Research

Papers sorted by area (in reverse chronological order here)

The 2D Muskat problem II: Stable regime small data singularity on the half-plane, preprint.
The 2D Muskat problem I: Local regularity on the half-plane, plane, and strips, preprint.
Local regularity and finite time singularity for the generalized SQG equation on the half-plane, preprint.
An improved regularity criterion and absence of splash-like singularities for g-SQG patches (with J. Jeon), Anal. PDE, to appear.
Uniqueness of positive vorticity solutions to the 2D Euler equations on singular domains (with Z. Han), Arch. Ration. Mech. Anal. 247 (2023), Article 84, 22pp.
Euler equations on general planar domains (with Z. Han), Ann. PDE 7 (2021), Article 20, 31pp.
The Euler equations in planar domains with corners (with C. Lacave), Arch. Ration. Mech. Anal. 234 (2019), 57-79.
On the rate of merging of vorticity level sets for the 2D Euler equations, J. Nonlinear Sci. 28 (2018), 2329-2341.
A note on stability shifting for the Muskat problem II: From stable to unstable and back to stable (with D. Córdoba and J. Gómez-Serrano), Anal. PDE 10 (2017), 367-378.
Local regularity for the modified SQG patch equation (with A. Kiselev and Y. Yao), Comm. Pure Appl. Math. 70 (2017), 1253–1315.
Finite time singularity for the modified SQG patch equation (with A. Kiselev, L. Ryzhik, and Y. Yao), Ann. of Math. (2) 184 (2016), 909-948.
Blow up for the 2D Euler equation on some bounded domains (with A. Kiselev), J. Differential Equations 259 (2015), 3490-3494.
A note on stability shifting for the Muskat problem (with D. Córdoba and J. Gómez-Serrano), Philos. Trans. A 373 (2015), 20140278.
Exponential growth of the vorticity gradient for the Euler equation on the torus, Adv. Math. 268 (2015), 396-403.
On discrete models of the Euler equation (with A. Kiselev), Int. Math. Res. Notices 2005, 2315-2339.

Reaction-Diffusion Equations, Homogenization, and Probability

Virtual linearity for KPP reaction-diffusion equations, preprint.
Quantitative homogenization for combustion in random media (with Y.P. Zhang), Ann. Inst. H. Poincaré Anal. Non Linéaire, to appear.
Homogenization for space-time-dependent KPP reaction-diffusion equations and G-equations (with Y.P. Zhang), Calc. Var. Partial Differential Equations 62 (2023), Article 248, 22pp.
Optimal estimates on the propagation of reactions with fractional diffusion (with Y.P. Zhang), Arch. Ration. Mech. Anal. 247 (2023), Article 93, 33pp.
Subadditive theorems in time-dependent environments (with Y.P. Zhang), Electron. J. Probab. 28 (2023), 1-23.
Homogenization for time-periodic KPP reactions, Nonlinearity 36 (2023), 1918-1927.
Long time dynamics for combustion in random media (with Y.P. Zhang), Arch. Ration. Mech. Anal. 243 (2022), 33-94.
Stochastic homogenization for reaction-diffusion equations (with J. Lin), Arch. Ration. Mech. Anal. 232 (2019), 813-871.
Multidimensional transition fronts for Fisher-KPP reactions (with A. Alwan, Z. Han, J. Lin, and Z. Tao), Nonlinearity 32 (2019), 927-941.
Propagation of reactions in inhomogeneous media, Comm. Pure Appl. Math. 70 (2017), 884-949.
Existence and non-existence of transition fronts for bistable and ignition reactions, Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), 1687-1705.
Transition fronts for inhomogeneous Fisher-KPP reactions and non-local diffusion (with T.S. Lim), Trans. Amer. Math. Soc. 368 (2016), 8615-8631.
Transition fronts for inhomogeneous monostable reaction-diffusion equations via linearization at zero (with T. Tao and B. Zhu), Nonlinearity 27 (2014), 2409-2416.
Speed-up of combustion fronts in shear flows (with F. Hamel), Math. Ann. 356 (2013), 845-867.
Generalized traveling waves in disordered media: Existence, uniqueness, and stability, Arch. Ration. Mech. Anal. 208 (2013), 447-480.
Transition fronts in inhomogeneous Fisher-KPP reaction-diffusion equations, J. Math. Pures Appl. 98 (2012), 89-102.
Existence and non-existence of Fisher-KPP transition fronts (with J. Nolen, J.-M. Roquejoffre, and L. Ryzhik), Arch. Ration. Mech. Anal. 203 (2012), 217-246.
Reaction-diffusion front speed enhancement by flows, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), 711-726.
Sharp asymptotics for KPP pulsating front speed-up and diffusion enhancement by flows, Arch. Ration. Mech. Anal. 195 (2010), 441-453.
Pulsating front speed-up and quenching of reaction by fast advection, Nonlinearity 20 (2007), 2907-2921.
KPP pulsating front speed-up by flows (with L. Ryzhik), Comm. Math. Sci. 5 (2007), 575-593.
Sharp transition between extinction and propagation of reaction, J. Amer. Math. Soc. 19 (2006), 251-263.
Quenching of combustion by shear flows (with A. Kiselev), Duke Math. J. 132 (2006), 49-72.
Quenching and propagation of combustion without ignition temperature cutoff, Nonlinearity 18 (2005), 1463-1475.

Drift-Diffusion Equations, Mixing, and Chemotaxis

Numerical evidence of exponential mixing by alternating shear flows (with L.-T. Cheng, F. Rajasekaran, and K.Y.J. Wong), Comm. Math. Sci. 21 (2023), 529-541.
On the fast spreading scenario (with S. He and E. Tadmor), Comm. Amer. Math. Soc. 2 (2022), 149-171.
Convection-induced singularity suppression in the Keller-Segel and other non-linear PDEs (with G. Iyer and X. Xu), Trans. Amer. Math. Soc. 374 (2021), 6039-6058.
Universal mixers in all dimensions (with T. Elgindi), Adv. Math. 356 (2019), 106807, 33pp.
Mixing and un-mixing by incompressible flows (with Y. Yao), J. Eur. Math. Soc. 19 (2017), 1911-1948.
Periodic orbits of the ABC flow with A=B=C=1 (with J. Xin and Y. Yu), SIAM J. Math. Anal. 48 (2016), 4087-4093.
Ballistic orbits and front speed enhancement for ABC flows (with T. McMillen, J. Xin, and Y. Yu), SIAM J. Appl. Dyn. Syst. 15 (2016), 1753-1782.
On the loss of continuity for super-critical drift-diffusion equations (with L. Silvestre and V. Vicol), Arch. Ration. Mech. Anal. 207 (2013), 845-877.
The Harnack inequality for a class of degenerate elliptic operators (with F. Hamel), Int. Math. Res. Notices 2013, 3732-3743.
On divergence-free drifts (with G. Seregin, L. Silvestre, and V. Šverák), J. Differential Equations 252 (2012), 505-540.
Exit times of diffusions with incompressible drifts (with G. Iyer, A. Novikov, and L. Ryzhik), SIAM J. Math. Anal. 42 (2010), 2484-2498.
Diffusion in fluid flow: Dissipation enhancement by flows in 2D, Comm. Partial Differential Equations 35 (2010), 496-534.
Relaxation enhancement by time-periodic flows (with A. Kiselev and R. Shterenberg), Indiana Univ. Math. J. 57 (2008), 2137-2152.
Diffusion and mixing in fluid flow (with P. Constantin, A. Kiselev, and L. Ryzhik), Ann. of Math. (2) 168 (2008), 643-674.

Spectral Theory, Orthogonal Polynomials, Graph Theory, etc.

On the high intensity limit of interacting corpora (with P. Constantin), Comm. Math. Sci. 8 (2010), 173-186.
Coefficients of orthogonal polynomials on the unit circle and higher order Szegö theorems (with L. Golinskii), Constr. Approx. 26 (2007), 361-382.
Sum rules for Jacobi matrices and divergent Lieb-Thirring sums, J. Funct. Anal. 225 (2005), 371-382.
Higher order Szegö theorems with two singular points (with B. Simon), J. Approx. Theory 134 (2005), 114-129.
Sparse potentials with fractional Hausdorff dimension, J. Funct. Anal. 207 (2004), 216-252.
Sum rules and the Szegö condition for orthogonal polynomials on the real line (with B. Simon), Comm. Math. Phys. 242 (2003), 393-423.
The Szegö condition for Coulomb Jacobi matrices, J. Approx. Theory 121 (2003), 119-142.
Note on regular embeddings of complete bipartite graphs (with R. Nedela and M. Škoviera), Discrete Math. 258 (2002), 379-381.
Bipartite maps, Petrie duality and exponent groups (with R. Nedela and M. Škoviera), Atti Sem. Mat. Fis. Univ. Modena 49 (2001), 109-133.
The diameter of lifted digraphs, Australas. J. Combin. 19 (1999), 73-82.
Construction of regular maps with multiple edges, International Scientific Conference on Mathematics. Proceedings, 155-160, Univ. Žilina, 1998.

Research supported in part by the NSF, Alfred P. Sloan Foundation, and Simons Foundation