KAM Theory and "KAM for PDEs"
Starting from year 2023, I am rewriting classical KAM theory and "KAM for PDEs" using paradifferential calculus. A powerful tool in harmonic analysis, paradifferential calculus provides delicate description of regularity of nonlinear mappings. Therefore, the classical KAM/Nash-Moser scheme used to deal with "small denominators" is replaced by a standard implicit function argument by paralinearization. Numbers of dynamical problems can be resolved in a fairly easy manner with this method. Outcomes of this project include the following:
Alazard, T., Shao, C. (2024). Paracomposition Operators and Paradifferential Reducibility. arXiv preprint, arXiv:2410.17211
Alazard, T., Shao, C. (2023). KAM via Standard Fixed Point Theorems. arXiv preprint, arXiv:2312.13971
Free Boundary Problems in Hydrodynamics
Alazard, T., Shao, C., Yang, H. (2025). Global well-posedness of a 2D fluid-structure interaction problem with free surface. arXiv preprint, arXiv:2504.00213
Shao, Chengyang. "On the Cauchy problem of spherical capillary water waves" Forum Mathematicum, 2025. https://doi.org/10.1515/forum-2024-0042
Shao, C. (2025). Toolbox of para-differential calculus on compact Lie groups. Journal of Functional Analysis, 111176. https://doi.org/10.1016/j.jfa.2025.111176
Shao, C. (2022). Longtime Dynamics of Irrotational Spherical Water Drops: Initial Notes. arXiv preprint, arXiv:2301.00115
My Ph.D. Thesis
Shao, C. (2022). Long Time Dynamics of Spherical Objects Governed by Surface Tension, Doctoral dissertation, Massachusetts Institute of Technology. The final version can be found here: Chengyang Shao Thesis
Shao, C. (2022) Long Time Behavior of a Quasilinear Hyperbolic System Modelling Elastic Membranes. Arch Rational Mech Ana, 243(2), 501-557. https://doi.org/10.1007/s00205-021-01730-8
Miscellaneous
Heller, G., Shao, C. (2021) Strichartz and Multi-linear Estimates for 1D Periodic Dysthe Equation. arXiv preprint, arXiv:2112.12734 . This is the MIT PRIMES research paper for year 2021.
Shao, C. (2018). Schauder Type Estimates for a Class of Hypoelliptic Operators. arXiv Preprint, https://arxiv.org/abs/1805.12283
Shao, C. (2017). On the convergence of massive harmonic explorers to mSLE(4) curves. This is my undergraduate thesis under the supervision of Professor Dmitry Chelkak. A version can be found here. Dmitry Chelkak and Yijun Wan cited this work as unpublished in the following paper: Chelkak, D., & Wan, Y. (2021). On the convergence of massive loop-erased random walks to massive SLE(2) curves. Electronic Journal of Probability, 26, 1-35.
Review Works
Journal of the European Mathematical Society
Archive for Rational Mechanics and Analysis