| Abstract |
The celebrated Fredholm alternative theorem works for the setting of
identity compact operators. This idea has been widely used to solve
linear partial differential equations. In this talk, we demonstrate a
generalized Fredholm theory in the setting of identity power compact
operators, which was suggested in Cercignani and Palczewski to solve
the existence of the stationary Boltzmann equation in a slab domain.
We carry out the detailed analysis based on this generalized Fredholm
theory to prove the existence theory of the stationary Boltzmann
equation in bounded three-dimensional convex domains. To prove that
the integral form of the linearized Boltzmann equation satisfies the
identity power compact setting requires the regularizing effect of the
solution operators. Once the existence and regularity theories for the
linear case are established, with suitable bilinear estimates, the
nonlinear existence theory is accomplished. This talk is based on a
collaborative work with Daisuke Kawagoe and Chun-Hsiung Hsia. |
