Chao Liu (刘 超)

Associate professor

Center for Mathematical Sciences and School of Mathematics and Statistics
Huazhong University of Science and Technology,
1037 Luoyu Road, 
Wuhan, Hubei Province,
China

Office: Math center 813, Enming Biulding

E-mail: chao_liu@hust.edu.cn

Self Introduction

  1. 2019.10-now. Associate professor, Center for Mathematical Sciences and School of Mathematics and Statistics, Huazhong University of Science and Technology
  2. 2017.10-2019.9. Postdoc, International Center for Mathematical Research (BICMR), Peking University. Supervisor: Prof. Gang Tian.
  3. 2014.2-2017.9. PhD, School of Mathematics, Monash University. Supervisors: Prof. Todd A. Oliynyk (main) and Dr. Julie Clutterbuck (associate).

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: 

  1. The emergence of nonlinear Jeans-type instabilities for quasilinear wave equations. arXiv:2409.02516. 

  2. Fully nonlinear gravitational instabilities for expanding spherical symmetric Newtonian universes with inhomogeneous density and pressure. arXiv:2305.13211. 

  3. Blowups for a class of second order nonlinear hyperbolic equations: A reduced model of nonlinear Jeans instability. arXiv:2208.06788. 

  4. (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. 
  5. Localized continuation criterion, improved local existence and uniqueness for the Euler-Poisson system in a bounded domain. arXiv:2111.11708. 

Publications: 

  1. (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. 

  2. 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. 
  3. 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. 

  4. (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. 

  5. (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. 

  6. (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 
  7. (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

  8. (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
  9. (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

  10. (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: 

  1. 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

  1. 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))
  2. Equations of Mathematical Physics (PHY0721): Spring 2021.
  3. Blowup for Nonlinear Hyperbolic Equations: Fall 2020 (Math Center).
  4. Minicourse on Sobolev Spaces: Jan. 2020 (Math center), (Notes).
  5. Graduate-Level Independent Study Courses: Fall 2020, Spring 2021, Fall 2021, Spring 2022, Fall 2022, Spring 2023, Fall 2023, Spring 2024, Fall 2024.

Postgraduate Students

  1. Yiqing Shi (时昳晴, PhD,2020.09-now);
  2. Weiqi Ji (纪玮琦, Master, 2022.09-2025,07). Thesis: The nonlinear Jeans-type instabilities for two families of quasilinear wave equations with sources (2025);
  3. Chihang He (何池航, Master, 2023.09-now);
  4. Chenxinyu Luo (罗辰新宇, Master, 2023.09-now);

Talks

  1. 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;
  2. 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;
  3. 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;
  4. 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 Unviersity, Wuhan, 1, March, 2024. Slides;
  5. On the nonlinear gravitational instabilities for Newtonian universes, Mathematical Relativity Seminar, Sun Yat-sen University, Guangzhou, 21, May, 2023. Slides;
  6. 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 Unviersity, Baoding, 10, November, 2022;
  7. Global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n ≥ 4, School of Science, Northwest Unviersity, Xi'an, 30, June, 2022;
  8. Global existence and stability of de Sitter-like solutions to the Einstein-Yang-Mills equations in spacetime dimensions n ≥ 4, School of Science, Tianjin Unviersity of Technology, Tianjin, 20, April, 2022;
  9. 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;
  10. Blowups and long-time evolution of irregularly-shaped molecular clouds, National University of Singapore, Singapore, 28, April, 2021;
  11. Blowups and long-time evolution of irregularly-shaped molecular clouds, Tianjin University, Tianjin, 23, April, 2021;
  12. Blowups and long-time evolution of irregularly-shaped molecular clouds, Sichuan Normal University, Chengdu, 15, January, 2021;
  13. Blowups and long-time evolution of irregularly-shaped molecular clouds, 2020 Conference on Geometric Analysis and Hyperbolic Equations, Guangxi University, Nanning, 21, December, 2020;
  14. Blowups and long-time evolution of irregularly-shaped molecular clouds, Hohai University, Nanjing, 24, October, 2020;
  15. Mathematical General Relativity and Cosmology, School of Physics, Huazhong University of Science and Technology, Wuhan, 18, September, 2020;
  16. Stability of FLRW metric for polytropic (Makino) fluids, Institut Mittag-Leffler, Djursholm, Sweden, 03, December, 2019. Slides;
  17. Cosmological Newtonian limits, Workshop on PDEs and harmonic analysis, Institute of Applied Physics and Computational Mathematics, Beijing, China, 13th, July, 2019. Slides;
  18. 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, 12th, May, 2019;
  19. Some aspects of Einstein-Euler equatons, Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, China, 15th, December, 2018;
  20. 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, 18th, October, 2018; 
  21. Cosmological global evolutions driven by dark energy--A model described by Einstein-Euler equations, Prof. Tian's seminar, BICMR, Peking University, Beijing, China, 25th, September, 2018;
  22. Cosmological Newtonian limits, Zhejiang Sci-Tech University, Hangzhou, China, 26th, June, 2018;
  23. Cosmological Newtonian limits on large scales, BICMR, Peking University, Beijing, China, 27th, December, 2017;
  24. Cosmological Newtonian limits on large scales, Xiamen University, Xiamen, China, 19th, September, 2017; 
  25. Long time cosmological Newtonian limits,  The 60th annual meeting of the AustMS, Australian National University, Canberra, Australia, 7th, December, 2016. 
  26. A time singular symmetric hyperbolic system with a singular parameter, APDEG seminar, Melbourne, Victoria, Australia, Monash University, 20th, September, 2016. ; 
  27. Long-Time Cosmological Newtonian Limits, Workshop on Analysis, Geometry and Mathematical Relativity, Melbourne, Victoria, Australia, Monash University, 22nd February, 2016 - 26th February, 2016. ;
  28. 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，5th January, 2016 - 9th January, 2016;
  29. Long time behavior of the Einstein-Euler system with Newtonian limit, 8th Australia New Zealand Mathematics Convention, University of Melbourne, Melbourne, Australia, 9th December, 2014.

Visiting Programmes

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

Links

My ORCID
My Researchgate
My ArXiv
My Google Scholar