integro-differential equation which is a model for the macroscopic crystal plasticity with density of dislocations. This leads to the formal derivation of the 1-D Groma-Balogh equations, a popular model describing the evolution of the density of  parallel straight dislocation lines.  One of the main difficulties is that we allow dislocations to have opposite orientation and so we have to deal with collisions of them. This is a joint work with Tharathep Sangsawang.

Orientation mixing in active suspensions
Michele Coti Zelati (06/01 2:00 pm - 2:50 pm)
We study a popular kinetic model introduced by Saintillan and Shelley for the dynamics of suspensions of active elongated particles. We focus on the linear analysis of incoherence, that is on the linearized equation around the uniform distribution, in the regime of parameters corresponding to spectral (neutral) stability. We show that in the absence of rotational diffusion, the suspension experiences a mixing phenomenon similar to Landau damping. We show that this phenomenon persists for small rotational diffusion, and is combined with an enhanced dissipation at time scale at a faster time scale than the diffusive one.

Chemotaxis phenomena and Hele-Shaw free boundary problems
Antoine Mellet (06/01 3:00 pm - 3:50 pm)
The classical (parabolic-elliptic) Keller-Segel model for chemotaxis leads to concentration and formation of regions with high density of organisms - and in many regimes to finite time blow-up. In order to study the long time dynamic of these high density regions, we consider a model in which blow-ups are prevented either by some nonlinear diffusion or by a hard incompressibility constraint (which requires the introduction of a pressure term, as in fluid mechanics). We then show that the resulting model leads to phase separation, that is the formation of patches (sets where the density is maximal). The evolution of these patches can then be described by some free boundary problems of Hele-Shaw type. In particular, we will prove that the effect of the attractive chemotaxis potential is (in some asymptotic regime) comparable to that of surface tension.

The strong Onsager conjecture
Matthew Novack (06/01 4:00 pm - 4:50 pm)
The phenomenon of anomalous dissipation in turbulence predicts the existence of solutions to the incompressible Euler equations that enjoy regularity consistent with Kolmogorov’s 4/5 law and satisfy a local energy inequality. The "strong Onsager conjecture" asserts that such solutions do indeed exist. In this talk, we will discuss the background and motivation behind the strong Onsager conjecture.  In addition, we outline a construction of solutions with regularity (nearly) consistent with the 4/5 law, thereby proving the conjecture in the natural L^3 scale of Besov spaces.  This is based on joint work with Hyunju Kwon and Vikram Giri.

Kinetic shock profiles for the Landau equation
Dallas Albritton (06/02 10:15 am - 11:15 am)
Compressible Euler solutions develop jump discontinuities known as shocks. However, physical shocks are not, strictly speaking, discontinuous. Rather, they exhibit an internal structure which, in certain regimes, can be represented by a smooth function, the shock profile. We demonstrate the existence of weak shock profiles to the kinetic Landau equation. Joint work with Jacob Bedrossian (UCLA) and Matthew Novack (Purdue University).