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.
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