arXiv subjects

Christopher Henderson

Publications and source records attributed to Christopher Henderson.

At least 19 recordsLinked to original sources

arXiv2026

Global smooth solutions to the inhomogeneous Landau-Fermi-Dirac equation

We consider the spatially inhomogeneous Landau-Fermi-Dirac equation with Coulomb potential, a quantum modification of the classical Landau equation for fermions. Mathematically, the Pauli exclusion principle manifests as an additional a priori $L^\infty$-bound for solutions. Using this bound, we propagate polynomial decay in velocity, yielding unconditional upper bounds on the local mass and energy densities, thereby ruling out the possibility of implosions in the hydrodynamic quantities. Combining this estimate with a modified mass spreading method that yields desaturation, we deduce the existence of global-in-time classical solutions for rough initial data with polynomial decay in velocity. This result stands in stark contrast to the theory for the classical Landau and Boltzmann equations, for which no comparable nonperturbative global existence result is known despite sustained effort. Our treatment is almost entirely self-contained, using only robust, generic estimates for linear kinetic equations. In particular, our proof of local existence, in contrast to prior works, more closely mirrors the theory for parabolic equations using weak solutions and simpler function spaces. It may provide a concise roadmap to organizing, adapting, and applying the various linear and nonlinear estimates to obtain well-posedness for kinetic equations.

math.AP↗

arXiv2026

Pointwise bounds and obstructions to blowup for the Landau and Boltzmann equations

We establish a new a priori estimate on solutions to the space-inhomogeneous Landau and Boltzmann equations. As a consequence, we prove a new continuation criterion, based on a weighted $L^\infty$-norm, without requiring bounds on the hydrodynamic quantities. This complements existing conditional regularity results from a rather different perspective. Consequently, we show that the singularities present in the fluid equations are largely incompatible with the Boltzmann and Landau equations. More precisely, we largely rule out ``lifting a singularity'' from the 3D Euler equations to the physical range of kinetic equations, a widely expected mechanism for singularity formation. Under general considerations, this mechanism is essentially excluded for soft potentials, whereas for hard potentials the situation is more nuanced: one cannot produce blowup through the standard hydrodynamic ansatz using known imploding solutions to the Euler equations.

math.AP↗

arXiv2026

Front Location for Go or Grow Models of Aerotaxis

We investigate the pushed-to-pulled transition for a minimal model for invasive fronts influence by ``aerotaxis,'' that is, when organisms follow oxygen gradients. We consider two singular reaction-advection-diffusion models for this. The version of primary interest arises as a hydrodynamic limit of a system of branching, rank-based interacting Brownian particles and features a nonlinear, nonlocal advection. The second version is introduced here as a local counterpart. We establish well-posedness for both models, with the local case requiring a novel use of the ``shape defect function.'' We further characterize the front location up to $O(1)$ precision in all cases, including the delicate boundary ``pushmi-pullyu'' case.

math.AP↗

arXiv2025

On Hydrodynamic Implosions and the Landau-Coulomb Equation

We study the inhomogeneous Landau equation with Coulomb potential and derive a new continuation criterion: a smooth solution can be uniquely continued for as long as it remains bounded. This provides, to our knowledge, the first continuation criterion based on a quantity not controlling the mass density. Consequently, we are able to rule out a potential singularity formation scenario known as tail fattening, in which an implosion occurs due to the loss of decay at large $v$. More generally, we are able to rule out all Type II approximately self-similar blow-up rates that are slower than the Type I blow-up rate, without any assumption of decay on the inner profile, complementing existing Type I blow-up analysis in the literature. Heuristically, this suggests that it should be impossible to directly use the hydrodynamic limit connection with the 3D compressible Euler equations to construct a singular solution to the Landau equation with Coulomb potential. Such a potential implosion scenario -- based on either an isentropic or nonisentropic implosion for the 3D Euler equations -- would naturally result in a slow Type II approximately self-similar blow-up scenario, falling well within the range our theorem. This preprint has been subsumed by a more recent work by the authors and Luis Silvestre titled ``Pointwise bounds and obstructions to blowup for the Landau and Boltzmann equations,'' arXiv:2605.20426. This manuscript will remain a permanent preprint; all references should be directed to the more recent work.

math.AP↗

arXiv2025

Global regularity and decay estimates for the relativistic Landau equation

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.

math.AP↗

arXiv2024

A Hamilton-Jacobi approach to road-field reaction-diffusion models

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.

math.AP↗

arXiv2024

A kinetic Nash inequality and precise boundary behavior of the kinetic Fokker-Planck equation

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.

math.AP↗

arXiv2023

Decay estimates and continuation for the non-cutoff Boltzmann equation

We consider the non-cutoff Boltzmann equation in the spatially inhomogeneous, soft potentials regime, and establish decay estimates for large velocity. In particular, we prove that pointwise algebraically decaying upper bounds in the velocity variable are propagated forward in time whenever the solution has finite weighted $L^\infty_{t,x} L^p_v$-norms for certain $p$. The main novelty is that these estimates hold for any decay exponent above $\max\{2,3 + \gamma\} +2s$, where $\gamma$ and $s$ are standard physical parameters such that $\gamma \in (-3,0)$ and $s\in (0,1)$. Our results are useful even for solutions with mild decay. As an application, we combine our decay estimates with recent short-time existence results to derive a continuation criterion for large-data solutions. Compared to past results, this extends the range of allowable parameters and weakens the requirements on smoothness and decay in velocity of solutions.

math.AP↗

arXiv2023

Front location determines convergence rate to traveling waves

We propose a novel method for establishing the convergence rates of solutions to reaction-diffusion equations to traveling waves. The analysis is based on the study of the traveling wave shape defect function introduced in [2]. It turns out that the convergence rate is controlled by the distance between the ``phantom front location'' for the shape defect function and the true front location of the solution. Curiously, the convergence to a traveling wave itself has a pulled nature, regardless of whether the traveling wave is of pushed, pulled, or pushmi-pullyu type. In addition to providing new results, this approach simplifies dramatically the proof in the Fisher-KPP case and gives a unified, succinct explanation for the known algebraic rates of convergence in the Fisher-KPP case and the exponential rates in the pushed case.

math.AP↗

arXiv2023

Traveling waves for the Keller-Segel-FKPP equation with strong chemotaxis

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.

math.AP↗

arXiv2022

Speed-up of traveling waves by negative chemotaxis

We consider the traveling wave speed for Fisher-KPP (FKPP) fronts under the influence of repulsive chemotaxis and provide an almost complete picture of its asymptotic dependence on parameters representing the strength and length-scale of chemotaxis. Our study is based on the convergence to the porous medium FKPP traveling wave and a hyperbolic FKPP-Keller-Segel traveling wave in certain asymptotic regimes. In this way, it clarifies the relationship between three equations that have each garnered intense interest on their own. Our proofs involve a variety of techniques ranging from entropy methods and decay of oscillations estimates to a general description of the qualitative behavior to the hyperbolic FKPP-Keller-Segel equation. For this latter equation, we, as a part of our limiting arguments, establish an explicit lower bound on the minimal traveling wave speed and provide a new construction of traveling waves that extends the known existence range to all parameter values.

math.AP↗

arXiv2022

Voting models and semilinear parabolic equations

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 the connection between the Fisher-KPP equation and BBM discovered by McKean in~\cite{McK}. 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 a few 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 pushmi-pullyu transition for a class of nonlinearities introduced by Ebert and van Saarloos.

math.AP↗

arXiv2022

Quantitative steepness, semi-FKPP reactions, and pushmi-pullyu fronts

We uncover a seemingly previously unnoticed algebraic structure of a large class of reaction-diffusion equations and use it, in particular, to study the long time behavior of the solutions and their convergence to traveling waves in the pulled and pushed regimes, as well as at the pushmi-pullyu boundary. One such new object introduced in this paper is the shape defect function, which, indirectly, measures the difference between the profiles of the solution and the traveling wave. While one can recast the classical notion of `steepness' in terms of the positivity of the shape defect function, its positivity can, surprisingly, be used in numerous quantitative ways. In particular, the positivity is used in a new weighted Hopf-Cole transform and in a relative entropy approach that play a key role in the stability arguments. The shape defect function also gives a new connection between reaction-diffusion equations and reaction conservation laws at the pulled-pushed transition. Other simple but seemingly new algebraic constructions in the present paper supply various unexpected inequalities sprinkled throughout the paper. Of note is a new variational formulation that applies equally to pulled and pushed fronts, opening the door to an as-yet-elusive variational analysis in the pulled case.

math.AP↗

arXiv2022

Classical solutions of the Boltzmann equation with irregular initial data

This article considers the spatially inhomogeneous, non-cutoff Boltzmann equation. We construct a large-data classical solution given bounded, measurable initial data with uniform polynomial decay of mild order in the velocity variable. Our result requires no assumption of strict positivity for the initial data, except locally in some small ball in phase space. We also obtain existence results for weak solutions when our decay and positivity assumptions for the initial data are relaxed. Because the regularity of our solutions may degenerate as $t \rightarrow 0$, uniqueness is a challenging issue. We establish weak-strong uniqueness under the additional assumption that the initial data possesses no vacuum regions and is H\"older continuous. As an application of our short-time existence theorem, we prove global existence near equilibrium for bounded, measurable initial data that decays at a finite polynomial rate in velocity.

math.AP↗

arXiv2022

Kinetic Schauder estimates with time-irregular coefficients and uniqueness for the Landau equation

We prove a Schauder estimate for kinetic Fokker-Planck equations that requires only H\"older regularity in space and velocity but not in time. As an application, we deduce a weak-strong uniqueness result of classical solutions to the spatially inhomogeneous Landau equation beginning from initial data having H\"older regularity in $x$ and only a logarithmic modulus of continuity in $v$. This replaces an earlier result requiring H\"older continuity in both variables.

math.AP↗

arXiv2021

A PDE hierarchy for directed polymers in random environments

For a Brownian directed polymer in a Gaussian random environment, with $q(t,\cdot)$ denoting the quenched endpoint density and \[ Q_n(t,x_1,\ldots,x_n)=\mathbf{E}[q(t,x_1)\ldots q(t,x_n)], \] we derive a hierarchical PDE system satisfied by $\{Q_n\}_{n\geq1}$. We present two applications of the system: (i) we compute the generator of $\{μ_t(dx)=q(t,x)dx\}_{t\geq0}$ for some special functionals, where $\{μ_t(dx)\}_{t\geq0}$ is viewed as a Markov process taking values in the space of probability measures; (ii) in the high temperature regime with $d\geq 3$, we prove a quantitative central limit theorem for the annealed endpoint distribution of the diffusively rescaled polymer path. We also study a nonlocal diffusion-reaction equation motivated by the generator and establish a super-diffusive $O(t^{2/3})$ scaling.

math.PR↗

arXiv2021

Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation

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 \cite{Bramson1,Bramson2}: $m_\beta(t) = 2t - (3/2)\log(t) + x_\infty + o(1)$ 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 $\beta<2$ rely on a new weighted Hopf-Cole transform that allows to control the advection term, when combined with additional steepness comparison arguments. The case $\beta>2$ 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.

math.AP↗

arXiv2021

Local well-posedness for the Boltzmann equation with very soft potential and polynomially decaying initial data

In this paper, we address the local well-posedness of the spatially inhomogeneous non-cutoff Boltzmann equation when the initial data decays polynomially in the velocity variable. We consider the case of very soft potentials $γ+ 2s < 0$. Our main result completes the picture for local well-posedness in this decay class by removing the restriction $γ+ 2s > -3/2$ of previous works. Our approach is entirely based on the Carleman decomposition of the collision operator into a lower order term and an integro-differential operator similar to the fractional Laplacian. Interestingly, this yields a very short proof of local well-posedness when $γ\in (-3,0]$ and $s \in (0,1/2)$ in a weighted $C^1$ space that we include as well.

math.AP↗