Research highlights

Papers sorted by area    (in reverse chronological order here)

Fluid Dynamics and Active Scalars

Drift-Diffusion Equations, Mixing, and Chemotaxis

Reaction-Diffusion Equations, Homogenization, and Probability

Spectral Theory, Orthogonal Polynomials, Graph Theory, etc.

• Well-posedness for low regularity solutions to the g-SQG equation with regular level sets (with J. Jeon), preprint.
  

• Well-posedness and finite time singularity for touching g-SQG patches on the plane (with J. Jeon), preprint.
  

• 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, Ann. PDE 11 (2026), Article 4, 62pp.
  

• Maximal double-exponential growth for the Euler equation on the half-plane, Invent. Math. 243 (2026), 117-126.
  

• Local regularity and finite time singularity for the generalized SQG equation on the half-plane, Duke Math. J. 174 (2025), 3493-3533.
  

• An improved regularity criterion and absence of splash-like singularities for g-SQG patches (with J. Jeon), Anal. PDE 17 (2024), 1005-1018.
  

• 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.
  

• 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.
  

• Virtual linearity for KPP reaction-diffusion equations, Adv. Math. 458 (2024), Article 109948, 17pp.
  

• Quantitative homogenization for combustion in random media (with Y.P. Zhang), Ann. Inst. H. Poincaré Anal. Non Linéaire 41 (2024), 853-876.
  

• 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.
  

• 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