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  • Jose A. Carrillo, Hailaing Liu, and Hui Yu
    Positivity-preserving and energy-dissipating discontinuous Galerkin methods for nonlocal Fokker-Planck equations. 
    Arxiv: 2403.15643
  • Yaoyao Chen, Hailiang Liu, Nianyu Yi and Peimeng Yin
    Unconditionally energy stable IEQ-FEMs for the Cahn-Hilliard equation and Allen-Cahn equation.
    ArXiv: 2402.02712
  • Xiaokai Huo, Hailiang Liu
    Inf-Sup neural networks for high-dimensional elliptic PDE problems
  • Xuping Tian, Baskar Ganapathysubramanian, Hailiang Liu
    Data-driven optimal control with neural network modeling of gradient flows
  • Tianxiang Gao, Xiaokai Huo, Hailiang Liu, Hongyang Gao
    Wide Neural Networks as Gaussian Processes: Lessons from Deep Equilibrium Models
  • Hailiang Liu, Levon Nurbekyan, Xuping Tian, Yunan Yang
    Adaptive Preconditioned Gradient Descent with Energy 
  • M. Bhtnagar, H. Liu 
    On critical thresholds for hyperbolic balance law systems
    arXiv:2302.12869  (published HYP2022 in Malaga)
  • M. Bhtnagar, H. Liu
    A complete characterization of sharp thresholds to spherically symmetric multidimensional pressureless Euler-Poisson systems
  • Hailiang Liu and Xuping Tian
    Dynamic behavior for a gradient algorithm with energy and momentum.
    ArXiv: 2203.12199, 2022; CMS.  



[167] Hailiang Liu, Jia-Hao He, Xuping Tian
Anderson acceleration of gradient methods with energy for optimization problems
ArXiv: 2211.08578
Communications on Applied Mathematics and Computation (CAMC)
published: 12/28/2023.

[166] Joshua Ballew, Gautam Iyer, C. David Levermore, Hailiang Liu, and Robert L. Pego. 
Global dynamics and photon loss in the Kompaneets equation.  
ArXiv: 2208.09755; SIAM J. Math. Anal. (2023)

[165] H. Liu and Xuping Tian
SGDEM: Stochastic gradient decent with energy and momentum.
Numerical Algorithms, 2023.

[164] Tianxiang Gao, Xiaokai Huo, Hailiang Liu, Hongyang Gao 
Exploring the Gaussian Process Nature of Wide Neural Networks: Insights from Deep Equilibrium Models
ArXiv: 2310.10767
 " The Thirty-Seventh Annual Conference on Neural Information Processing Systems (NeurIPS 2023), New Orleans, Louisiana,  2023.

[163] M. Bhtnagar, H. Liu and C.-H. Tan
Critical Thresholds in the Euler-Poisson-alignment system 
ArXiv: 2111.11999, 2021.  
Journal of Differential Equations, Volume 375 (5), December 2023, Pages 82-119 

[162] H. Liu and W. Maimaitiyiming
A dynamic mass transport method for Poisson-Nernst-Planck equations
Journal of Comput. Phys,  473, 111699, 2023. 

[161] H. Liu and P.-M. Yin 
On the SAV-DG method for a class of fourth order gradient flows
Numerical Methods in PDEs 39(2), 1185--1200, 2023 (Open Access) 

[160] H. Liu and X.P. Tian
AEGD: adaptive gradient decent with energy. 
arXiv:  2010.05109
Numerical Algebra, Control and Optimization  April 2023. Doi: 10.3934/naco.2023015 

[159] H. Liu and X.P. Tian, 
Data-driven optimal control of a SEIR model for COVID-19
arXiv:2102.00698, 2021. 
Commm. Pure and Appl. Anal., 22(1): 19--39, 2023. DOI:10.3934/cpaa.2021093. 


[158] H. Liu and Xuping Tian, 
 An adaptive gradient method with energy and momentum
arXiv:2203.12191, 2022.
Annals of Applied Mathematics, 38(2): 1-40, 2022.

[157] Tianxiang Gao, Hailiang Liu, Jia Liu, Hridesh Rajan, and Hongyang Gao
A global convergence theory for deep implicit learning via over-parametrization
arXiv:2110.05645, 2021.
ICLR’22: The 10th International Conference on Learning Representations (Apr. 2022).

[156] M. Bhatnagar and H. Liu,
 Global dynamics of the Euler-alignment system with weakly singular kernel
ArXiv: 2110.10314 (2021) 
Appl. Math. Letters, 128: 107856, 2022. 

[155] H. Liu, Z.M. Wang, P.M. Yin, and H. Yu,
Positivity-preserving third order DG schemes for Poisson–Nernst–Planck equations
J. Comput. Phys. 452: 110777, 2022.

[154] H. Liu and P. M. Yin 
High order unconditionally energy stable RKDG schemes for the Swift-Hohenberg equation
J. Comput. Appl. Math., Vol 407, 114015, 2022. 

[153] H. Liu and J. Shin
Necessary conditions for blow-up solutions to the restricted Euler-Poisson equations
Communication in Mathematical Sciences (CMS), 20(2): 327--357, 2022. 
ArXiv.2107.0415 (2021) 


[152] X Huo, H Liu, AE Tzavaras, S Wang
An energy stable and positivity-preserving scheme for the Maxwell-Stefan diffusion system.
SIAM J. Numer. Anal. 59(5): 2321--2345, 2021. 
arXiv preprint arXiv:2005.08062.
[151] Alberto Bressan, Yi Jiang, Hailiang Liu,
Numerical Study of Non-uniqueness for 2D Compressible Isentropic Euler Equations.
J. Compt. Phys. 445: 110588, 2021. 
DOI: 10.1016/ arXiv preprint arXiv:  2009.09494

[150] H. Liu and  F. Thein,
On the invariant region for compressible Euler equations with a general equation of state. 
Commm. Pure and Appl. Anal., 20(7\& 8): 2751--2763, 2021. DOI:10.3934/cpaa.2021084.
in honor of Prof. Chen Shuxing on the occasion of his 80th birthday.

[149] Manas Bhatnagar and Hailiang Liu 
Sharp critical thresholds in a hyperbolic system with relaxation.
Discrete & Continuous Dynamical Systems (DCDS), 41(11): 5271--5289, 2021.
DOI: 10.3934/dcds.2021098. arXiv preprint arXiv:2012.06804. 

[148] Manas Bhatnagar and Hailiang Liu,
Well-posedness and critical thresholds in a nonlocal Euler system with relaxation.
Discrete & Continuous Dynamical Systems (DCDS), 2021.
DOI: 10.3934/dcds.2021076,  arXiv preprint arXiv: 2010.02362 

[147]   N.-Y. Yi and H. Liu
A mass- and energy-conserved DG method for the Schr\"{o}dinger-Poisson equation
Numerical Algorithms, June 2021.
DOI: 10.1007/s11075-021-01139-0.

[146]  X.-K. Huo and H. Liu
A positivity-preserving and energy stable scheme for a quantum diffusion equation.
Numer. Methods Partial Differ. Eq. (2021), 1–27.
DOI: 10.1002/num.22809. arXiv preprint ArXiv: 1912.00813

[145] H. Liu,
Analysis of direct discontinuous Galerkin methods for multi-dimensional convection-diffusion equations,
Numerische Mathematik, 147(4), 839-867, 2021. 

[144] H. Liu and W. Maimaitiyiming
Efficient, positive, and energy stable schemes for multi-D Poisson-Nernst-Planck systems.
J. Sci. Comput. 87: 92, 2021. 
DOI:10.1007/s10915-021-01503-1.  arXiv preprint ArXiv:2001.08350 

[143] W.-L. Cai and H. Liu
Evolution of species trait in a discrete system of resource competition.
Communication in Mathematical Sciences, 19(3), 737--760, 2021. 
DOI:10.4310/CMS.2021.V19.N3.A8.  arXiv preprint ArXiv:2007.12596

[142] Matthias Kunik, Hailiang Liu, and Gerald Warnecke
Radially symmetric solutions of the ultra-relativistic Euler equations.
to appear in Methods and Applications of Analysis, 28(4): 401--422, 2021.
Dedicated to Professor Hsiao Ling on the occasion of her 80th birthday.
arXiv preprint arXiv:2002.01181

[141] Hailiang Liu and Peimeng Yin
Unconditionally energy stable discontinuous Galerkin schemes for the Cahn-Hilliard equation.
Journal of Computational and Applied Mathematics, 390: 113375, 2021. 
DOI:10.1016/ arXiv preprint ArXiv:1912.10197


[140] Manas Bhatnagar and Hailiang Liu,
Critical thresholds in 1D pressureless Euler-Poisson systems with varying background.
Physica D, 414:132728, 2020. DOI: 10.1016/j.physd.2020.132728
arXiv preprint arXiv:2004.06807

[139] H. Liu and P. Markowich
Selection dynamics for deep neural networks
J. Diff. Equ., 269: 11540--11574, 2020. DOI: 10.1016/j.jde.2020.08.041.
arXiv preprint ArXiv:1905.09076 

[138] H. Liu and W. Maimaitiyiming
Positive and free energy satisfying schemes for diffusion with interaction potentials
J. Comput. Phys. 419:109483, 2020. DOI: 10.1016/
ArXiv preprint Arxiv: 1910.00151.

[137] M. Bhatnagar and H. Liu, Critical thresholds in one dimensional damped Euler-Poisson systems
Math. Mod. Meth. Appl. Sci. (M3AS), 30(5): 891--916, 2020. . DOI: 10.1142/S0218202520500189.

[136] H. Liu and W. Maimaitiyiming
Unconditional positivity-preserving and energy stable schemes for a reduced Poisson-Nernst-Planck system.
Commun. Comput. Phys., 27: 1505--1529,  2020. DOI: 10.4208/cicp.OA-2019-0063
Arxiv: 1909.13161.

[135] H. Liu
On structure-preserving high order methods for conservation laws
XVII International conference on: Theory, Numerics and Applications of Hyperbolic Problems, College Park, June 25-29, 2018. 
AIMS on Applied Mathematics, Vol 10: 203--213, 2020. 


[134] X.-B. Feng, H. Liu and S. Ma, Mass-and energy-conserved numerical schemes for nonlinear Schroedinger equations
Commun. Comput. Phys. 26:1365--1396, 2019.

[133] W.-L. Cai, P. Jabin, and H. Liu
Time-asymptotic convergence rates towards discrete steady states of a nonlocal selection-mutation model,
Math. Mod. Meth. Appl. Sci. (M3AS), 29(11), 2063--2087.

[132] H. Liu, J. Ralston and P.-M. Yin
General superpositions of gaussian beams and propagation errors
Math. Comp. 89:322, 675--697, 2019.

[131] H. Liu and P-M. Yin
Unconditional energy stable DG methods for the Swift-Hohenberg equation
Journal of Scientific Computing,  81(2): 789--819, 2019.

[130] N. Yi, Y. Huang and H. Liu
A conservative discontinuous Galerkin method for nonlinear electromagnetic Schroedinger equations
SIAM J. Sci. Comput., 41(6): B1389--B1411, 2019.

[129] H. Yu and H. Liu
Third order maximum-principle-satisfying DG schemes for convection-diffusion problems with anisotropic diffusivity
Journal Comp. Physics. 391: 14–36, 2019.

[128] H. Liu, R. Groepler, and G. Warnecke
A high order positivity preserving DG method for coagulation-fragmentation equations
SIAM J. Sci. Comput., 41(3): B448--B465, 2019.

[127] Yi Jiang and H. Liu
An invariant region preserving limiter for DG schemes to isentropic Euler equations
Numerical methods for PDEs, 35(1):5--33, 2019.

[126] H. Liu and L.-R. Qian
Alternating evolution methods for static Hamilton-Jacobi equations
Journal of Computational and Applied Mathematics, 351: 270--289, 2019.

[125] H. Liu, Y.-Q. Huang, W.-Y. Lu and N.Y. Yi
On accuracy of the mass preserving DG method to multi-dimensional Schroedinger equations
IMA Journal of Numerical Analysis, 39:760--791, 2019.


[124] H. Liu and P-M. Yin
A mixed discontinuous Galerkin method without interior penalty for time-dependent fourth order problems,
J. of Scientific Computing. 77(1), 467--501, 2018.

[123] Yi Jiang and H. Liu
Invariant-region-preserving DG methods for multi-dimensional hyperbolic conservation law systems, with an application to compressible Euler equations,
Journal of Comput. Phys. 373(15), 385--409, 2018. arXiv:1804.08829 

[122] H. Liu and H.-R. Wen
Error estimates of the third order Runge-Kutta alternating evolution discontinuous Galerkin method for convection-diffusion problems,
ESAIM: Mathematical Modelling and Numerical Analysis (M2AN), 52(5): 1709--1732, 2018.

[121] Yi Jiang and H. Liu
An Invariant-region-preserving (IRP) limiter to DG methods for compressible Euler equations
HYP 2016: Theory, Numerics and Applications of Hyperbolic Problems II, 71–83, 2018.
The Springer Proceedings in Mathematics and Statistics book series (PROMS, volume 237).

[120] N.-Y. Yi and H. Liu
An energy conserving discontinuous Galerkin method for a nonlinear variational wave equation, Comm. Comput. Phys. 23(3), 747--772, 2018.

[119] P.-M. Yin, Y.-Q. Huang and H. Liu
Error estimates for the iterative discontinuous Galerkin method to the nonlinear Poisson-Boltzmann equation, Comm. Comput. Phys. 23(1):168--197, 2018.

[118] H. Liu and H.-R. Wen,
Error estimates of the AEDG method for one-dimensional linear convection-diffusion equations, Math Comput. 87(309): 123--148, 2018.


[117] P-E. Jabin and H. Liu
On a nonlocal selection-mutation model with a gradient flow structure, Nonlinearity, 30:4220--4238, 2017.

[116] W.-L. Cai and H. Liu,
A finite volume method for nonlocal competition-mutation equations with a gradient flow structure,
ESAIM: M2AN, 51(4): 1223--1243, 2017.

[115] H. Liu and M. Pryporov
Error Estimates for Gaussian beam methods applied to symmetric strictly hyperbolic systems, Wave Motion, 73:57--75, 2017.

[114] H. Liu and Z.-M. Wang
A free energy satisfying discontinuous Galerkin method for Poisson-Nernst-Planck systems, J. Comput. Phys. 238: 413--437, 2017.

[1113] W.-X. Cao, H. Liu and Z.-M. Zhang
Superconvergence of the direct discontinuous Galerkin method for convection-diffusion equations
Numerical Methods for Partial Differential Equations, 33(1): 290--317, 2017.


[112] T. Li, H. Liu and L.-H. Wang
On traveling wave solutions to the Keller-Segel model of mixed type, J. Differential Equations, 261: 7080--7098, 2016.

[111] H. Liu, O. Runborg and N. Tanushev
Sobolev and Max Norm Error Estimates for Gaussian Beam Superpositions, Comm. Math. Sci. 14(7): 2041--2076, 2016.

[110] D. Levermore, H. Liu and R. Pego
Global dynamics of Bose-Einstein condensation for a model of the Kompaneets equation, SIAM J. MATH. ANAL. 48(4): 2454--2494, 2016.

[109] H. Liu and Z.-M. Wang
An entropy satisfying discontinuous Galerkin method for nonlinear Fokker-Planck equations J Sci Comput., 68:1217--1240, 2016.

[108] H. Liu and Y.-L. Xing
An invariant preserving discontinuous Galerkin methods for the Camassa-Holm equation, SIAM J. Sci. Comput. 38(4): A1919--1934, 2016.

[107] H. Liu and N.-Y. Yi,
A Hamiltonian preserving discontinuous Galerkin method for the generalized Korteweg-de Vries equation, J. Comp. Phys, 15: 776--796, 2016.

[106] H. Liu and M. Pollack,
Alternating evolution Galerkin methods for convection-diffusion equations, J. Comp. Phys. 307 (2016), 574--592.

[105] H. Liu and T. Pendleton, On invariant-preserving finite difference schemes for the Camassa-Holm equation,
Communication in Computaitonal Physics, 19(04):1015--1041, 2016.


[104] Y. Lee and H. Liu
Threshold for shock formation in the hyperbolic Keller--Segel model, Appl. Math. Letters. 50 (2015), 56--63.

[103] H. Liu, W.-L. Cai and N. Su
Entropy satisfying schemes for computing selection dynamics in competitive interactions, SIAM J. Numer. Anal. 53(3) (2015), 1393--1417.

[102] W.-L. Cai, P. Jabin and H. Liu
Time-asymptotic convergence rates towards the discrete evolutionary stable distribution, Mathematical Models and Methods in Applied Sciences
(M3AS), 25(8) (2015), 1589--1616.

[101] W.-Y. Lu, Y.-Q Huang and H. Liu
Mass preserving direct discontinuous Galerkin methods for Schr\"{o}dinger equations, J. Comp. Phys., 282(1) (2015), 210--226.

[100] H. Liu
Optimal error estimates of the directdiscontinuous Galerkin method for convection-diffusion equations,  Math. Comp. 84 (2015), 2263--2295.

[99] Y. Lee and H. Liu
Thresholds for shock formation in traffic flow models with Arrhenius look-ahead dynamics,  Discrete and Continuous Dynamical Systems --Series A, 35(1) (2015), 323--339.

[98] Tae-Gab Ha and H. Liu
On traveling wave solutions of the theta-class of dispersive equations,  J. Math. Anal. Appl. 421(1) (2015), 399--414

[97] H. Liu and M. Pryporov
Error estimates of the Bloch band-based Gaussian beam superposition for the Schroedinger equation, Contemporary math. 640 (2015), 87--114.

[96] H. Liu and N. Ploymaklam
A local discontinuous Galerkin method for the Burgers-Poisson equation, Numer. Math. 129 (2) (2015), 321--351.

[95] H. Liu and H. Yu,
The entropy satisfying discontinuous Galerkin method for Fokker-Planck equations, J. Sci. Comput. 62 (2015), 803--830.


[94] H. Liu and H. Yu,
Maximum-principle-satisfying third order discontinuous Galerkin schemes for Fokker-Planck equations, SIAM J. Sci. Comput. 36(5)(2014), A2296--A2325.

[93] H. Liu and Z.-M. Wang, 
A free energy satisfying finite difference method for Poisson-Nernst-Planck equations, J. Comput. Phys. 268 (2014), 363--376.

[92] Peimeng Yin, Yunqing Huang and H. Liu,
A direct discontinuous Galerkin (DDG) method for solving the nonlinear Poisson-Boltzmann equation, Commun. Comput. Phys. 16(2) (2014), 491--515.

[91] H. Liu,  J. Ralston, O. Runborg and N. Tanushev
Gaussian beam methods for the Helmholtz equation, SIAM J. Appl. Math. 74(3) (2014), 771--793.

[90] H. Liu, Y.-Q.  Huang and N.-Y. Yi
A direct discontinuous Galerkin method for the Degasperis-Procesi equation, Methods and Applications of Analysis. 21(1) (2014), 83--106.

[89] H. Liu and M. Pollack
Alternating evolution DG methods for Hamilton-Jacobi equations, J. Comput. Phys. 258 (2014), 32-46.

[88] H. Liu and Hui Yu
Entropy/Energy stable schemes for evolutionary dispersal models,  J. Comp. Phys.  256 (2014), 656--677.


[87] H. Liu
The alternating evolution methods for first order nonlinear partial differential equations
Communications in Information and Systems (CIS), 13(3) (2013), 291--325.

[86] Yongki Lee and H. Liu
Thresholds in three-dimensional restricted Euler-Poisson equations, Phys. D. 262 (2013), 59--70.

[85] Nianyu Yi, Yunqing Huang and H. Liu
A direct discontinuous Galerkin method for the Korteweg-de Vries equation: energy conservation and boundary effect, J. Comp. Phys. 242 (2013), 351--366.

[84] H. Liu, M. Pollack and H. Saran
Alternating evolution schemes for Hamilton-Jacobi equations, SIAM J. Sci. Comput. 35(1) (2013), 122--149.

[83] H. Liu,  O. Runborg and N. Tanushev
Error estimates for Gaussian beam superpositions,  Math Comp. 82 (2013), 919--952.


[82] H. Liu and J. Shin
The Cauchy-Dirichlet problem for the FENE dumbbell model of polymeric flows, SIAM J. Math. Anal. 44(5) (2012), 3617--3648.

[81] Hailiang Liu and Hui Yu
An entropy satisfying conservative method for the Fokker Planck equation of FENE dumbbell model. SIAM Journal on Numerical Analysis 2012, Vol. 50, No. 3, pp. 1207-1239 

[80] Yunqing Huang, Hailiang Liu and Nianyu Yi
Recovery of normal derivatives from the piecewise L^2 projection, J. Comp. Phys. 231 (2012), 1230--1243.  

[79] Jaemin Shin and H. Liu
 Global well-posedness for the microscopic FENE model with a sharp boundary condition,  Journal Diff. Equ.  252 (2012), 641--662.

[78] P. Degond, H. Liu, D. Savelief and M.-H. Vignal
Numerical approximation of the Euler-Poisson-Boltzmann model in the quasineutral limit,  J. Sci. Comput.  51(1) (2012), 59--86.

 [77]  Haseena Saran and H. Liu
Alternating evolution (AE) schemes for hyperbolic conservation laws,  SIAM J. on Scientific Computing. 33(6) (2011), 3210--3240.

[76]  H. Liu and Z.-Y. Yin
Global regularity, and wave breaking phenomena in a class of nonlocal dispersive equations. Contemporary Mathematics, 526 (2011), 274--294. Nonlinear Partial Differential Equations and Hyperbolic Wave Phenomena. Helge Holden and Kenneth H. Karlsen, Editors.

[75] H. Liu, Z.-M. Wang and R. Fox
Level set approach for dilute non-collisional fluid-particle flows, Journal of Computational Physics 230 (4) (2011), 920--936.

[74] H. Liu and J. Yan
The Direct Discontinuous Galerkin (DDG) method for diffusion with interface corrections, Commun. Comput. Phys. 8(3) (2010), 541--564.

[73] H. Liu and J. Ralston
Recovery of high frequency wave fields from phase space based measurements, Multiscale Model. Simul. 8(2) (2010), 622--644.


[72]H. Liu and J. Ralston
Recovery of high frequency wave fields for the acoustic wave equation, Multiscale Model. Simul. 8(2) (2009), 428--444.

[71] P. Degond and H. Liu
Kinetic models for polymers with inertial effects, Networks and Heterogeneous Media. 4(4)(2009), 625--647.

[70] H. Liu, E. Tadmor and D. Wei
Global regularity of the 4D Restricted Euler equations, Physics D. published online July 2009.

[69] Tong Li and H. Liu

Critical thresholds in hyperbolic relaxation systems, J. Diff. Equ. 247(1) (2009), 33--48.

[67] H. Liu and Zhongming Wang

A Bloch band based level set method for computing the semi-classical limit  of Schroedinger equations, J. Comp. Phys.  228(9) (2009), 3326--3344.

[66] Tong Li and H. Liu
Critical thresholds in a relaxation system with resonance of characteristic speeds, Discrete and Continuous Dynamical Systems, Series A. 24(2) (2009), 511-521.

[65]  H. Liu and J. Yan
The Direct Discontinuous Galerkin (DDG) methods for diffusion problems, SIAM Journal on Numerical Analysis,  47(1) (2009), 675--698.


[64] H. Liu
On discreteness of the Hopf equationActa Math. Appl. Sin. Engl. Ser. 24 (2008), no. 3, 423--440.

[63] H. Liu and Z.M. Wang 
Superposition of multi-valued solutions in high frequency wave dynamics, Journal of Scientific Computing, 35(2-3) (2008), 192--218.

[62] H. Liu and C. Sparber

Rigorous Derivation of the Hydrodynamical Equations for Rotating Superfluids, Mathematical Models and Methods in Applied Sciences, 18(5) (2008), 689�706.

[61] H. Liu
An alternating evolution approximation to systems of hyperbolic conservation laws,  Journal of Hyperbolic Differential Equations, 5(2) (2008), 1--27.

[60] T. Li and H. Liu
Critical thresholds in a relaxation model for travel flows, Indiana Univ. Math. Journal,57(3) (2008), 1409--1430.

[59]  M. D. Francesco, K. Fellner and H. Liu
A non-local conservation law with nonlinear `radiation�inhomogeneity. J. Hyperbolic Differ. Equ.  5  (2008),  no. 1, 1--23.

[58] C. Liu and H. Liu,

Boundary conditions for the microscopic FENE models, SIAM J. Appl. Math. 68 (2008),no. 5, 1304--1315.


[57] H. Liu
Global orientation dynamics for liquid crystalline polymers, Physica D. 228 (2007), 122-129.

[56] H. Liu and Z.M. Wang,
A field space-based level set method for computing multi-valued solutions to 1D Euler-Poisson equations, Journal of Computational Physics, 225 (2007), 591--614.

[55] H. Liu and Z.M. Wang
Computing multi-valued velocity and electric fields for 1D Euler-Poisson equations,
Applied Numerical Mathematics, 57 (2007), 821�836.

[54] H. Liu, S. Osher and R. Tsai, Multi-valued solution and level set methods in computational  high frequency wave propagation, Commun. Comput. Phys. 1(5) (2006), 765-804.

[53] H.L. Liu, L.-T. Cheng and S. Osher, A level set framework for capturing  multivalued solutions to nonlinear first-order equations, J. Sci. Comput. 29(3)  (2006), 353--373.

[52] H. Liu
Wave Breaking in a class of nonlocal dispersive wave equations,  Journal of Nonlinear Math Phys. 13 (3), (2006), 441-466.

[51] H. Liu and J. Yan
A local discontinuous {G}alerkin method for the {K}orteweg-de {V}ries equation with boundary effect, JCP 215 (2006), 197-218.

[50] H. Liu
Critical Thresholds in the Semiclassical Limit of  2-D Rotational Schr\"{o}dinger Equations, ZAMP. 57 (2006), 42-58.

[49] H. Liu, H. Zhang and P.W. Zhang

 Axial Symmetry and Classification of Stationary Solutions of Doi-Onsager Equation on the Sphere with Maier-Saupe Potential, Comm. Math. Sci. Vol 3 (2), (2005), 201-218.

[48] S. Jin,  H.L. Liu, S. Osher and R. Tsai

Computing multi-valued physical observables for the high frequency limit of symmetric hyperbolic systems, JCP, 210(2005), 498-518.

[47] S. Jin,  H.L. Liu, S. Osher and R. Tsai (2005)

Computing multi-valued physical observables for the semiclassical limit of the Schroedinger equation, JCP, 205 (2005), 222-241.

[46] T.  Li  and  H. Liu

Stability of a traffic flow model with  nonconvex relaxation, Comm. Math. Sci., Vol 3 (2), (2005), 101--118.


[45]  H. Liu  
Relaxation Dynamics, Scaling Limits and Convergence of Relaxation Schemes,  (2005) Springer Verlag.

[44] H.L. Liu and E. Tadmor
Rotation Prevents Finite Time BreakdownPhysica D 188 (2004) 262-276.

[43]G.-Q. Chen  and  H.L. Liu 
Concentration and Cavitation in Solutions of the Euler equations for nonisentropic fluids as the pressure vanishes Physics. D.  189 (2004), 141--165.

[42] H.L. Liu and M. Slemrod
KdV  Dynamics in the Plasma-Sheath Transition,   Appl. Math. Lett.  17  ( 2004)   401--419.

[41] S. Jin,  H.L. Liu, S. Osher and R. Tsai
Computing multivalued physical observables for the semiclassical limit of the Schrodinger equations,  JCP, 205 (1) (2005), 222-241.

[40] L.T. Cheng, H.L. Liu and S. Osher
Computational high frequency wave propagation using the level set method,  with applications to the semi-classical  limit of Schroedinger equations,  Comm. Math. Sci. Vol 1, No. 3 (2003), 593--621.

[39] L.T. Cheng, H.L. Liu and S. Osher
High frequency wave propagation in Schroedinger equations using the Level Set method, preprint, 2003.

[38] H.L. Liu and E. Tadmor
Critical Thresholds in 2-D Restricted Euler-Poisson Equations,  SIAM J. Appl. Math. 63 (6) (2003), 1889--1910.

[37]  H.L. Liu  
Asymptotic Stability of Relaxation Shock Profiles for Hyperbolic Conservation Laws,   J. Diff. Equ.  192 (2003), 285--307.

[36] G.-Q. Chen  and  H.L. Liu 
Formation of Delta-Shocks and Vacuum States in the Vanishing Pressure Limit of Solutions to the Isentropic Euler Equations
 SIAM J. Math Anal.  34 (2003), 925--938.

[35] J. Pan and H.L. Liu
Convergence rates to traveling waves of viscous conservation laws with dispersion 
J. Diff. Equ.  187 (2003), 337--358.

[34] H. L. Liu
The l^1 Global Decay to Discrete Shocks for Scalar Monotone Schemes,   Math. Comp. 72 (2003), 227-245.


[33] H.L. Liu and E. Tadmor
Semi classical Limit of the Nonlinear Schrodinger-Poisson Equation with Subcritical Initial Data
 Methods and Applications of Analysis, Vol 9, No. 4 (2002), 517--532.

[32]  H.L. Liu and E. Tadmor (2002)
Critical Thresholds and Conditional Stability for Euler Equations and Related Models, Proceedings
of the Ninth International Conference on ''Hyperbolic Problems: Theory, Numerics,  Applications",
 Editors: T.Y. Hou and E. Tadmor, Springer,  pp227--240.

[31] H.L. Liu and E. Tadmor 
Spectral Dynamics of the Velocity Gradient Field in Restricted  Fluid Flows
 Commun. Math. Phys. 228 (2002), 435--466.  

[30] H.L. Liu and E. Tadmor
Critical Thresholds in a Convolution Model for Nonlinear Conservation Laws,
  SIAM J. Math. Anal. 33 (2002), 930--945.


[29] H.T.  Fan and  H.L. Liu  (2001)
Pattern Formation, Wave Propagation and Stability in Conservation Laws  with  Slow Diffusion and Fast Reaction
 UCLA CAM report 01-24. submitted

[28] S. Engelberg, H.L. Liu and E. Tadmor,
Critical Thresholds in Euler-Poisson equations
Indiana University Mathematics Journal,  50 (2001),  109--157.

[27] H.L.  Liu  and  W.-A. Yong  
Time-Asymptotic  stability of boundary-layers for a hyperbolic relaxation system
Comm. Partial  Diff. Equ., 26 (2001), 1323--1343.

[26]  H.L. Liu 
Asymptotic decay to the relaxation shock fronts in two dimensions,    Proceedings of the Royal Society of  Edinburgh: Section A  131A (2001), 1385--1410.

[25] H. L. Liu, J. Wang and G. Warnecke
Convergence of a splitting scheme applied to the R-W model of the Boltzmann equation,  J. Comp. Appl. Math.  134 (2001), 343--367.

[24] H.L. Liu, J. Wang and G. Warnecke
Convergence rate to discrete shocks for nonconvex conservations,  Numerische Mathematik,  88 (2001), 513-541.

[23] H. Liu and R. Natalini
Long-Time Diffusive Behavior of Solutions to a Hyperbolic Relaxation System,  Asymptotic Analysis,  25  (2001) 21-38.

[22] H.L. Liu
The L^p Stability of Relaxation Rarefaction Profiles,  J. Differ Equations,  171, (2001),  397-411.


[21] H. Liu, J. Wang and G. Warnecke
The lip^+ Stability and Error Estimates for a Relaxation Scheme,  SIAM J. Numer. Anal. 38 (2000), no. 4, 1154--1170

[20] S. Jin and H. L. Liu
A Diffusive Subcharacteristic Condition for Hyperbolic Systems with Diffusive Relaxation, Transport Theory and Statistical Physics, 29 (3-5), (2000), 583--593.

[19] H.L. Liu, G. Warnecke
Convergence rates for relaxation schemes approximating conservation laws, SIAM J. Numer. Anal. 37, No. 4, (2000), 1316--1337.

[18] H.L. Liu
Convergence rate to the discrete traveling wave for relaxation schemes,  Math. Comp. 69 (2000), 583--608.


[17]H.L. Liu, J. Wang and T. Yang
Nonlinear stability and existence of stationary discrete traveling waves for the relaxing schemes, Japan J. Indust. Appl. Math. 16 (1999), 195--224.


[16] J. Pan, H.L. Liu
On stability of travelling waves of Burgers-Fisher equation, Ann. Differential Equations, 14 (1998), 37--47.

[15] H.L. Liu
Nonlinear stability of shock profiles for non-convex model equations with degenerate shock, J. Partial Differential Equations 11(1998), 209--230.

[14] S. Jin and H. L. Liu
Diffusion limit of a hyperbolic system with relaxation, Methods Appl. Anal. 5 (1998) 317-334.

[13] H.L. Liu, J. Wang and T. Yang
Stability for a relaxation model with a nonconvex flux, SIAM J. Math. Anal. 29 (1998), 18-29.

[12] H.L. Liu and J. Wang
Asymptotic stability of stationary discrete shocks of Lax-Friedrichs scheme for non-convex conservation laws, Japan J. Indust. Appl. Math. 15 (1998), 1-16.


[11] H.L. Liu, C. W. Woo and T. Yang
Decay rate for traveling waves of a relaxation model, J. Diff. Equ. 134 (1997), 343-367.

[10]H. L. Liu, J. Wang and T. Yang
Existence of the discrete traveling waves for a relaxing scheme, Appl. Math. lett., 10 (1997), 117-122.

[9]H.L. Liu
Asymptotic stability of shock profiles for non-convex convection diffusion equation, Appl. Math. Lett. 10 (1997), 129-134.

[8] H.L. Liu, J. Wang
Decay rate for perturbations of stationary  discrete shocks for convex scalar conservation laws, Math. Comp. 66 (1997), 69-84.


[7] H. L. Liu
Asymptotic properties of solutions to nonconvex hyperbolic conservation laws.(Chinese), Gaoxiao Yingyong Shuxue Xuebao Ser. A 11 (1996), 277--282.

[6] H.L. Liu, J. Wang
Asymptotic stability of traveling wave solution for a hyperbolic system with relaxation terms, Beijing Math. 2 (1996), 119-130.

[5] H. L. Liu, J. Wang
Nonlinear stability of stationary discrete profiles of non-convex scalar conservation laws, Math. Comp. 65 (1996), 1137-1153.


[4] H. Liu and J. Pan
Decay rate for perturbations of viscous shock profiles for non-convex convection-diffusion equation, Appl. Functional Anal. 2 (1995), 171-176.

[3] H. L. Liu
An existence theorem for radial positive solutions of nonlinear elliptic equations, Sys. Sci. Math. Sci. 7 (1994), 1-4.

[2] H. L. Liu
The interactions of shock waves of nonstrictly hyperbolic systems, Acta Math. Scientia, 12 (1992), 312-336.

[1] H. L. Liu
Large time behavior of solutions of the porous medium equation with convection, Acta Mathematicae Appl. Sinica 15 (2) (1992), 239-256.