Regularization of Measure Valued Solutions to the Boltzmann Equation and Some Related Problems

In recent years, a lot of progress has been made on the Boltzmann equation without angular cutoff based on the effort by many researchers in 1990s or even earlier. One of the main differences between the Boltzmann equation without angular cutoff and the one with Grad’s angular cutoff is about the regularizing effect on the solutions because of the angular singularity. Now this regularizing effect on the solutions is well-known and has been extensively analyzed in the framework in which the solutions have some kind of initial regularity, for example, in some weighted Sobolev spaces, in general. On the other hand, since the Boltzmann equation is about the time evolution of particle distribution function, one natural function space is the space of probability measure with some moment constraint that corresponds to, for example the momentum and energy. For the spatially homogeneous Boltzmann equation with the Maxwellian molecule type cross section, the existence and large time behavior of this kind of solution was studied in the early work by Tanada in 1978 using probability theory, and then Toscani- Villani, Pulvirenti-Toscani using Wasserstein metrics, till recently, the works on solutions with infinite energy by Cannone-Karch and Morimoto motivated by the selfsimilar solutions constructed by Cercignani-Toscani. Even though the solutions given in these works have little regularity requirement on the initial data, it was shown in our recent work with Morimoto that they become arbitrarily smooth in any positive time as long as the initial datum is not a single Dirac mass which is exactly what Villani proposed on this problem. These works lead to the study on the problems proposed in this project that aim to further develop the mathematical theories on this interesting phenomenon. Here, we mention the recent work by Lu-Mouhot on the existence of measure valued solutions for the hard potentials, but without analysis on the regularizing effect. In this project, we plan to work on the following problems. First of all, since the regularizing effect for the measure valued solutions has been proved only for Maxwellian molecule type cross section in the spatially homogeneous case that benifits a lot by using the Bobylev formula, one natural question is how about the general cross sections, for example, the hard and soft potentials. In these cases, so far there is no result on the measure valued solutions because of the complicated structure of the collision operator with or without using the Bobylev formula. And the second question is about the large time behavior. Even though the convergence to either an equilibrium or a selfsimilar solution has been studied by several authors, but more is needed to be clarified as we will discuss later. In fact, the convergence should be “strong” in some sense as the solutions become arbitrarily smooth in any positive time. With the strong enough convergence to the Maxwellian, we can then further study the well-posedness of the Boltzmann equation for the spatially inhomogeneous problem. Finally, we will explore the corresponding regularizing effect problems for the spatially inhomogeneous Boltzmann equation.