Regularity and Singularity of the Landau Equation

This thesis primarily concerns the regularity estimates and singularity formation for the Landau equation. The Landau equation models the evolution of charged particles in a plasma through their density at given time, position, and velocity. A common simplification is the spatially homogeneous setting, where there is no dependency in x-variable. In recent years, there has been significant progress in the understanding of the space-homogeneous Landau equation, particularly in the Coulomb potential case, which is the most physically relevant case. We begin by studying the dissipation of entropy for the spatially homogeneous Landau equation with Coulomb potential. We improve the weight in entropy dissipation estimate obtained by Desvillettes and prove its optimality. Then, motivated by the breakthrough of Guillen and Silvestre, who showed that Fisher information is non-increasing for the Landau-Coulomb equation, we study the dissipation of Fisher information. In addition, we prove monotonicity of the Fisher information for an isotropic modification of the Landau-Coulomb equation, sometimes referred to as the Krieger-Strain model. We also investigate the Bakry-Émery Gamma_2 inequality, originally introduced as a criterion for the logarithmic Sobolev inequality. This function inequality played a crucial role in the work of Guillen and Silvestre. We improve the known lower bound for 3D and extend it to arbitrary dimensions. Finally, we turn to the space-inhomogeneous Landau equation with very hard potentials. By lifting a blow-up profile from the compressible Euler equation, we construct smooth initial data that develops a finite-time implosion singularity. This provides the first example of a collisional kinetic model that is globally well-posed in the homogeneous setting, yet exhibits finite-time singularity in the inhomogeneous case.