Current Research Focus
Bacteriophage DNA: Continuum and stochastic modeling of DNA configuration and dynamics in viral capsids; model calibration using cryo-EM data; and analysis of DNA structures using topological invariants and topological data analysis.
Alzheimer's Disease: Modeling tau-protein propagation over the cortical surface and along fiber tracts to investigate Alzheimer's disease progression, in collaboration with Wenrui Hao (Penn State) and Guorong Wu's group (UNC--Chapel Hill); and machine-learning-based model calibration using real patient data mapped onto an atlas brain surface.
Structure-Preserving Numerical Methods: Integrating variational principles in developing numerical methods for PDEs. Variational moving-mesh methods for gradient flows, preserving energy dissipation while resolving steep gradients and interfaces; and for hyperbolic conservation laws, ensuring entropy stability while accurately capturing shock formation and propagation.
Broader Research Interests
Mathematical Biology: Mathematical modeling and analysis of biological systems across scales, including DNA, proteins, ion channels, and neurotransmitters, with extensions to cellular, tissue, and network dynamics.
Complex Fluids: Energetic variational modeling and simulation of complex fluid systems, including liquid crystals, polymers, and electrolytes, grounded in the principles of statistical mechanics and thermodynamics, with applications to materials science and energy devices.
Scientific Machine Learning: Neural-network-based modeling and data-driven calibration of biological and complex fluid systems; integration of neural networks with numerical methods.
Grants and Funding
Principal Investigator, NSF DMS Award 2318053 Collaborative Research: Topology and Infection Dynamics of Bacteriophage Viruses}. July 2023 -- June 2027
Collaborative award with UC Davis (J. Arsuaga) and the University of Minnesota (M.C. Calderer).
Professional Service
Panelist and reviewer, NSF Division of Mathematical Sciences (DMS), Spring 2026
Panelist and reviewer, NSF Division of Mathematical Sciences (DMS), Spring 2025
Guest editor, special issue "Recent Advances in Variational Modeling: Theory, Computation, and Applications", Studies in Applied Mathematics (Wiley), 2026
Reviewer, Banff International Research Station (BIRS) workshop proposal, 2025
Submitted and Under Review
D. Siebel-Cortopassi, P. Liu, Numerical study of single-core and split-core defects in nematic liquid crystals, revised manuscript submitted to Advances in Applied Mathematics and Mechanics. arXiv:2510.26215
P. Liu, T. Christiani, Z. Wang, F. Guo, M. Vazquez, M.C. Calderer, J. Arsuaga, Twist dominates over bending in the liquid crystal organization of bacteriophage DNA, submitted, arXiv:2510.04408.
P. Liu, Asymptotic error analysis of Gauss quadrature for nonsmooth functions, submitted. arXiv:2509.23532.
Y. Zhou, P. Liu, S. Zhou, A local approximation model for biomolecular solvation based on molecular density functional theory, submitted.
P. Liu, Energetic variational arbitrary Lagrangian--Eulerian method for gradient flow: Discrete gauge symmetry breaking, submitted.
P. Liu, M.C. Calderer, C. Liu, Energetic variational modeling and analysis of cascade gating at aqueous--hydrophobic interfaces, submitted.
P. Liu, D. Banerjee, D. Siebel-Cortopassi, S. Song, M.C. Calderer, Modeling and Analysis of Cooling-Induced Isotropic–Nematic Transitions in Pharmaceutical Liquid Crystals, submitted
Published:
P. Liu, Z. Wang, T. Christiani, M. Vazquez, M.C. Calderer, J. Arsuaga, Knotted DNA configurations in bacteriophage capsids: A liquid crystal theory approach, SIAM Journal on Applied Mathematics, 86 (2026), pp. 1570--1592.
X. Ji, C. Liu, P. Liu, S. Zhou, Energetic variational approach for prediction of thermal electrokinetics in charging and discharging processes of electrical double layer capacitors, Journal of Power Sources, 551 (2022), 232184.
P. Liu, J. Arsuaga, M.C. Calderer, D. Golovaty, M. Vazquez, S. Walker, Helical organization of DNA-like liquid crystal filaments in cylindrical viral capsids, Proceedings of the Royal Society A, 478 (2022), 20220047.
P. Liu, J. Arsuaga, M.C. Calderer, D. Golovaty, M. Vazquez, S. Walker, Ion-dependent DNA configuration in bacteriophage capsids, Biophysical Journal, 120 (2021), pp. 3292--3302.
C.-Y. Hsieh, T.-C. Lin, C. Liu, P. Liu, Global Existence of the Non-isothermal Poisson-Nernst-Planck-Fourier System. Journal of Differential Equations, 269 (2020), pp. 7287--7310.
J. Liang, P. Liu, Z. Xu, A fast Poisson solver of arbitrary order accuracy through the harmonic surface mapping algorithm. Communications in Computational Physics, 30 (2021), pp. 210--226.
Y. Wang, C. Liu, P. Liu, B. Eisenberg, Field theory of reaction-diffusion: Mass action with an energetic variational approach . Physical Review E, 102 (2020), 062147.
P. Liu, S. Wu, C. Liu, Non-Isothermal Electrokinetics: Energetic Variational Approach. Comm. Math. Sci., 5 (2018), pp. 1451–1463.
Q. Zhao , P. Liu, Z. Xu, A fast method for evaluating Green's function in irregular domains with application to charge interaction in a nanopore. Commun. Comput. Phys., 24 (2018), pp. 1241-1258.
L. Ji, P. Liu, Z. Xu, S. Zhou, Asymptotic Analysis on Dielectric Boundary Effects of Modified Poisson--Nernst--Planck Equations . SIAM J. Appl. Math., 78(3): 1802--1822, 2018.
Z. Wu, P. Liu, Y. Liu, W. Wei, X. Zhang, P. Wang, Z. Xu, H. Xiong, Regulating sequence distribution of polyethers via ab initio kinetics control in anionic copolymerization. Polym. Chem.,8: 5673--5678, 2017.
P. Liu, X. Ji, Z. Xu, Modified Poisson-Nernst-Planck model with accurate Coulomb correlation in variable media. SIAM J. Appl. Math., 78(1), 226--245, 2017.
P. Liu, C. Liu, Z. Xu, Generalized Shockley-Ramo Theorem in electrolytes . Comm. Math. Sci., 15(2): 555--564, 2017.
P. Liu, M. Ma, Z. Xu, Understanding depletion induced like-charge attraction from self-consistent field model. Comm. Comp. Phys., 22(1): 95--111, 2017.
Z. Xu, M. Ma, P. Liu, Self-energy modified Poisson-Nernst-Planck equations: WKB approximation and finite-difference approaches. Phys. Rev. E., 90(1), 013307, 2014.
B. Li, P. Liu, S. Zhou, Z. Xu, Ionic size effects: Generalized Boltzmann distributions, counterion stratification, and modified Debye length. Nonlinearity, 26(10): 2899--2922, 2013. Selected in the journal's Highlights of 2013.