https://sites.google.com/site/Chenmin

Microlocal analysis, Control for Partial Differential Equations

Damped wave equations, Propagation of singularities and Observability:

Geometric condition for observability of electromagnetic Schr\"odinger operators, with [K. Le Balc'h and J. Niu ], submitted, [arXiv ].

2. Quantitative observability for one-dimensional Schr\"odinger equations with potentials, with [Pei Su and Xu Yuan ], J. Funct. Analysis (2025), Vol.288, 2, [arXiv].

3. Sharp resolvent estimate for the Baouendi-Grushin operator and applications, [with V. Arnaiz], Comm. Math. Phys. 400 (2023), no. 1, 541–637. [Arxiv ] Slide

Remark: This work contains the detailed construction and propagation of semiclassical microlocal measures in the subelliptic regime for quasimodes of the Baouendi-Grushin operator.

4. Sharp decay rate for the damped wave equations with convex-shaped damping, Int. Math. Res. Not. IMRN(2023), no. 7, 5905–5973.

[Arxiv ] Slide

5. Decays rates for Kelvin-Voigt damped wave equations II: the geometric control condition, [with N. Burq ], Proc. Amer. Math. Soc. 150 (2022), 1021-1039 , arxiv.org/abs/2010.05614

6. Decays for Kelvin-Voigt damped wave equation: Piece-wise smooth damping, [with N. Burq ], J. London. Math. Soc.(2), 106 (2022), no. 1, 446-483.

arxiv.org/abs/2007.12994

7. Time-optimal controllability and observability for Grushin Schrödinger equation, [with N. Burq],] Analysis & PDE, Vol. 15 (2022), No. 6, 1487-1530.

arxiv.org/abs/1910.03691,

8. Observability of Bouendi-Grushin type equations through resolvent estimates, [with C. Letrouit ], J. Inst. Math. Jussieu. 22, No. 2, 541-579 (2023), arxiv.org/abs/2010.05540

9. Semi-classical propagation of singularities for Stokes system, Comm. Partial Differential Equations, 45:8, 970-1030, [Arxiv]

10. On the stabilization of a hyperbolic Stokes system under geometric control condition. [with F.-W. -Chaves-Silva] , Z. Angew. Math. Phys. 71, 139 (2020). [Arxiv]

11. Internal controllability for the Kadomtsev-Petviashvili II equation. [with I. Rivas] , SIAM Journal on Control and Optimization 58(3):1715-1734, [Arxiv]

12. Exact controllability of linear KP-I equation. (16 pages), permanent preprint, [Arxiv] :

Remark: This manuscript is a chapter of my PhD thesis and is a natural supplementary of the controllability for the KP-II paper. The result is truth worthy, but for some reason, I intend not to publish it in a journal.

My PhD thesis

Nonlinear dispersive equations

Dispersive equations with randomness:

Probabilistic well-posedeness for the nonlinear Schrödinger equation on the 2d sphere I: positive regularities , [with N. Burq, N. Camps, N. Tzvetkov], arXiv .

2. The Second Picard iteration of NLS on the 2d sphere does not regularize Gaussian random initial data, [with N. Burq, N. Camps M. Latocca, N. Tzvetkov ],

to appear in EMS Survey 2025, arXiv .

3. Quasi-invariance of Gaussian measures for the 3d energy critical nonlinear Schr\"odinger equations, [with N. Tzvetkov], to appear in Comm. Pure Appl. Math, arXiv .

4. Refined probabilistic well-posedness for the weakly dispersive NLS, [with N. Tzvetkov], Nonlinear Analysis, 213(11): 112530., arxiv.org/pdf/2010.13065 .

5. Gibbs measure dynamics for the fractional nonlinear Schrödinger equation, [with N. Tzvetkov], SIAM J. Math. Anal., 52(5), 4638–4704. arxiv.org/abs/1912.07303,

6. New examples of probabilistic well-posedness for nonlinear wave equations.[with N. Tzvetkov] , J. Funct. Analysis. 278 (2020), 108322, [Arxiv]

7. Weak universality for a class of nonlinear wave equations, [with N. Tzvetkov and Weijun Xu], to appear in Ann. Inst, Fourier. , arXiv .

Related overview Universality results for a class of nonlinear wave equations and their Gibbs measures, Séminaire Laurent Schwartz — EDP et applications (2021-2022), Exposé no. 15, 10 p.

8. Probabilistic well-posedness for supercritical wave equations with periodic boundary condition on dimension three. [with B. Xia] , Illinois J. Math. 60(2016), no.2, 481-503 [Arxiv]

Miscellaneous:

Hyperbolic nonlinear Schrödinger equations on R \times T, [[with E. Başakoğlu, N. Tzvetkov, Yuzhao Wang], preprint.

Local well-posedness for the periodic Boltzmann equation with constant collision kernel, [with E. Başakoğlu, N. Tzvetkov, Yuzhao Wang], preprint

3. On the pathological set in probabilistic well-posedness of nonlinear wave equations, [with N. Tzvetkov], Comptes Rendus Mathematique, Tome 358 (2020) no.9-10, pp. 989-999. arxiv.org/abs/2001.10293 Slide

Remark: A full proof of $G_{\delta}$ dense structure of the pathological set is contained in the updated arXiv version. A better understanding of this type of strong ill-posedness is recently achieved by Camps-Gassot where they proved the result for NLS.

4. Low regularity blowup solutions for the mass-critical NLS in higher dimensions. [with J. Zheng] , Journal de Mathématiques Pures et Appliquées, 134 (2020) 255-298, [Arxiv]

5. Scattering below ground state of focusing fractional nonlinear Schrödinger equation with radial data. [with H. Wang,X. Yao,J. Zheng], Discrete and Continuous Dynamical Systems, 2018,38(4):2207-2228 [Arxiv]

Other writings: expository notes, slides, ...

Second semi-classical microlocal measure: An example of how to use 2-semi-classical microlocal measure to prove the observability for the Schr\"odinger equation on the torus.
Note on the Gibbs measure for the mass-critical focusing NLS
Ipad note on the derivation of the linear Boltzmann equation from random Schr\"odinger equation (after Erdös-Yau) : Part I , Part II Part III Part IV