CONVERGENCE OF THE CHILD-LANGMUIR ASYMPTOTICS OF THE BOLTZMANN-EQUATION OF SEMICONDUCTORS

In a previous paper, the Child-Langmuir asymptotics of the Boltzmann e quation of semiconductors were presented, a limit problem was then der ived, and its well posedness was analyzed. In this paper, the converge nce of the perturbed problem to the limit one is proved. The proof is done in three steps. In the first step, we prove uniform bounds by usi ng supersolution and support estimates. The second step involves combi ning the supersolution technique and semiexplicit formulas derived fro m the phase portrait of the Boltzmann equation to improve the regulari ty results. Finally, the limit equation is integrated and the converge nce is proved in the third step.