Shijie DONG - Research Interest

Shijie DONG

I am interested in analysis and PDEs, especially in nonlinear wave equations, nonlinear Klein-Gordon equations, and their coupled systems. I mainly focus on: 1) establishing the small global-in-time solutions to such equations for certain kinds of (quadratic) nonlinearities; 2) building global existence results and unified pointwise decay results, for the Klein-Gordon equations, which are uniform with respect to the possibly vanishing mass parameter. The study is mainly motivated by the Einstein-Klein-Gordon model, the electroweak standard model, and the Klein-Gordon-Zakharov model from mathematical physics, which is mathematically interesting and challenging.

I am planning to write a more detailed introduction on what I am working on. TBA

Possible future interest: pure wave equations, more types of dispersive equations, probability, fluid mechanics.

Publications and Preprints

25. Large data global existence for coupled massive-massless wave-type systems, with Yuan Cai, Kuijie Li, and Jingya Zhao, preprint arXiv:2406.05762 (2024).

24. Global smooth solutions to 2D semilinear wave equations with large data, with Bingbing Ding and Gang Xu, preprint arXiv:2312.08018 (2023).

23. Cubic Dirac equations with a class of large data, with Kuijie Li and Jingya Zhao, preprint arXiv:2312.07880 (2023).

22. Generically sharp decay for quasilinear wave equations with null condition, with Siyuan Ma, Yue Ma, and Xu Yuan, preprint arXiv:2212.12115 (2022).

21. Global solution to the 3D Dirac--Klein-Gordon system with uniform energy bounds, with Kuijie Li and Xu Yuan, preprint arXiv:2208.14131 (2022), Calc. Var. Partial Differential Equations 62 (2023), no. 5, Paper No. 146, 42 pp.

20. Global Behavior of Small Data Solutions for The 2D Dirac-Klein-Gordon Equations, with Kuijie Li, Yue Ma, and Xu Yuan, preprint arXiv:2205.12000 (2022), to appear in Transactions of the American Mathematical Society.

19. Asymptotic behavior of 2D Wave-Klein-Gordon coupled system under null condition, with Yue Ma and Xu Yuan, preprint arXiv:2202.08139 (2022), Bull. Sci. Math. 187 (2023), Paper No. 103313.

18. Revisit of the Faddeev Model in Dimension Two, with Zhen Lei, Chin. Ann. Math. Ser. B 43, 797–818 (2022) (This is a special issue of CAM in memory of the late Professor Gu Chaohao 1926-2012, who is a highly distinguished mathematician.)

17. Global Existence and Scattering of the Klein-Gordon-Zakharov System in Two Space Dimensions, with Yue Ma, preprint arXiv:2111.00244 (2021), to appear in Peking Math. J..

16. Global solution to the cubic Dirac equation in two space dimensions, with Kuijie Li, preprint arXiv:2111.04048(2021), J. Differential Equations 331 (2022), 192–222.

15. Hidden structure and sharp asymptotics for the Dirac--Klein-Gordon system in two space dimensions, with Zoe Wyatt, preprint arXiv:2105.13780 (2021), to appear in Annales de l’Institut Henri Poincaré C. Analyse Non Linéaire.

14. The top-order energy of quasilinear wave equations in two space dimensions is uniformly bounded, with Philippe LeFloch and Zhen Lei, preprint arXiv:2103.07867 (2021), to appear in Fundamental Research.

13. Stability of some two dimensional wave maps, with Zoe Wyatt, preprint arXiv:2103.05318 (2021), to appear in Differential and Integral Equations.

12. Global solution to the Klein-Gordon-Zakharov equations with uniform energy bounds, preprint arXiv:2101.02927 (2021), SIAM Journal on Mathematical Analysis 54 (2022), no. 1, 595–615.

11. Two dimensional wave--Klein-Gordon equations with semilinear nonlinearities, with Zoe Wyatt, preprint arXiv:2011.11990 (2020), NoDEA Nonlinear Differential Equations Appl. 30 (2023), no. 5, Paper No. 59, 32 pp.

10. Asymptotic Behavior of the Solution to the Klein-Gordon-Zakharov Model in Dimension Two, preprint arXiv:2006.04443, (2020), Communications in Mathematical Physics 384 (2021), no. 1, 587–607.

9. Global solution to the wave and Klein-Gordon system under null condition in dimension two, preprint arXiv:2005.04767, (2020), Journal of Functional Analysis 281 (2021), no. 11, Paper No. 109232, 29 pp.

8. The zero mass problem for Klein-Gordon equations: quadratic null interactions, preprint arXiv:2004.10467, (2020), Forum of Mathematics, Sigma 10 (2022), Paper No. e27, 23 pp.

7. Stability of a wave and Klein-Gordon system with mixed coupling, preprint arXiv:1912.05578, (2019), to appear in Tohoku Mathematical Journal.

6. Stability of a class of semilinear waves in 2+1 dimension under null condition, preprint arXiv:1910.09828, (2019).

5. Stability of Quasilinear Waves in 1+1 Dimension Under Null Condition, (2019), Journal of Dynamics and Differential Equations 33 (2021), no. 2, 961–970.

4. The zero mass problem for Klein-Gordon equations, preprint arXiv:1905.08620, (2019), Commun. Contemp. Math. 25 (2023), no. 7, Paper No. 2250029, 20 pp.

3. Global evolution of the U(1) Higgs Boson: nonlinear stability and uniform energy bounds, with Philippe G. LeFloch and Zoe Wyatt, preprint arXiv:1902.02685, (2019), Annales Henri Poincare 22 (2021), no. 3, 677–713.

2. The finite volume method on a Schwarzschild background, with Philippe G. LeFloch, preprint arXiv:1901.10973, (2019), ESAIM Math. Model. Numer. Anal. 53 (2019), no. 5, 1459–1476.

1. Stability of a coupled wave-Klein-Gordon system with quadratic nonlinearities, with Zoe Wyatt, preprint arXiv:1811.10022, (2018), Journal of Differential Equations 269 (2020), no. 9, 7470–7497.

Refereed journals:

JEMS, Advances in Maths., JDE, IMRN, and some others.

Online seminar:

Analysis and PDE Seminar

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