Local existence, lower mass bounds, and smoothing for the Landau equation
Christopher Henderson, Stanley Snelson, Andrei Tarfulea · arXiv (Cornell University) · 2017
We consider the spatially inhomogeneous Landau equation with soft potentials, including the case of Coulomb interactions. We establish the existence of solutions for a short time, assuming the initial data is in a fourth-order Sobolev space and has Guassian decay in the velocity variable. We also show, using an argument based on an associated stochastic process, that the equation instantaneously spreads mass, providing a lower bound on the mass density at every point in the domain. This allows us to apply the prior work of the first two authors to conclude that our solution is $C^\infty$ in all three variables, and also that blow-up cannot occur as a result of vanishing mass, but instead, must coincide with the mass, energy, or entropy density becoming unbounded from above. Our proof makes essential use of the nonlocality of the Landau equation.