This is the current schedule for the Nonlinear PDE seminar in Texas A&M University.
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Please contact Minh-Binh Tran at minhbinh_at_tamu_dot_edu or Xin Liu at xliu23_at_tamu_dot_edu for any question.
Title: The Fuzzy Landau equation and the Fisher information.
Abstract: The Landau equation was introduced by Lev Landau in 1936 as a modification of the Boltzmann equation to specific applications in plasma physics, and describes the interactions and collisions among charged particles in a plasma. The mathematical investigation of the Landau equation has been active for several decades, with researchers exploring various aspects of its behavior and properties. While the homogeneous version of this equation has been fully understood, the inhomogeneous equation remains a very difficult problem. In this talk I will present the first global well-posedness theory for a fuzzy version of the inhomogeneous Landau equation. Fuzzy equations allow delocalized collision. We will see that this delocalization offers a wider range of analytical tools and provides additional structure that can be exploit to show smoothness and global existence. I will also show examples of new Lyapunov functions of Fisher type.
Title: Fluctuations around the mean-field limit for Coulomb/Riesz gases
Abstract: This talk will report on recent developments in the description of log/Coulomb/Riesz gases beyond the leading order mean-field description, focusing on central limit theorems for fluctuations and asymptotics of correlation functions in varying temperature regimes. We focus on techniques based on the modulated (free) energy and estimates for their variations, known as commutator estimates, which have been successfully used to analyze the mean-field limit and may also be used to prove convergence of the fluctuations. The talk will highlight an important interplay between the initial size of fluctuations, the singularity of the interaction, and the scaling of the temperature that arises in analyzing the fluctuations. Time permitting, we will discuss both the equilibrium and non-equilibrium cases. Based on joint work with Jiaoyang Huang and Sylvia Serfaty.
Title: Carleman-Weighted Global Reconstruction for 3D Inverse Scattering with Partial Boundary Data.
Abstract: Inverse scattering problems arise in many applications where one wants to locate and characterize hidden objects from wave measurements, for example, in medical imaging, non-destructive testing, or subsurface exploration. This talk focuses on a three-dimensional frequency-domain scattering model governed by the Helmholtz equation with an unknown dielectric coefficient. The available data consist of multi-frequency backscattered measurements on one-sixth of the boundary generated by a single point source. We first describe a reduction in the frequency variable. Using a logarithmic change of variables and an expansion in a suitable polynomial–exponential (generalized Fourier) basis, we transform the original equation, which depends on both space and frequency, into a finite system of coupled quasi-linear elliptic equations that depend only on the spatial variables. To solve this nonlinear system, we apply the Carleman convexification method. Based on a weighted Carleman estimate, we construct a Tikhonov-type functional that is globally strictly convex on bounded sets. This structure proves that a simple gradient descent algorithm converges globally to the unique minimizer of the functional, regardless of the initial guess. The talk concludes with three-dimensional numerical reconstructions from both simulated and experimental data. These examples demonstrate that the method can accurately recover the locations and shapes of multiple types of scatterers.
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