Dissipation in Wigner-Poisson systems

The Wigner-Poisson (WP) system (or quantum Vlasov-Poisson system) is modified to include dissipative terms in the Hamiltonian. By utilizing the equivalence of the WP system to the Schrodinger-Poisson system, global existence and uniqueness are proved and regularity properties are deduced. The proof differs somewhat from that for the nondissipative case treated previously by Brezzi-Markowich and Illner et al.; in particular the Hille-Yosida Theorem is used since the linear evolution is not unitary, and a Liapunov function is introduced to replace the energy, which is not conserved.