Research

My research is focused on the theory and applications of partial differential equations (PDEs). Specific areas include:

• Kinetic theory. This classical area of physics seeks to understand the macroscopic properties of matter by studying statistical averages of its constituent particles. This leads to a differential equation for the particle density function. I am especially interested in two kinetic equations known as the Boltzmann and Landau equations, which have proven difficult to understand mathematically despite their long history.

• Shape optimization. A class of geometric optimization problems that arise in areas such as structural design. Mathematically, the goal is to minimize quantities that depend on the shape of a geometric region, subject to volume, perimeter, or other constraints. One important class of problems involves finding  minimal domains for eigenvalues of the Laplacian or other elliptic differential operators.

• Nonlinear waves. Wave propagation that features nontrivial self-interaction leads to a wave equation with a nonlinear term. Depending on the form of the nonlinearity, a wide variety of phenomena can be modeled in this way. These models sometimes give rise to solitary waves and other special solutions, and the stability or instability of these solutions is a key ingredient in understanding the long-time behavior of the system.

My research is partially supported by NSF grant DMS-2213407. I am also grateful for past support from the Simons Foundation and Oak Ridge Associated Universities.

Publications and preprints

Here is a reverse chronological list. Click here to see my papers organized by subject.

• Regularity and nondegeneracy for nonlocal Bernoulli problems with variable kernels (with E. Teixeira)
  Submitted for publication.

• Decay estimates and continuation for the non-cutoff Boltzmann equation (with C. Henderson and A. Tarfulea)
  Submitted for publication.

• A continuation criterion for the Landau equation with very soft and Coulomb potentials (with C. Solomon)
  Submitted for publication.

• On the Bernoulli problem with unbounded jumps (with E. Teixeira)
  Submitted for publication.

• Classical solutions of the Boltzmann equation with irregular initial data (with C. Henderson and A. Tarfulea)
  Annales Scientifiques de l'École Normale Supérieure, to appear.

• Optimal domains for elliptic eigenvalue problems with rough coefficients (with E. Teixeira)
  SIAM Journal on Mathematical Analysis, to appear.

• Existence and stability of near-constant solutions of variable-coefficient scalar-field equations (with M. Alammari)
  Differential and Integral Equations, to appear.

• Global existence for an isotropic modification of the Boltzmann equation
  Journal of Functional Analysis 286(12), Paper No. 110423, 2024.

• Solutions to the non-cutoff Boltzmann equation uniformly near a Maxwellian (with L. Silvestre)
  Mathematics in Engineering 5(2) 1–36, 2023.

• Non-existence of some approximately self-similar singularities for the Landau, Vlasov-Poisson-Landau, and Boltzmann equations (with J. Bedrossian and M. Gualdani)
  Transactions of the AMS 375(3) 2187–2216, 2022.

• Linear and orbital stability analysis for solitary-wave solutions of variable-coefficient scalar field equations (with M. Alammari)
  Journal of Hyperbolic Differential Equations 19(1) 175–201, 2022.

• C∞ smoothing for weak solutions of the inhomogeneous Landau equation (with C. Henderson)
  Archive for Rational Mechanics and Analysis 236(1) 113–143, 2020.

• Local solutions of the Landau equation with rough, slowly decaying initial data (with C. Henderson and A. Tarfulea)
  Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire 37(6) 1345–1377, 2020.

• Gaussian bounds for the inhomogeneous Landau equation with hard potentials
  SIAM Journal on Mathematical Analysis 52(2) 2081–2097, 2020.

• Self-generating lower bounds and continuation for the Boltzmann equation (with C. Henderson and A. Tarfulea)
  Calculus of Variations and PDE 59(6) 191, 2020.

• Local well-posedness of the Boltzmann equation with polynomially decaying initial data (with C. Henderson and A. Tarfulea)
  Kinetic & Related Models 13(4) 837–867, 2020.

• Velocity decay estimates for Boltzmann equation with hard potentials (with S. Cameron)
  Nonlinearity 33(6) 2941–2959, 2020.

• Local existence, lower mass bounds, and a new continuation criterion for the Landau equation (with C. Henderson and A. Tarfulea)
  Journal of Differential Equations 266(2–3) 1536–1577, 2019.

• Asymptotic stability for odd perturbations of the the stationary kink in the variable-speed φ4 model
  Transactions of the AMS 370(10) 7437–7460, 2018.

• Global a priori estimates for the inhomogeneous Landau equation with moderately soft potentials (with S. Cameron and L. Silvestre)
  Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire 35(3) 625–642, 2018.

• Linear stability of the Skyrmion (with M. Creek, R. Donninger, and W. Schlag)
  International Mathematics Research Notices 2017(8) 2497–2537, 2017.

• An integro-differential equation without continuous solutions (with L. Silvestre)
  Mathematical Research Letters 24(4) 1157–1166, 2016.

• Regularity and long-time behavior of nonlocal heat flows
  Calculus of Variations and PDE 54(2) 1705–1723, 2015.

• Matrix probing: a randomized preconditioner for the wave-equation Hessian (with L. Demanet, P.D. Létourneau, N. Boumal, H. Calandra, and J. Chiu)
  Applied and Computational Harmonic Analysis 32(2) 155–168, 2012.