Microlocal analysis, Control for Partial Differential Equations
I am currently interested in understanding how the local information (control region) of PDEs propagate, in order to influence the dynamics of the equation. For hyperbolic and dispersive waves, this is related to the propagation of singularities, and the basic tool is the microlocal (semi-classical) analysis.
Damped wave equaitons, Propagation of singularites and Observability:
Quantitative observability for one-dimensional Schr\"odinger equations with potentials, with [Pei Su and Xu Yuan ], Submitted, preprint, arXiv
2. 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.
3. Sharp decay rate for the damped wave equations with convex-shaped damping, Int. Math. Res. Not. IMRN(2023), no. 7, 5905–5973.
4. 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
5. 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.
6. Time-optimal controllability and observability for Grushin Schrödinger equation, [with N. Burq],] Analysis & PDE, Vol. 15 (2022), No. 6, 1487-1530.
7. 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
8. Semi-classical propagation of singularities for Stokes system, Communications in Partial Differential Equations, 45:8, 970-1030, [Arxiv]
9. 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]
Control for the Kadomtsev-Petviashvili equations:
1. Internal controllability for the Kadomtsev-Petviashvili II equation. [with I. Rivas] , SIAM Journal on Control and Optimization 58(3):1715-1734, [Arxiv]
2. Exact controllability of linear KP-I equation. (16 pages), [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
Random data theory for nonlinear dispersive equations
There are two main motivations of studying the random data theory for dispersive equations. The first one is to provide a macroscopic description of the flow of Hamiltonian systems (in the infinite-dimensional space). The second one, from the PDE viewpoint, is to extend the Cauchy-theory for generically ill-posed data.
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],
preprint, 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 ],
preprint, arXiv .
3. Quasi-invariance of Gaussian measures for the 3d energy critical nonlinear Schr\"odinger equations, [with N. Tzvetkov], preprint, 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] , Journal of Functional 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]
Other results for dispersive equations:
1. 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.
2. 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]
3. 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.
Colloque Talk amss 2021 : Observation, Stabilization and Resolvent estimate: Beyond the geometric control condition.
Note on the Gibbs measure for the mass-critical focusing NLS