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

AbstractIn this paper, we study the L1 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 L1 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 L1-stable:||f(·,·,t)−f̄(·,·,t)||L1(R2)⩽G||f0(·,·)−f̄0(·,·)||L1(R2),where G is a positive constant independent of time t, f and f̄ are mild solutions corresponding to initial data f0 and f̄0 in L1(R2)∩L+∞(R2), respectively