Chao Liu - Math

Self Introduction

2019.10-present: Associate Professor, Center for Mathematical Sciences and School of Mathematics and Statistics, Huazhong University of Science and Technology
2017.10-2019.9: Postdoc, International Center for Mathematical Research (BICMR), Peking University. Supervisor: Prof. Gang Tian.
2014.2-2017.9: PhD, School of Mathematics, Monash University. Supervisors: Prof. Todd A. Oliynyk (main) and Dr. Julie Clutterbuck (associate).
2010.9-2013.7: MS, School of Mathematics, Hebei University. Supervisors: Prof. Hongya Gao.
2004.9-2008.7: BEng, School of Chemical Engineering, North University of China.

Research Interests

Nonlinear PDEs; Mathematical General Relativity; Hyperbolic PDEs; Fluid Mechanics; Singularity Formation; Einstein-matter Equations; Euler-Poisson Equations; Cosmology; A-harmonic Equations.

Publications and Preprints

Preprints

(With Chihang He) Existence and bounds of nonlinear singularity-free cosmological solutions in a string-inspired gravity. arXiv:2512.00455.
(With Yiqing Shi) The emergence of nonlinear Jeans-type instabilities for quasilinear wave equations. II: Generalizations. arXiv:2511.06289.
(With Chihang He, Jinhua Wang) Proofs of singularity-free solutions and scalarization in nonlinear Einstein-scalar-Gauss-Bonnet cosmology. arXiv:2507.15304.
The emergence of nonlinear Jeans-type instabilities for quasilinear wave equations. arXiv:2409.02516.
Fully nonlinear gravitational instabilities for expanding spherical symmetric Newtonian universes with inhomogeneous density and pressure. arXiv:2305.13211.
(With Jinhua Wang) A new symmetric hyperbolic formulation and the local Cauchy problem for the Einstein-Yang-Mills system in the temporal gauge. arXiv:2111.04540v2.
Localized continuation criterion, improved local existence and uniqueness for the Euler-Poisson system in a bounded domain. arXiv:2111.11708.

Publications

Blowups for a class of second order nonlinear hyperbolic equations: a reduced model of nonlinear Jeans instability. Mathematische Annalen (MAAN), 2025, 393, 317-363. DOI: 10.1007/s00208-025-03260-0; arXiv:2208.06788.
(With Todd A. Oliynyk, Jinhua Wang) Future global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n≥4. Journal of the European Mathematical Society (JEMS), 2024, published online first. DOI: 10.4171/JEMS/1556; arXiv:2202.05432.
Fully nonlinear gravitational instabilities for expanding Newtonian universes with inhomogeneous pressure and entropy: Beyond the Tolman's solution. Physical Review D (PRD), 2023, 107(12): 123534. DOI: 10.1103/PhysRevD.107.123534; arXiv:2210.04657.
Blowups and longtime developments with near-boundary mass accretions of irregularly-shaped Euler-Poisson dominated molecular clouds in astrophysics. SIAM Journal on Mathematical Analysis (SIMA), 2023, 55, 2553-2594. DOI: 10.1137/22M1469420; arXiv:2102.11550v2.
(With Yiqing Shi) Rigorous proof of slightly nonlinear Jeans instability in the expanding Newtonian universe. Physical Review D (PRD), 2022, 105(4): 043519. DOI: 10.1103/PhysRevD.105.043519; arXiv:2201.01199.
(With Changhua Wei) Future stability of the FLRW spacetime for a class of perfect fluids. Annales Henri Poincaré (AHP), Springer Nature, 2021, 22(3), 715-770. DOI: 10.1007/s00023-020-00987-1; arXiv:1810.11788.
(With Todd A. Oliynyk) Newtonian limits of isolated cosmological systems on long time scales. Annales Henri Poincaré (AHP), Springer Nature, 2018, 19, 2157-2243. DOI: 10.1007/s00023-018-0686-2; arXiv: 1701.03975.
(With Todd A. Oliynyk) Cosmological Newtonian limits on large spacetime scales. Communications in Mathematical Physics (CMP), 2018, 364(3) 1195-1304. DOI:10.1007/s00220-018-3214-9; arXiv:1711.10896.
(With Hongya Gao, Hong Tian) A Generalization of Exponential Class and Its Applications. Abstract and Applied Analysis, 2013, Article ID 476309. DOI:10.1155/2013/476309; arXiv:1812.07843.
(With Hongya Gao, Hong Tian) Remarks on a Paper by Leonetti and Siepe. Journal of Mathematical Analysis and Applications (JMAA), 2013 (401), 881-887. DOI:10.1016/j.jmaa.2012.12.037; arXiv:1812.07740.
(With Hongya Gao, Junwei Li) Hölder Continuity and Differentiability Almost Everywhere of (K₁, K₂)-Quasiregular Mappings. Acta Mathematica Sinica Chinese Series, 2012,55(4), 721-726. Journal; arXiv: 1812.07779.

Books

Methods and Ideas in Linear Partial Differential Equations. Preprint (Draft, 381 pages).

PhD Thesis

Cosmological Newtonian Limits on Large Scales, Ph.D. Thesis (2018), Monash University, Melbourne, Australia. Available on figshare (216 pages).

Teachings

Equations of Mathematical Physics and Special Functions (MAT0701): Spring 2020 (Videos), Spring 2021, Spring 2022, Spring 2023 (Videos), Fall 2023 (Videos), Spring 2024 (Videos), Spring 2025 (Course Website (Notes and Videos)), Spring 2026 (Notes and Videos)
Equations of Mathematical Physics (PHY0721): Spring 2021.
Blowup for Nonlinear Hyperbolic Equations: Fall 2020 (Math Center).
Minicourse on Sobolev Spaces: Jan. 2020 (Math Center), (Notes).
Graduate-Level Independent Study Courses: Fall 2020, Spring and Fall 2021-2025.

Postgraduate Students

PhD Supervision

Haozhe Ju (鞠浩哲，PhD, 2026-present, Lead supervisor).
Chihang He (何池航, MS and PhD, 2023.09-present, Lead supervisor).
Yiqing Shi (时昳晴, PhD, 2020.09-2025.12, Lead supervisor). Thesis: The nonlinear Jeans-type instabilities for two families of hyperbolic equations (2025).

Master Supervision

Xiangcheng Meng (孟祥呈, MS, 2025.09-present, Lead supervisor).
Chenxinyu Luo (罗辰新宇, MS, 2023.09-2026.07, lead supervisor). Thesis: Nonlinear Jeans Instability for an Euler–Poisson System with Source Terms (2026).
Weiqi Ji (纪玮琦,MS, 2022.09-2025.07, Lead supervisor). Thesis: The nonlinear Jeans-type instabilities for two families of quasilinear wave equations with sources (2025).

Awards, Grants and Fellowships

HUST Frontier Scientific Research Achievement Award, 2025.
Key Project of the Mathematics Special Program under the "Fundamental Research Support Program" of HUST, PI, 2026-2028.
NSFC General Program‌, No. 12571234, PI, 2026-2029.
NSFC General Program‌, No. 11971503, Participant, 2020-2023.
Fundamental Research Funds for the Central Universities, HUST: 2020kfyXJJS037, PI, 2020-2022.
Mittag-Leffler Fellowship, 2019.
China Postdoctoral Science Foundation Grant, No. 2018M641054, PI, 2018-2019.

Conference Organisation

2025 workshop on geometric analysis and hyperbolic PDE, Xiamen University, Xiamen, China, 24 - 27 October 2025. Organised joint with Jiongyue Li (Sun Yat-sen University), Jinhua Wang, Chao Xia (Xiamen University) and Xiao Zhang (Academy of Mathematics and Systems Science, CAS)

Visiting Programmes

Institut Mittag-Leffler "GENERAL RELATIVITY, GEOMETRY, AND ANALYSIS: BEYOND THE FIRST 100 YEARS AFTER EINSTEIN", Djursholm, Sweden, September – December 2019.
Erwin Schrödinger Institute "GEOMETRY AND RELATIVITY PROGRAM ", Vienna, Austria, July – August 2017.
Institut Henri Poincaré "MATHEMATICAL GENERAL RELATIVITY PROGRAM ", Paris, France, September – November 2015.

Talks

Nonlinear singularity-free cosmological solutions and scalarization in ESGB theories Hunan University, Changsha, 12 December, 2025. Slides.
The emergence of nonlinear Jeans-type instabilities for quasilinear wave equations, ``Structures and Dynamics in Cosmology'' Conference, TSIMF, Sanya, 13-17 January, 2025. Video; Slides.
The emergence of nonlinear Jeans-type instabilities for quasilinear wave equations, International Conference on Ricci Curvature and Geometric Analysis, Xi'an Jiaotong University, Xi'an, 12-16 August, 2024. Slides.
Future global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n ≥ 4, Teachers forum, School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 25 April, 2024.
Future global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n ≥ 4, Tianyuan Mathematical Center in Central China, Wuhan University, Wuhan, 1 March, 2024. Slides.
On the nonlinear gravitational instabilities for Newtonian universes, Mathematical Relativity Seminar, Sun Yat-sen University, Guangzhou, 21 May, 2023. Slides.
Global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n ≥ 4, College of Mathematics and Information Science, Hebei University, Baoding, 10 November, 2022.
Global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n ≥ 4, School of Science, Northwest University, Xi'an, 30 June, 2022.
Global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n ≥ 4, School of Science, Tianjin University of Technology, Tianjin, 20 April, 2022.
Global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n ≥ 4, Black Hole Initiative, Harvard University, CMSA General Relativity Seminar, 14 April, 2022. Video; Slides.
Blowups and long-time evolution of irregularly-shaped molecular clouds, National University of Singapore, Singapore, 28 April, 2021.
Blowups and long-time evolution of irregularly-shaped molecular clouds, Tianjin University, Tianjin, 23 April, 2021.
Blowups and long-time evolution of irregularly-shaped molecular clouds, Sichuan Normal University, Chengdu, 15 January, 2021.
Blowups and long-time evolution of irregularly-shaped molecular clouds, 2020 Conference on Geometric Analysis and Hyperbolic Equations, Guangxi University, Nanning, 21 December, 2020.
Blowups and long-time evolution of irregularly-shaped molecular clouds, Hohai University, Nanjing, 24 October, 2020.
Mathematical General Relativity and Cosmology, School of Physics, Huazhong University of Science and Technology, Wuhan, 18 September, 2020.
Stability of FLRW metric for polytropic (Makino) fluids, Institut Mittag-Leffler, Djursholm, Sweden, 03 December, 2019. Slides.
Cosmological Newtonian limits, Workshop on PDEs and harmonic analysis, Institute of Applied Physics and Computational Mathematics, Beijing, China, 13 July, 2019. Slides.
How a Star is Born: Makino’s conjecture and freely falling boundary problem, Workshop on PDEs and mathematical physics 2019, Guizhou Minzu University, Guiyang, China, 12 May, 2019.
Some aspects of Einstein-Euler equatons, Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, China, 15 December, 2018.
FLRW-like cosmology can be approximated by Newtonian universe, Workshop on geometric analysis and hyperbolic equations 2018, University of Chinese Academy of Sciences, Beijing, China, 18 October, 2018.
Cosmological global evolutions driven by dark energy--A model described by Einstein-Euler equations, Prof. Tian's seminar, BICMR, Peking University, Beijing, China, 25 September, 2018.
Cosmological Newtonian limits, Zhejiang Sci-Tech University, Hangzhou, China, 26 June, 2018.
Cosmological Newtonian limits on large scales, BICMR, Peking University, Beijing, China, 27 December, 2017.
Cosmological Newtonian limits on large scales, Xiamen University, Xiamen, China, 19 September, 2017.
Long time cosmological Newtonian limits, The 60th annual meeting of the AustMS, Australian National University, Canberra, Australia, 7 December, 2016.
A time singular symmetric hyperbolic system with a singular parameter, APDEG seminar, Melbourne, Victoria, Australia, Monash University, 20 September, 2016.
Long-Time Cosmological Newtonian Limits, Workshop on Analysis, Geometry and Mathematical Relativity, Melbourne, Victoria, Australia, Monash University, 22-26 February, 2016.
Long Time Existence of Cosmological Newtonian Limits for Einstein-Euler Systems with Positive Cosmological Constant (Conformal Method), Conference in Mathematical General Relativity 2016, Tsinghua Sanya International Mathematics Forum, Sanya, Hainan, China，5-9 January, 2016.
Long time behavior of the Einstein-Euler system with Newtonian limit, 8th Australia New Zealand Mathematics Convention, University of Melbourne, Melbourne, Australia, 9 December, 2014.

Chao Liu (刘超)

Associate Professor