A brief description of my research:
My research focuses on four topics: 1. Long time dynamics for NLS (nonlinear Schr\"odinger equations) on waveguide manifolds; 2. Long time dynamics for NLS on Euclidean spaces; 3. Many body problems (and other Mathematical Physics models); 4. Long time behavior for stochastic dispersive PDE; 5. Decaying property of nonlinear dispersive equations
Papers and Preprints:
[1] Global well-posedness and scattering for the defocusing cubic Schr{\"o}dinger equation on waveguide $\mathbb{R}^2 \times \mathbb{T}^2$, Journal of Hyperbolic Differential Equations, Vol. 16, No. 1 (2019) 1-57. (arXiv: 1710.09702)
[2] On scattering for the defocusing high dimensional inter-critical NLS (joint with C. Gao), Journal of Differential Equations, Volume 267, Issue 11, 15 November 2019, Pages 6198-6215. (arXiv: 1809.04211)
[3] On scattering for the defocusing quintic nonlinear Schr\"odinger equation on the two dimensional cylinder} (joint with X. Cheng and Z. Guo), SIAM Journal on Mathematical Analysis, 52 (2020), no. 5, 4185-4237. (arXiv: 1809.01527)
[4] Well-posedness for energy-critical NLS on waveguide manifold} (joint with X. Cheng and J. Zheng), Journal of Mathematical Analysis and Applications, Volume 494, Issue 2, 15 February 2021. (arXiv: 1911.00324)
[5] On scattering for the defocusing nonlinear Schr{\"o}dinger equation on waveguide $ \mathbb{R}^m \times \mathbb{T}$ (when $m=2,3$), Journal of Differential Equations, Volume 275, 25 February 2021, Pages 598-637. (arXiv: 1712.01266)
[6] Decay estimates for nonlinear Schr\"odinger equations}, (joint with C. Fan), Discrete and Continuous Dynamical Systems-A, 41 (2021), no. 8, 3973–3984. (arXiv: 2007.04491)
[7] Global well-posedness for the focusing, cubic NLS on product space $\mathbb{R} \times \mathbb{T}^3$, (joint with X. Yu and H. Yue), SIAM Journal on Mathematical Analysis, Vol. 53, Issue. 2; Pages. 2243-2274. (arXiv: 2009.01116)
[8] Long time dynamics for defocusing cubic NLS on three dimensional product space, (joint with J. Zheng), SIAM Journal on Mathematical Analysis, Vol. 53, No. 3, pp. 3644--3660. (arXiv: 2003.00665)
[9] Global estimates for the Hartree-Fock-Bogoliubov Equation, (joint with J. Chong, M. Grillakis and M. Machedon), Comm. in PDE, 46 (2021), no. 10, 2015–2055. (arXiv: 2008.01753)
[10] Dynamical Hartree-Fock-Bogoliubov approximation of interacting Bosons, (joint with J. Chong), Ann. Henri Poincar\'e, 23 (2022), no. 2, 615–673. (arXiv: 1711.00610)
[11] A note on decay property of nonlinear Schr\"odinger equations (joint with C. Fan), Proceedings of AMS, Volume 151, Number 6, June 2023, Pages 2527–2542.
[12] On the decay property of cubic fourth order Schr\"odinger equation (joint with X. Yu and H. Yue), Proceedings of AMS, Volume 151, Number 6, June 2023, Pages 2619–2630.
[13] On scattering asymptotics for the 2D cubic resonant system (joint with K. Yang), Journal of Differential Equations, Vol. 345, 5 February 2023, Pages 447-484. (arXiv:2207.01185)
[14] Dynamics of subcritical threshold solutions for energy-critical NLS (joint with Q. Su), Dynamics of Partial Differential Equations, Vol. 20, No. 1 (2023), pp. 37-72. (arXiv: 1811.07239)
[15] Singular Levy processes and dispersive effects of generalized Schr\"odinger equations (joint with Y. Sire, X. Yu and H. Yue), Dynamics of Partial Differential Equations, Volume 20, Number 2, 2023, Pages: 153-178. (It is a stronger version of arXiv:2207.00485 which incorporates another research note).
[16] On scattering for NLS on waveguide manifolds: a short survey, (joint with X. Yu), to appear in Birkh\"auser Series: Research Perspectives Ghent Analysis and PDE Center.
[17] Global well-posedness and scattering for fourth order Schr\"odinger equation on waveguide manifold (joint with X. Yu and H. Yue), arXiv: 2111.09651.
[18] On long time behavior for stochastic nonlinear Schr\"odinger equations with a multiplicative noise, (joint with C. Fan), arXiv: 2010.11045, submitted.
[19] Long time behavior of stochastic NLS with a small multiplicative noise (joint with C. Fan and W. Xu), arXiv:2111.07212, submitted.
[20] Global uniform in $N$ estimates for solutions of a system of Hartree-Fock-Bogoliubov type in the case $\beta<1$ (joint with J. Chong, X. Dong, M. Grillakis and M. Machedon), arXiv: 2203.05447, submitted.
[21] On scattering for generalized NLS on waveguide manifolds (joint with Y. Sire, X. Yu and H. Yue), arXiv:2207.00485.
[22] On decaying properties of nonlinear Schr\"odinger equations (joint with C. Fan and G. Staffilani), arXiv: 2211.03124, submitted.
[23] Global well-posedness for cubic-quintic NLS on tori (joint with Y. Luo, X. Yu and H. Yue), arXiv: 2301.13433, submitted.
[24] On Strichartz estimate for many body Schr\"odinger equation in the waveguide setting, arXiv: 2301.13429, submitted.
[25] On Strichartz estimate for many body Schr\"odinger equation in the periodic setting (joint with X. Yu and J. Zheng), arXiv:2304.00882, submitted.
Arxiv Link for my works: https://arxiv.org/search/?query=Zhao%2C+Zehua&searchtype=all&abstracts=show&order=-announced_date_first&size=50.
Welcome to contact me by email on research! :)