L1 stability estimate for a one-dimensional Boltzmann equation with inelastic collisions

In this paper, we study the L-1 stability of a one-dimensional Boltzmann equation on the line with inelastic collisions in Rend. Sem. Mat. Fis. Milano 67 (1997) 169-179. Under the suitable assumptions on the initial data, we construct a nonlinear functional H(t) which measures L-1 distance between two mild solutions, and is nonincreasing in time t. Using the time-decay estimate of H(t), we show that mild solutions are L-1-stable: \\f(.,.,t) - (f) over bar(.,.,t)\\(L1(R2)) less than or equal to G\\f(0)(.,.) - (f) over bar (0) (.,.) \\(L1 (R2)), where G is a positive constant independent of time t,f and (f) over bar are mild solutions corresponding to initial data f(0) and (f) over bar (0) in L-1 (R-2) boolean AND L-+(infinity) (R-2), respectively. (C) 2002 Elsevier Science (USA). All rights reserved.