Christopher Henderson

We consider the relativistic Landau equation in the spatially inhomogeneous, far-from-equilibrium regime. We establish regularity estimates of all orders, implying that solutions remain smooth for as long as some zeroth-order conditional bounds hold. We also prove that polynomial and exponential decay in the momentum variable is propagated forward in time. As part of our proof, we establish a Schauder estimate for linear relativistic kinetic equations, that may be of independent interest.

We consider the road-field reaction-diffusion model introduced by Berestycki, Roquejoffre, and Rossi. By performing a "thin-front limit," we are able to deduce a Hamilton-Jacobi equation with a suitable effective Hamiltonian on the road that governs the front location of the road-field model. Our main motivation is to apply the theory of strong (flux-limited) viscosity solutions in order to determine a control formulation interpretation of the front location. In view of the ecological meaning of the road-field model, this is natural as it casts the invasion problem as one of finding optimal paths that balance the positive growth rate in the field with the fast diffusion on the road. Our main contribution is a nearly complete picture of the behavior on two-road conical domains. When the diffusivities on each road are the same, we show that the propagation speed in each direction in the cone can be computed via those associated with one-road half-space problem. When the diffusivities differ, we show that the speed along the faster road is unchanged, while the speed along the slower road can be enhanced. Along the way we provide a new proof of known results on the one-road half-space problem via our approach.

In this paper, we prove a kinetic Nash type inequality and adapt it to a new functional inequality for functions in a kinetic Sobolev space with absorbing boundary conditions on the half-space. As an application, we address the boundary behavior of the kinetic Fokker-Planck equations in the half-space. Our main result is the sharp regularity of the solution at the absorbing boundary and grazing set.

We consider the long time behavior of the solutions to the Burgers-FKPP equation with advection of a strength \beta\in\mathbb{R} . This equation exhibits a transition from pulled to pushed front behavior at \beta_{c}=2 . We prove convergence of the solutions to a traveling wave in a reference frame centered at a position m_{\beta}(t) and study the asymptotics of the front location m_{\beta}(t) . When \beta < 2 , it has the same form as for the standard Fisher-KPP equation established by Bramson: m_{\beta}(t) = 2t - (3/2)\log t + x_{\infty} + o(1) as t\to\infty . This form is typical of pulled fronts. When \beta > 2 , the front is located at the position m_{\beta}(t)=c_{*}(\beta)t+x_{\infty}+o(1) with c_{*}(\beta)=\beta/2+2/\beta , which is the typical form of pushed fronts. However, at the critical value \beta_{c} = 2 , the expansion changes to m_{\beta}(t) = 2t - (1/2)\log t + x_{\infty} + o(1) , reflecting the “pushmi-pullyu” nature of the front. The arguments for \beta2 relies on standard pushed front techniques. The proof in the case \beta=\beta_{c} is much more intricate and involves arguments not usually encountered in the study of the Bramson correction. It relies on a somewhat hidden viscous conservation law structure of the Burgers-FKPP equation at \beta_{c}=2 and utilizes a dissipation inequality, which comes from a relative entropy type computation, together with a weighted Nash inequality involving dynamically changing weights.

We show that there exist traveling wave solutions of the Keller-Segel-FKPP equation, which models a diffusing and logistically growing population subject to chemotaxis. In contrast to previous results, our result is in the strong aggregation regime; that is, we make no smallness assumption on the parameters. The lack of a smallness condition makes $L^\infty$-estimates difficult to obtain as the comparison principle no longer gives them ``for free." Instead, our proof is based on suitable energy estimates in a carefully tailored uniformly local $L^p$-space. Interestingly, our uniformly local space involves a scaling parameter, the choice of which is a crux of the argument. Numerical experiments exploring the stability, qualitative properties, and speeds of these waves are presented as well.

We present probabilistic interpretations of solutions to semi-linear parabolic equations with polynomial nonlinearities in terms of the voting models on the genealogical trees of branching Brownian motion (BBM). These extend McKean’s connection between the Fisher–KPP equation and BBM (McKean 1975 Commun. Pure Appl. Math. 28 323–31). In particular, we present ‘random outcome’ and ‘random threshold’ voting models that yield any polynomial nonlinearity f satisfying f ( 0 ) = f ( 1 ) = 0 and a ‘recursive up the tree’ model that allows to go beyond this restriction on f. We compute several examples of particular interest; for example, we obtain a curious interpretation of the heat equation in terms of a nontrivial voting model and a ‘group-based’ voting rule that leads to a probabilistic view of the pushed-pulled transition for a class of nonlinearities introduced by Ebert and van Saarloos.