Dong Li

Chair Professor, Department of Maths, HKU
Email: mathdl@hku.hk

Research Interests

Analysis and PDE; Applied math.

Selected papers

D. Li and R. Shao. An intrinsic flow for elliptic regularity. Preprint.
This new paper presents a completely new proof of the celebrated De Giorgi-Nash-Moser theorem, six decades after its original formulation.
D. Li and K. Yang. On the gap property of a linearized NLS operator. Adv. Math. 492 (2026): 110904.
This proves the spectral gap property of linearized 3D cubic NLS operators $L_+$ and $L_-$.
D. Li and P. Zhang. A theory of refined analyticity radius for 3D Navier-Stokes equations. Preprint.
This develops a novel framework to establish refined analyticity for 3D Navier-Stokes equations in the general $L^p$ context which gives the first significantly improved lower bound after Lemarié-Rieusset 2000.
D. Li and P. Zhang. On the refined analyticity radius of 3D generalized Navier-Stokes equations. Adv. Math. 487 (2026): 110769.
This settles a decade-long open problem by Herbst and Skibsted about the local refined analyticity radius for 3D Navier-Stokes equations.
D. Li. Optimal Gevrey regularity for supercritical quasi-Geostrophic equations. Comm. Math. Phys. 405 (2024), no. 2, Paper No. 30. This solves an open problem by Biswas-Martinez-Silva (JFA 2015) concerning the sharp Gevrey regularity for supercritical SQG.
D. Li. Uniform estimates for 2D quasilinear wave. Adv. Math. Vol 428 (2023), 109157 This settles Alinhac's ``blow-up-at-infinity" conjecture.
D. Li and Y. Sire. Remarks on the Bernstein inequality for higher order operators and related results. Trans. Amer. Math. Soc. 376 (2023), no. 2, 945–967.
This solves a folklore open problem concerning the lack of Bernstein inequality for higher-order Laplacians.
D. Li. Global well-posedness of hedgehog solutions for the $(3+1)$ Skyrme model. Duke Math. J. 170 (2021), no. 7, 1377–1418.
This introduces an original strategy to prove large data global wellposedness for the energy-supercritical Skyrme model which has been open since 1961.

✦ The following series of five papers (joint with J. Bourgain) develops several original methods to settle the longstanding open problem of strong ill-posedness of incompressible Euler equations in borderline spaces.

J. Bourgain and D. Li. Strong ill-posedness of the incompressible Euler equation in borderline Sobolev spaces. Invent. Math. 201 (2015), no. 1, 97–157.
J. Bourgain and D. Li. Strong ill-posedness of the 3D incompressible Euler equation in borderline spaces. Int. Math. Res. Not. IMRN (2021), no. 16, 12155–12264. [30th Anniversary Collection]
J. Bourgain and D. Li. Strong illposedness of the incompressible Euler equation in integer $C^m$ spaces. Geom. Funct. Anal. 25 (2015), no. 1, 1–86.
J. Bourgain and D. Li. Galilean boost and non-uniform continuity for incompressible Euler. Comm. Math. Phys. 372 (2019), no. 1, 261–280.
J. Bourgain and D. Li. On an endpoint Kato-Ponce inequality. Diff. Inte. Eqn. 27 (2014), no. 11–12, 1037–1072.
D. Li. On Kato-Ponce and fractional Leibniz. Rev. Mat. Iberoam. 35 (2019), no. 1, 23–100.
This settles an open problem in Kato-Ponce's seminal '88 work and gives a deep generalization of the celebrated Kenig-Ponce-Vega estimate to all higher order Laplacian operators. Many well-known commutator estimates (including difficult end-point situations) are special cases of this new fractional Leibniz rule.
D. Li. Why large time-stepping methods for the Cahn-Hilliard equation is stable. Math. Comp. 91 (2022), no. 338, 2501–2515.
This solves a longstanding open issue of the stability of large-time-stepping methods in phase field models where maximum principle is no longer in force.
X. Cheng, H. Kwon and D. Li. Non-uniqueness of steady-state weak solutions to the surface quasi-geostrophic equations. Comm. Math. Phys. 388 (2021), 1281–1295.
This introduces a new twist of the convex integration scheme to prove non-uniqueness of stationary SQG. In addition we settle the weak rigidity conjecture by Isett-Vicol.
D. Li. Effective maximum principles for spectral methods. Ann. Appl. Math. 37 (2021), no. 2, 131–290.
This work develops a novel effective-maximum principle and stability framework for $L^2$-based spectral methods which were long believed to admit only energy bounds (due to issues such as Gibbs oscillation and lack of maximum principle).

The following series of four papers (joint work with Cai, Huang and Yang) settle the $\mathcal O(N)$ conjecture due to Sun-Qu-Wright (Found. Comput. Math. 2018) in phase retrieval.

J. Cai, M. Huang, D. Li and W. Yang. Nearly optimal bounds for the global geometric landscape of phase retrieval. Inverse Problems 39 (2023), no. 7, Paper No. 075011.
J. Cai, M. Huang, D. Li and W. Yang. Solving phase retrieval with random initial guess is nearly as good as by spectral initialization. Appl. Comput. Harmon. Anal. 58 (2022), 60–84.
J. Cai, M. Huang, D. Li and W. Yang. The global landscape of phase retrieval II: quotient intensity models. Ann. Appl. Math. 38 (2022), 62–114.
J. Cai, M. Huang, D. Li and W. Yang. The global landscape of phase retrieval I: perturbed amplitude models. Ann. Appl. Math. 37 (2021), 437–512.

The following series of works develop a new stability analysis framework for implicit-explicit methods in phase field equations (open since Chen-Shen, Comput. Phys. Comm, 1998).

Stability and convergence analysis for the implicit-explicit method to the Cahn-Hilliard equation. Math. Comp. 91 (2022), no. 334, 785–809.
Stability and convergence of Strang splitting. Part I: Scalar Allen-Cahn equation. J. Comput. Phys. 458 (2022), No. 111087.
Stability of the semi-implicit method for the Cahn-Hilliard equation with logarithmic potentials. Ann. Appl. Math. 37 (2021), 31–60.
On second order semi-implicit Fourier spectral methods for 2D Cahn-Hilliard equations. J. Sci. Comput. 70 (2017), no. 1, 301–341.
Characterizing the stabilization size for semi-implicit Fourier-spectral method to phase field equations. SINUM 54 (2016), no. 3, 1653–1681.
B. Han, Z. Lei, D. Li and N. Zhao. Sharp one component regularity for Navier–Stokes. Arch. Ration. Mech. Anal. 231 (2019), no. 2, 939–970.
This work settles the sharp one-component regularity in $\dot H^{\frac 12+ \frac 2p}$ for all $p\in [2, \infty[$ which was open after Chemin-Zhang 2016 (for $p \in [2,\, 4]$).
D. Li and Y. Wu. The Cauchy problem for the two dimensional Euler-Poisson system. J. Eur. Math. Soc. 16 (2014), no. 10, 2211–2266.
This settles the global wellposedness of 2D Euler-Poisson system which was open since Y. Guo's work in 3D (1998 CMP).
D. Li, X. Yu and Z. Zhai. On the Euler-Poincaré equation with non-zero dispersion. Arch. Ration. Mech. Anal. 210 (2013), no. 3, 955–974.
J. Cai, X. Chen and D. Li. A quadratic lower bound for the permanent and determinant problem over any characteristic $\not = 2$. STOC 2008 (40th ACM Symposium on Theory of Computing).
W. E and D. Li. On the crystallization of 2D hexagonal lattices. Comm. Math. Phys. 286 (2009), no. 3, 1099–1140.
H. Dong and D. Li. Spatial analyticity of the solutions to the subcritical dissipative quasi-geostrophic equations. Arch. Ration. Mech. Anal. 189 (2008), no. 1, 131–158.
W. E and D. Li. The Andersen thermostat in molecular dynamics. Comm. Pure Appl. Math. 61 (2008), no. 1, 96–136.
This gives for the first time a rigorous proof of the ergodicity of the Andersen thermostat process which has been widely used in molecular dynamics simulations.
Last updated: March 2026