Boltzmann Equation with Debye-Yukawa Potential

The Boltzmann equation is a key stone in statistical physics. A lot of work has been done under the Grad’s cutoff assumption. Without angular cutoff, the collision operator is an integral operator with singular kernel. Recently, work has been done on the non-cutoff potentials of inverse power laws where the collision operator behaves like a fraction of the Laplacian on the velocity variable.The main concern of this project is to study the Boltzmann equation for the Debye-Yukawa potential where the collision operator behaves like a logarithm of the Laplacian to some power that is a weaker regular operator. The study involves investigating the hypoellipticity of the operator and its smoothing effect on the solutions so that the techniques from the pseudo-differential operators and Bony’s paralinear operator methods become useful.