UPPER BOUNDS AND APPROXIMATION FOR THE SOLUTION OF A NONLINEAR PARTICLE TRANSPORT PROBLEM WITH MAXWELL TYPE BOUNDARY CONDITIONS

Abstract

We consider a nonlinear particle transport problem in a slab of thickness 2a with Maxwell-type boundary conditions. The nonlinearity is due to the presence of a scattering term, which is a quadratic function of the (unknown) particle density u(t)=u(x, y; t). We first prove that, under suitable assumptions, the integral version of the transport equation has a unique strongly continuous solution u=u(t)∈L1∩L∞, defined over an arbitrary fixed time interval [0, t0]. We also evaluate upper bounds for the norms ||u(t)|| and ||u(t)||∞, which are useful to understand the mathematical and the physical properties of the particle transport problem under consideration. Finally, we discretize the angle variable y in the unknown function u(t)=u(x, y; t), (i.e., we use the “discrete ordinate method”) and we prove the convergence of the discretized solution to the exact solution by using a generalization of Trotter’s theory on sequences of Banach spaces approximating a given Banach space.