Christopher Henderson

Interests

Teaching

Mathematical analysis, applied math

Research

Mathematical analysis, applied mathematics, partial differential equations, mathematical physics, mathematical biology, probability, reaction-diffusion equations, kinetic equations

Courses

Scholarly Contributions

Journals/Publications

• Gu, Y., & Henderson, C. (2023). Long-time behaviour for a nonlocal model from directed polymers. Nonlinearity, 36(2), 902--954.
• Henderson, C., & Wang, W. (2023). Kinetic Schauder estimates with time-irregular coefficients and uniqueness for the Landau equation.
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  We prove a Schauder estimate for kinetic Fokker-Planck equations thatrequires only H\"older regularity in space and velocity but not in time. As anapplication, we deduce a weak-strong uniqueness result of classical solutionsto the spatially inhomogeneous Landau equation beginning from initial datahaving H\"older regularity in $x$ and only a logarithmic modulus of continuityin $v$. This replaces an earlier result requiring H\"older continuity in bothvariables.[Journal_ref: ]
• An, J., Henderson, C., & Ryzhik, L. (2022). Quantitative steepness, semi-FKPP reactions, and pushmi-pullyu fronts.
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  We uncover a seemingly previously unnoticed algebraic structure of a largeclass of reaction-diffusion equations and use it, in particular, to study thelong time behavior of the solutions and their convergence to traveling waves inthe pulled and pushed regimes, as well as at the pushmi-pullyu boundary. Onesuch new object introduced in this paper is the shape defect function, which,indirectly, measures the difference between the profiles of the solution andthe traveling wave. While one can recast the classical notion of `steepness' interms of the positivity of the shape defect function, its positivity can,surprisingly, be used in numerous quantitative ways. In particular, thepositivity is used in a new weighted Hopf-Cole transform and in a relativeentropy approach that play a key role in the stability arguments. The shapedefect function also gives a new connection between reaction-diffusionequations and reaction conservation laws at the pulled-pushed transition. Othersimple but seemingly new algebraic constructions in the present paper supplyvarious unexpected inequalities sprinkled throughout the paper. Of note is anew variational formulation that applies equally to pulled and pushed fronts,opening the door to an as-yet-elusive variational analysis in the pulled case.[Journal_ref: ]
• An, J., Henderson, C., & Ryzhik, L. (2022). Voting models and semilinear parabolic equations. arxiv.
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  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. [Journal_ref: ]
• Calvez, V., Henderson, C., Mirrahimi, S., Turanova, O., & Dumont, T. (2022). Non-local competition slows down front acceleration during dispersal evolution. Ann. H. Lebesgue, 5, 1--71.
• Griette, Q., Henderson, C., & Turanova, O. (2022). Speed-up of traveling waves by negative chemotaxis. arxiv.
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  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. [Journal_ref: ]
• Henderson, C. (2022). Slow and fast minimal speed traveling waves of the FKPP equation with chemotaxis. J. Math. Pures Appl. (9), 167, 175--203.
• Henderson, C., & Wang, W. (2022). Local well-posedness for the Boltzmann equation with very soft potential and polynomially decaying initial data. SIAM J. Math. Anal., 54(3), 2845--2875.
• Henderson, C., Snelson, S., & Tarfulea, A. (2022). Classical solutions of the Boltzmann equation with irregular initial data.
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  This article considers the spatially inhomogeneous, non-cutoff Boltzmannequation. We construct a large-data classical solution given bounded,measurable initial data with uniform polynomial decay of mild order in thevelocity variable. Our result requires no assumption of strict positivity forthe initial data, except locally in some small ball in phase space. We alsoobtain existence results for weak solutions when our decay and positivityassumptions for the initial data are relaxed. Because the regularity of oursolutions may degenerate as $t \rightarrow 0$, uniqueness is a challengingissue. We establish weak-strong uniqueness under the additional assumption thatthe initial data possesses no vacuum regions and is H\"older continuous. As anapplication of our short-time existence theorem, we prove global existence nearequilibrium for bounded, measurable initial data that decays at a finitepolynomial rate in velocity.[Journal_ref: ]
• An, J., Henderson, C., & Ryzhik, L. (2021). Pushed, pulled and pushmi-pullyu fronts of the Burgers-FKPP equation. arXiv preprint arXiv:2108.07861, accepted Journal of the European Mathematical Society.
• Bouin, E., & Henderson, C. (2021). The Bramson delay in a Fisher-KPP equation with log-singular nonlinearity. Nonlinear Anal., 213, Paper No. 112508, 30.
• Bramburger, J. J., & Henderson, C. (2021). The speed of traveling waves in a FKPP-Burgers system. Arch. Ration. Mech. Anal., 241(2), 643--681.
• Gu, Y. u., & Henderson, C. (2021). A PDE hierarchy for directed polymers in random environments. Nonlinearity, 34(10), 7335--7370.
• Henderson, C. (2020). Local well-posedness of the Boltzmann equation with polynomially decaying initial data. Kinetic and Related Models. doi:http://dx.doi.org.ezproxy1.library.arizona.edu/10.3934/krm.2020029
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  We consider the Cauchy problem for the spatially inhomogeneous non-cutoffBoltzmann equation with polynomially decaying initial data in the velocityvariable. We establish short-time existence for any initial data with thisdecay in a fifth order Sobolev space by working in a mixed $L^2$ and $L^\infty$space that allows to compensate for potential moment generation and obtainingnew estimates on the collision operator that are well-adapted to this space.Our results improve the range of parameters for which the Boltzmann equation iswell-posed in this decay regime, as well as relax the restrictions on theinitial regularity. As an application, we can combine our existence result withthe recent conditional regularity estimates of Imbert-Silvestre(arXiv:1909.12729 [math.AP]) to conclude solutions can be continued for as longas the mass, energy, and entropy densities remain under control. Thiscontinuation criterion was previously only available in the restricted range ofparameters of previous well-posedness results for polynomially decaying initialdata.[Journal_ref: ]
• Henderson, C. (2020). Self-generating lower bounds and continuation for the Boltzmann equation. Calculus of Variations and Partial Differential Equations.
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  For the spatially inhomogeneous, non-cutoff Boltzmann equation posed in thewhole space $\mathbb R^3_x$, we establish pointwise lower bounds that appearinstantaneously even if the initial data contains vacuum regions. Our lowerbounds depend only on the initial data and upper bounds for the mass and energydensities of the solution. As an application, we improve the weakest knowncontinuation criterion for large-data solutions, by removing the assumptions ofmass bounded below and entropy bounded above.[Journal_ref: ]
• Henderson, C., & Souganidis, P. E. (2020). Brownian fluctuations of flame fronts with small random advection. Mathematical Models and Methods in Applied Sciences, 30(07), 1375-1406. doi:10.1142/s0218202520500256
• Henderson, C., Bouin, E., & Ryzhik, L. (2020). The Bramson delay in the non-local Fisher-KPP equation. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 37(1), 51-77. doi:10.1016/j.anihpc.2019.07.001
• Henderson, C., Snelson, S., & Tarfulea, A. (2020). Local solutions of the Landau equation with rough, slowly decaying initial data. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 37(6), 1345-1377. doi:10.1016/j.anihpc.2020.04.004
• Hamel, F., & Henderson, C. (2019). Propagation in a Fisher-KPP equation with non-local advection. Journal of Functional Analysis, 108426.
• Henderson, C., & Snelson, S. (2019). C\^{}∞ Smoothing for Weak Solutions of the Inhomogeneous Landau Equation. Arch. Ration. Mech. Anal., 151.
• Henderson, C., Snelson, S., & Tarfulea, A. (2019). Local existence, lower mass bounds, and a new continuation criterion for the Landau equation. Journal of Differential Equations, 266(2-3), 1536--1577.
• Henderson, C., Snelson, S., & Tarfulea, A. (2020). Local Solutions of the Landau Equation with Rough, Slowly Decaying Initial Data. Annales de l'Institut Henri Poincaré C, Analyse non linéaire. doi:https://doi-org.ezproxy1.library.arizona.edu/10.1016/j.anihpc.2020.04.004

Proceedings Publications

• Bouin, E., Henderson, C., & Ryzhik, L. (2019). The Bramson delay in the non-local Fisher-KPP equation. In Annales de l'Institut Henri Poincar\'e C, Analyse non lin\'eaire.