Free boundary problems in Fluid and Mean Field limit (Nguyen Quoc Hung)

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01 13, 2023

  In 2021, we proved that the 2d-muskat equation is local/global well-posed where initial data belongs to Sobolev space H^(3/2+[log]^(2/3)).In 2022 in [2], we extended this result to the 3d-muskat equation. In [4], we proved first local wellposedness result for the 2d/3d-muskat equations with initial data f_0 \in C^1. Finally, in [1], we solved the 2d-muskat equation with H^(3/2) initial data. This result is optimal with respect to the scaling of the equation and for unbounded slopes. Our proof is the first in which a null-type structure is identified for the Muskat equation, allowing to compensate for the degeneracy of the parabolic behavior for large slopes. In [3], we used this method to prove the wellposedness of a nonlocal nonlinear equation of the roots of polynomials under differentiation. 

  In 2018, Sylvia Serfaty established the mean field convergence for Coulomb potential and super-Coulombic Riesz potential of points evolving along the gradient flow of their interaction energy. Serfaty's proof is based on a modulated energy method using a Coulomb or Riesz distance; and the Caffarelli-Silvestre extension theorem. In [5], we extended this result to general kernels where the Caffarelli-Silvestre extension is not available for these kernels. Moreover, our assumption on kernel contains Lenard-Jones type potentials. To get our result, we established new commutator estimates.

  

  Publications:
  [1]    Thomas Alazar, Quoc-Hung Nguyen, Endpoint Sobolev theory for the Muskat equation, Commun. Math. Phys. (2022). https://doi.org/10.1007/s00220-022-04514-7
  [2]    Thomas Alazar, Quoc-Hung Nguyen, Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem, Advances in Math, Volume 399, 16 April 2022, 108278, https://doi.org/10.1016/j.aim.2022.108278
  [3]    Thomas Alazard, Omar Lazar and Quoc-Hung Nguyen, On the dynamics of the roots of polynomials under differentiation, Journal de mathematiques pures et Appliqu'ees, V 162, June 2022, Pages 1-22,https://doi.org/10.1016/j.matpur.2022.04.001,
  [4]    Ke Chen, Quoc-Hung Nguyen, and Yiran Xu, The Muskat problem with C^1 data, Trans. Amer. Math. Soc. 375 (2022), 3039-3060
  [5]    Quoc-Hung Nguyen, Matthew Rosenzweig, and Sylvia Serfaty, Mean-field limits of
  [6]  Riesz-type singular flows, Ars Inveniendi Analytica (2022), Paper No. 4, 45 pp, DOI: https://doi.org/10.15781/nvv7-jy87
  
  Authors:
  Nguyen Quoc Hung
  Email: qhnguyen@amss.ac.cn

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