## Cinetic Equations Problem

**Cinetic Models Application**

Consider the following BGK type model

where

and

**I** – Let be the solution of with .

1. Prove that .

2. Prove that .

3. Prove that if then .

**II** – Let be the operator defined on which assigns to the function where is the solution of

4. Let . Prove that maps to and that is a convex closed subset of .

5. Prove that for every in we have

6. Prove that if we denote by the solutions of corresponding to we have for every that

7. Prove that is a contraction from to and that has a unique solution.

**III** Choose bounded in , and denote the solution of which corresponds to .

8. Prove that if , then . (we can prove that ).

9. Prove that and are bounded in .

10. Prove that is compact in for every ball of .

11. Prove that, up to a subsequence, we ca pass to the limit in the sense of distributions in with as .