A Non-linear Example: The Boltzmann Equation

Abstract

The purpose of this chapter is to show an application of the concepts of integration by parts introduced so far in an explicit example. With this motivation in mind, we have chosen as a model the Boltzmann equation. The Boltzmann equation is a non-linear equation related to the density of particles within the gas. We consider an imaginary situation where particles collide in a medium and we observe a section of it, say $\mathbb {R}^2$. These particles collide at a certain angle $\theta$ and velocity v, which generates a certain force within a gas. The Boltzmann equation main quantity of interest, $f_t(v)$, describes the density of particles traveling at speed v at time $t>0$ supposing an initial distribution $f_0$. We assume that these densities are equal all over the section and therefore independent of the position within the section. The feature of interest to be proven here is that even if $f_0$ is a degenerate law in the sense that it may be concentrated at some points, the noise in the corresponding equation will imply that for any $t>0$ $f_t$ will be a well-defined function with some regularity. The presentation format in this section follows closely the one presented in [7] with some simplifications. This field of research is growing very quickly and therefore even at the present time the results presented here may be outdated. Still, our intention is to provide an explicit example of application of the method presented in the previous chapter. For the same reason, not all proofs are provided and some facts are relegated to exercises with some hints given in Chap. 14.