Nonlinear Partial Differential Equations and Harmonic Analysis. In particular, the study of nonlinear dispersive PDEs such as nonlinear Schrödinger equations, nonlinear wave equations, and the KdV equation by using techniques from PDEs, Harmonic Analysis, and Probability theory. Mainly, well-posedness (existence, uniqueness, and stability of solutions) in both deterministic and probabilistic settings, existence of invariant measures, Strichartz estimates in different settings, etc. Also, interested in Fourier restriction theory and decoupling theory.
(with J. Forlano) Invariant Gibbs dynamics for the nonlinear Schrödinger equations on the disc, arXiv:2509.14861.
(with T. Oh, E. Başakoğlu) Sharp unconditional well-posedness of the 2-$d$ periodic cubic hyperbolic nonlinear Schrödinger equation, arXiv:2509.01650.
(with Y. Deng, H. Wang, Z. Zhao) On restricted-type Strichartz estimates and the applications, arXiv:2508.18827.
(with T. Oh) Revisiting Bourgain's probabilistic construction of solutions to the 2-$d$ cubic NLS, arXiv:2505.24271.
(with E. Başakoğlu, N. Tzvetkov, C. Sun) Hyperbolic nonlinear Schrödinger equations on $\mathbb R \times \mathbb T$, arXiv:2504.15836.
(with E. Başakoğlu, N. Tzvetkov, C. Sun) Local well-posedness for the periodic Boltzmann equation with constant collision kernel, arXiv:2411.12140.
(with V.D. Dinh, N. Rougerie, L. Tolomeo) Statistical mechanics of the radial focusing nonlinear Schrödinger equation in general traps, arXiv:2312.06232.
(with R. Liu, N. Tzvetkov) Existence, uniqueness, and universality of global dynamics for the fractional hyperbolic $\Phi^4_3$-model, arXiv:2311.00543.
(with D. Greco, G. Li, T. Oh, R. Liang) Optimal divergence rate of the focusing Gibbs measure, arXiv:2310.08783.
(with R. Liang) Gibbs Dynamics for the weakly dispersive nonlinear Schrödinger equations, Comm. Math. Phys. 405 (2024), no. 10, Paper No. 250, 69 pp.
(with C. Ilya, T. Oh) Norm inflation for the cubic nonlinear heat equation above the scaling critical regularity, arXiv:2205.14488, to appear in Funkcialaj Ekvacioj (2024).
(with T. Oh, L. Tolomeo, G. Zheng) Hyperbolic $P(\Phi)_2$-model on the plane, arXiv:2211.03735.
(with T. Robert, K. Seong, L. Tolomeo) Focusing Gibbs measures with harmonic potential, Ann. Inst. Henri Poincaré Probab. Stat. 61 (2025), no. 1, 571–-598.
(with Y. Zine) Norm inflation for the derivative nonlinear Schrödinger equation, C. R. Math. Acad. Sci. Paris 362 (2024), 1857--1871.
(with R. Liang) Gibbs measure for the focusing fractional NLS on the torus, SIAM J. Math. Anal. 54 (2022), no. 6, 6096--6118.
(with T. Oh, Y. Zine) Three-dimensional stochastic cubic nonlinear wave equation with almost space-time white noise, Stoch. Partial Differ. Equ. Anal. Comput. 10 (2022), no. 3, 898--963.
(with T. Oh, T. Robert) On the parabolic and hyperbolic Liouville equations, Comm. Math. Phys. 387 (2021), no. 3, 1281–1351.
(with T. Oh, T. Robert, and P. Sosoe) Invariant Gibbs dynamics for the dynamical sine-Gordon model, Proc. Roy. Soc. Edinburgh Sect. A 151 (2021), no. 5, 1450–1466.
(with T. Oh, T. Robert, and P. Sosoe) On the two-dimensional hyperbolic stochastic sine-Gordon equation, Stoch. Partial Differ. Equ. Anal. Comput. 9 (2021), no. 1, 1--32.
(with T. Oh) On global well-posedness of the modified KdV equation in modulation spaces, Discrete Contin. Dyn. Syst. 41 (2021), no. 6, 2971--2992.
(with T. Oh) Normal form approach to the one-dimensional periodic cubic nonlinear Schrödinger equation in almost critical Fourier-Lebesgue spaces, J. Anal. Math. 143 (2021), no. 2, 723--762.
(with T. Oh, T. Robert, N. Tzvetkov) Stochastic quantization of Liouville conformal field theory, arXiv:2004.04194, 77 pages.
(with T. Oh, N. Tzvetkov) Solving the 4NLS with white noise initial data, Forum Math. Sigma 8 (2020), Paper No. e48, 63 pp.
(with T. Oh) Global well-posedness of the one-dimensional cubic nonlinear Schr\"dinger equation in almost critical spaces, J. Differential Equations 269 (2020), no. 1, 612--640.
(with W. Wang) Liouville-type theorems for the stationary MHD equations in 2D, Nonlinearity 32 (2019), no. 11, 4483--4505.
(with T. Oh, O. Pocovnicu) On the stochastic nonlinear Schrödinger equations with non-smooth additive noise, Kyoto J. Math. 60 (2020), no. 4, 1227–1243.
(with O. Pocovnicu) An Lp-theory for almost sure local well-posedness of the nonlinear Schrödinger equations. Comptes Rendus Mathematique. 356 (2018), no. 6, 637--643.
(with T. Oh) Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces. Forum of Mathematics, Sigma. 6 (2018), E5. doi:10.1017/fms.2018.4. arXiv:1707.02013.
(with T. Oh) On the ill-posedness of the cubic nonlinear Schrödinger equation on the circle. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 64 (2018), no. 1, 53--84.
(with J. Xiao) A Liouville problem for the stationary fractional Navier-Stokes-Poisson system. J. Math. Fluid Mech. (2017). https://doi.org/10.1007/s00021-017-0330-9
(with Z. Guo, Y. Sire, L. Zhao) On the energy-critical fractional Schrödinger equation in the radial case, (arXiv link) Dyn. Partial Differ. Equ. 15 (2018), no. 4, 265--282.
(with J. Xiao) Well/ill-posedness for the dissipative Navier-Stokes system in generalized Carleson measure spaces, Adv. Nonlinear Anal. https://doi.org/10.1515/anona-2016-0042
(with J. Xiao) A constructive approach to positive solutions of $\Delta_p f + f(u) \le 0$ on Riemannian manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 33 (2016), no. 6, 1497--1507.
(with J. Xiao) A uniqueness principle for $u^p \le (-\Delta)^{\alpha} u$ in the Euclidean space, Commun. Contemp. Math. 18 (2016), no. 6, 1650019, 17 pp.
(with Y. Liu, J. Xiao) Nonnegative solutions of a fractional sub-Laplacian differential inequality on Heisenberg group, Dyn. Partial Differ. Equ. 12 (2015), no. 4, 379--403.
(with J. Xiao) Homogeneous Campanato-Sobolev classes, Appl. Comput. Harmon. Anal. 39 (2015), no. 2, 214--247.
(with Z. Guo, T. Oh) Strichartz estimates for Schrödinger equations on irrational tori, Proc. Lond. Math. Soc. 109 (2014), no. 4, 975--1013.
(with Z. Guo) Improved Strichartz estimates for a class of dispersive equations in the radial case and their applications to nonlinear Schrödinger and wave equations. J. Anal. Math. 124 (2014), 1--38.
(with L. Molinet) Dispersive limit from the Kawahara to the KdV equation, J. Differential Equations 255, (2013), 2196--2219.
Periodic nonlinear Schrödinger equation in critical $H^s(\T^d)$ space, SIAM J. Math. Anal. 45, (2013), 1691--1703.
Periodic Cubic Hyperbolic Schrödinger equation on $H^s(\T^2)$, J. Funct. Anal. 265 (2013), 424--434.
Local well-posedness for hyperbolic-elliptic Ishimori equation, Nonlinear Anal. 75 (2012), 2534--2541.
Quadratic dispersive generalized Benjamin-Ono equation, J. Math. Anal. Appl. 387 (2012), 844--856.
Global well-posedness and scattering for derivative Schrödinger equation, Comm. Partial Differential Equations 36 (2011), 1694--1722.
(with Z. Guo, L. Peng, B. Wang) Uniform well-posedness and inviscid limit for the Benjamin-Ono-Burgers equation, Adv. in Math. 228 (2011), 647--677.
(with Z. Guo) On the well-posedness of the Schrödinger-KdV system, J. Differential Equations 249 (2010), 2500--2520.
The Cauchy problem for the elliptic-hyperbolic Davey-Stewartson system in Sobolev space, J. Math. Anal. Appl. 367 (2010), 174--192.
Mini Course