Large-Time Behavior of Solutions for the Boltzmann Equation with Hard potentials

Abstract

We study the quantitative behavior of the solutions of the one-dimensional Boltzmann equation for hard potential models with Grad’s angular cutoff. Our results generalize those of [5] for hard sphere models. The main difference between hard sphere and hard potential models is in the exponent of the collision frequency $\nu(\xi)\approx (1+|\xi|)^\gamma$. This gives rise to new wave phenomena, particularly the sub-exponential behavior of waves. Unlike the hard sphere models, the spectrum of the Fourier operator $-i\xi^1\eta+L$ is non-analytic in η for hard potential models. Thus the complex analytic methods for inverting the Fourier transform are not applicable and we need to use the real analytic method in the estimates of the fluidlike waves. We devise a new weighted energy function to account for the sub-exponential behavior of waves.