Transport equations in weak topologies of dual banach spaces

Abstract

Nonlinear transport equations are studied, in which the nonlinearity, arising from the collision operator, is well behaved in the weak topology of a weakly compactly generated Banach space. The Cauchy problem is posed for general semilinear evolution equations, which can model a variety of diffusion and kinetic equations. Local existence theorems are obtained for such spaces. In particular, the results are applicable to transport equations in {ital L}{sup {infinity}} with appropriate weak (i.e., {ital L}{sup 1}) continuity properties.