Schedule for: 22w5101 - Recent Progress in Kinetic and Integro-Differential Equations

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

This talk concerns the relativistic Boltzmann equation without angular cutoff. Global in time unique solutions close to equilibrium were built in Jang-Strain (Ann. PDE, 2022). However the non-negativity of those solutions remained an open problem. Now we establish local wellposedness and non-negativity for solutions to the special relativistic Boltzmann equation without angular cutoff. The solution lies in an appropriate fractional Sobolev type space. In addition, as a corollary our results provide a rigorous proof for the non-negativity of the classical solutions of Jang-Strain (Ann. PDE, 2022) in the perturbative setting nearby the relativistic Maxwellian. This is a joint work with Jin Woo Jang.

In this talk, we revisit the grazing collision limit from the Boltzmann equation to the Landau equations utilizing their recent reinterpretations as gradient flows. We utilize the framework of Γ-convergence of gradient flows technique introduced by Sandier and Serfaty.

We consider an infinitely extended (screened) Vlasov-Poisson plamsa on $\R3$ coupled to a point charge. The well-posedness of this problem has been studied thoroughly in recent years, while little is known about Landau damping in this setting. Contrary to other results in nonlinear Landau damping, the dynamics are driven by the non-trivial electric field $E[F]$ of the plasma, even for large times $t\gg 1$. We rigorously prove the validity of the `stopping power theory' in physics, which predicts a decrease of the velocity $V(t)$ of the point charge given by $\rh{\dot{V} \sim -|V|^{-3} V}$. This formula was first predicted by Bohr (1915), and has since become a standard tool in physics. Our result holds for all initial velocities $V_0>0$ larger than a threshold value $\ol{V}>0$ and remains valid until (i) the particle slows down to velocity $\ol{V}$, or (ii) the time is exponentially long compared to the velocity of the charge, i.e. $T= \exp(V(T))$.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will discuss some recent results on Sobolev estimates for fractional elliptic and parabolic equations with or without weights. We considered equations with time fractional derivatives of the Caputo type, or with nonlocal derivatives in the space variables, or both. This is based on joint work with Doyoon Kim, Pilgu Jung (Korea University), and Yanze Liu (Brown).

The non-cutoff Boltzmann equation is known to have a regularizing effect on solutions because of the fractional diffusion induced by the collision operator. This suggests that classical solutions should exist even for irregular (e.g. lying in a zeroth-order space) initial data. Such an existence result has previously been shown in the close-to-equilibrium and space-homogeneous settings, and in this talk, we discuss the extension to the general case: spatially varying initial data that is far from equilibrium. Our classical solutions have initial data in an L^\infty(R^6) space with mild polynomial weight in the velocity variable. This talk is based on joint work with Henderson and Tarfulea.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, we present a recent result that shows an operator splitting scheme for the multiscale Levy-Fokker-Planck equation is asymptotic preserving (AP). The analysis is carried out by separating the parameter domain, which generalizes the traditional AP method when estimates are performed in a uniform way.

The kinetic description of granular media is through a Boltzmann type equation with a nonlocal nonlinear collision operator that is of the same form as in the continuum equation for aggregation diffusion dynamics. While the singular behavior of the continuum equation is well studied in the literature, the extension to the kinetic equation is highly nontrivial. The main question is whether the singularity formed in velocity direction will be enhanced or mitigated by the shear. Here we present some preliminary study by a careful numerical investigation and a heuristic argument.

In this talk, I'll describe a deterministic particle method for the weighted porous medium equation. The key idea behind the method is to approximate the PDE via certain highly nonlocal continuity equations. The formulation of the method and the proof of its convergence rely on the Wasserstein gradient flow formulation of the aforementioned PDEs. This is based on joint work with Katy Craig, Karthik Elamvazhuthi, and Matt Haberland.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We provide a new boundary estimate on the vorticity for the incompressible Navier-Stokes equation endowed with no-slip boundary condition. The estimate is rescalable through the inviscid limit. It provides a control on the layer separation at the inviscid Kato double limit, which is consistent with the Layer separation predictions via convex integration.

The Landau and Boltzmann equations are nonlocal, nonlinear equations for which (large data) global well-posedness is an extremely difficult problem that is nearly completely open. Recently, with Snelson and Tarfulea, we have pursued a program to understand a more tractable, related question: what are the weakest conditions for which local well-posedness holds and what quantities prevent blow-up at finite times? I will discuss two pieces of this program that were established in collaboration with Weinan Wang: (1) existence of solutions starting from initial data that decays "slowly" in velocity -and- (2) "irregular" Schauder estimates and their application to uniqueness of solutions with almost no initial regularity in velocity. The talk will focus on interesting pieces of the proofs (instead of slogging through a priori estimates!) such as a simple version of the proof in (1) in certain parameter regimes.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk deals with the space-homogenous Landau equation with very soft potentials, including the Coulomb case. This nonlinear equation is of parabolic type with diffusion matrix given by the convolution product of the solution with the matrix $a_{ij}(z)=|z|^\gamma (|z|^2\delta_{ij}−z_iz_j)$ for $\gamma \in [−3,−2)$. We derive local truncated entropy estimates and use them to establish two facts. Firstly, we prove that the set of singular points (in time and velocity) for the weak solutions constructed as in [C. Villani, Arch. Rational Mech. Anal. 143 (1998), 273-307] has zero $P^{m_∗}$ parabolic Hausdorff measure with $m_∗ :=1+\frac{5}{2} |2+\gamma|$. Secondly, we prove that if such a weak solution is axisymmetric, then it is smooth away from the symmetry axis. In particular, radially symmetric weak solutions are smooth away from the origin.

In this talk we will discuss a generalization of the Boltzmann equation that takes into account both binary and ternary interactions of particles. In particular, the first part of the talk will focus on the joint work with Ioakeim Ampatzoglou on a derivation of a binary-ternary Boltzmann equation describing the kinetic properties of a dense hard spheres gas, where particles undergo either binary or ternary instantaneous interactions, while preserving momentum and energy. This will be followed by a discussions of a recent result obtain in collaboration with Ioakeim Ampatzoglou, Irene Gamba and Maja Tasković on generation and propagation in time of polynomial and exponential moments, as well as global well-posedness of the homogeneous binary-ternary Boltzmann equation. In particular, we show that this equation is “better behaved” compared to the homogeneous version of the (binary) Boltzmann equation or the homogeneous version of the purely ternary Boltzmann equation.

Building on an improved local existence (and smoothing) result for the inhomogeneous Boltzmann equation, we show that, under mild assumptions on the initial data, the smooth solution obtained is unique in a broad class of weak solutions (L^1 in time, L^\infty(R^6) with polynomial velocity weights). We show this by propagating a H\"older modulus of continuity in space and velocity. This then yields H\"older continuity in time by controlling time differences. Finally, we obtain weak-strong uniqueness through a suitable energy method. This talk is based on joint work with Henderson and Snelson.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in the PDC front desk for a guided tour of The Banff Centre campus.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

We consider a kinetic Fokker-Planck equation with rough coefficients and the spatial variable restricted to a bounded domain. There are recent results concerning interior Holder estimates for this class of equations following techniques by De Giorgi, Nash and Moser. In this talk, we discuss the regularity of the solutions on the boundary.

In a work in collaboration with Ricardo Alonso, Véronique Bagland and Bertrand Lods, we investigate the properties of the spatially homogeneous Landau equation for fermions, in the case of soft potentials. We propose regularity and large time behavior results which are uniform with respect to the quantum parameter, so that they also hold in the classical case

Kinetic equations arising in biology and social sciences often have non-explicit steady states unlike the ones coming from mathematical physics such as Boltzmann-type equations. This makes it hard to use classical hypocoercivity techniques to study the long-time behaviour of these equations. Particularly, it is difficult to obtain Poincaré-type inequalities. Harris-type theorems present an alternative approach since they are based on controlling the behaviour of moments rather than Poincaré-type inequalities, thus we look at the point-wise bounds rather than integral controls of operators. I will talk about Harris-type theorems with a couple of examples from mathematical physics and biology.

BIRS staff will meet you at the TCPL foyer to take a group photo.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

We study the Neumann problem for nonlocal operators in bounded domains with prescribed data in the complement of the domain. We introduce corresponding function spaces and trace resp. extension results. Finally, we present the probabilistic interpretation of the problem.

This talk introduces the stationary radiative transfer equation coupled with a non-local temperature equation in the local thermodynamic equilibrium. Via the system, we study the temperature distribution of a body when the heat is transmitted only by radiation. The heat transferred by convection and conduction is ignored. We prove that the system with the incoming boundary condition admits a solution in a generic case when both the emission/absorption and the scattering of interacting radiation are considered. We also introduce the entropy production formula of the system for the uniqueness of the solution. This is joint work with Juan J. L. Velázquez.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.