Well/ill-posedness bifurcation for the Boltzmann equation with constant collision kernel

Abstract

We consider the 3D Boltzmann equation with the constant collision kernel. We investigate the well/ill-posedness problem using the methods from nonlinear dispersive PDEs. We construct a family of special solutions, which are neither near equilibrium nor self-similar, to the equation, and prove that the well/ill-posedness threshold in H s Sobolev space is exactly at regularity s " 1, despite the fact that the equation is scale invariant at s " 1 2 .Contents. Introduction 1 2. Loss term bilinear estimate and its sharpness 7 3. Gain term bilinear estimate 10 4. Well-posedness 13 5. Ill-posedness 14 Appendix A. Remarks on scaling and Strichartz estimates 27 References 28