A personal view of mathematical research:

• On variational formulation of entropy solutions to nonlinear conservation laws
  
• IMS Public Lecture The Mathematics of Scientific Computation
  
• Velocity averaging, kinetic formulations and regularizing effects in conservation laws and related PDEs
  
• On a multiscale representation of images as hierarchy of edges
  
• Processing discontinuous spectral data
  
• Critical thresholds and Eulerian dynamics
  
• Burgers-type equations and vanishing hyper-viscosity
  
• Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity
  
• On a new scale of borderline regularity spaces for Euler equations
  
• Detection, reconstruction and approximate evolution of piecewise smooth solutions [ICM 2002 Lecture]

Selected Publications    Other lists of publications: [by chronological order] [by subject classification] [significant publications]

• On the fast spreading scenario (with S. He and A. Zlatoš) ArXiv:2104.00701 (2021).
  
• Multi-species Patlak-Keller-Segel system (with S. He) Indiana U. Math Journal 70(4) (2021) 1577-1624.
  
• On the mathematics of swarming: emergent behavior in alignment dynamics Notices of the AMS 68(4) (2021) 493-503.
  
• Multi-flocks: emergent dynamics in systems with multi-scale collective behavior (with R. Shvydkoy) MMS 19(2) (2021) 1115-1141.
  
• Anticipation breeds alignment (with R. Shu) ARMA 240 (2021) 203-241.
  
• A game of alignment: collective behavior of multi-species (with S. He) AIHP (c) Non Linear Analysis 38(4) (2021) 1031-1053.
  
• Optimal regularity in time and space for the porous medium equation (with B. Gess & J. Sauer) Analysis & PDE 13(8) (2020) 2441-2480.
  
• Flocking hydrodynamics with external potentials (with R. Shu) ARMA 238 (2020) 347-381.
  
• Topologically-based fractional diffusion and emergent dynamics with short-range interactions
  (with R. Shvydkoy) SIMA 52(6) (2020) 5792-5839.
  
• Global regularity of two-dimensional flocking hydrodynamics (with S. He), Comptes rendus - Mathématique Ser. I 355 (2017) 795–805.
  
• On the computation of measure-valued solutions (with U. Fjordholm and S. Mishra), Acta Numerica 25 (2016) 567-679.
  
• Hierarchical construction of bounded solutions in critical regularity spaces CPAM 69(6) (2016) 1087-1109.
  
• Heterophilious dynamics enhances consensus (with S. Motsch), SIAM Review 56(4) (2014) 577–621 (with Introduction by D. J. Higham)
  
• Stability and spectral convergence of Fourier method for nonlinear problems. On the shortcomings of the 2/3 de-aliasing method
  (with C. Bardos), Numerische Mathematik 129 (2014) 749-782.
• A review of numerical methods for nonlinear partial differential equations, Bull. AMS 49(4) (2012) 507-554.
  
• ENO reconstruction and ENO interpolation are stable (+errata) (with U. Fjordholm & S. Mishra), FoCM 13(2) (2012) 139-159.
  
• A new model for self-organized dynamics and its flocking behavior (with S. Motsch), J. Stat. Physics 144(5) (2011) 923-947.
  
• Energy-preserving and stable approximations for the two-dimensional shallow water equations (with W. Zhong) in "Mathematics and Computation - A Contemporary View", Proc. of the Third Abel Symposium (H. Munthe-Kaas & B. Owren eds.), Springer 2008, pp. 67-94.
  
• L²-stability analysis of the central Discontinuous Galerkin method and a comparison between the central and regular Discontinuous Galerkin methods (with Y.-J. Liu, C.-W. Shu  & M. Zhang) M2AN 42 (2008) 593-607 [highlighted paper].
• Multiscale hierarchical decomposition of images with applications to deblurring, denoising and segmentation
  (with S. Nezzar & L. Vese) Comm. Math. Sciences 6(2) (2008) 281-307.
  
• Recovery of edges from spectral data with noise---a new perspective (with S. Engelberg) SINUM 46(5) (2008), 2620-2635.
  
• Long time existence of smooth solutions for the rapidly rotating shallow-water and Euler equations
  (with B. Cheng) SIAM J. Math. Anal. 39(5) (2008) 1668-1685.
  
• Filters, mollifiers and the computation of the Gibbs phenomenon Acta Numerica 16 (2007) 305-378.
  

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• Velocity averaging, kinetic formulations and regularizing effects in quasilinear PDEs (with T. Tao) CPAM 60 (2007), 1488-1521.
  
• Entropy stability theory for difference approximations of nonlinear conservation laws and related time dependent problems
  Acta Numerica 12 (2003), 451-512.
  
• High resolution methods for time dependent problems with piecewise smooth solutions "International Congress of Mathematicians", Proceedings of the ICM02 Beijing 2002 (Li Tatsien, ed.), Vol. III: Invited lectures, Higher Education Press (2002) 747-757.
  
• On a new scale of regularity spaces with applications to Euler's equations Nonlinearity 14 (2001) 513-532.
  
• Strong stability-preserving high-order time discretization methods (with S. Gottlieb & C.-W. Shu) SIAM Review 43 (2001) 89-112.
  
• New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations (with A. Kurganov)
  JCP, 160 (2000) 241-282.
  
• Approximate solutions of the incompressible Euler equations with no concentrations (with M. Lopes Filho & H. J. Nussenzveig) Annal. de l'institut Henri Poincare (c) Non Linear Analysis, 17 (2000) 371-412.
  
• Approximate solution of nonlinear conservation laws and related equations "Recent Advances in Partial Differential Equations and Applications" 1996 Venice Conference R. Spigler and S. Venakides eds.), AMS Proc. Symp. Appl. Math. 54 (1998) 325-368.
  
• A kinetic formulation of multidimensional scalar conservation laws and related equations (with P.-L. Lions & B. Perthame)
  J. Amer. Math. Soc. 7 (1994) 169-191.
  
• Local error estimates for discontinuous solutions of nonlinear hyperbolic equations SINUM 28 (1991) 891-906.
  
• A kinetic equation with kinetic entropy functions for scalar conservation Laws (with B. Perthame) Comm. Math. Phys. 136 (1991) 501-517.
  
• Non-oscillatory central differencing for hyperbolic conservation laws (with H. Nessyahu) JCP 87 (1990) 408-463.
  
• An O(N²) method for computing the eigensystem of N x N symmetric tridiagonal matrices by the divide and conquer approach (with D. Gill) SIAM J. Scientific Computing 11 (1990) 161-173.
  
• Convergence of spectral methods for nonlinear conservation laws SINUM 26 (1989) 30-44.
  
• Numerical viscosity and the entropy condition for conservative difference schemes Math. Comp. 43 (1984) 369-381
  
• The equivalence of L²-stability, the resolvent condition and strict H-stability LAA 41 (1981) 151-159.
  
• Scheme-independent stability criteria for difference approximations of hyperbolic initial-boundary value problems. II (with M. Goldberg)
  Math. Comp. 36 (1981) 603-626.
  

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