Kinetic Theory of Boundary Layer Problems without Angular Cut-off

The boundary layer problems arise in many physical considerations and they provide many challenges in mathematical studies. This kind of problems can be formulated in different physical scales. In this research project, the researchers will concentrate on some problems in the mesoscopic level, that is, problems on the kinetic equations. As a fundamental equation in kinetic theories and a key stone in statistical physics, the Boltzmann equation attracts a lot of research investigations since its derivation in 1872. Mathematically, a lot of works have been done under the Grad’s cutoff assumption to avoid the non-integrable angular singularity in the cross-sections. Under this assumption, the half-space (initial) boundary value problem of the nonlinear Boltzmann equation by assigning the Dirichlet data for the outgoing particles at the boundary and a Maxwellian in the far field has been extensively studied. This kinetic boundary layer problem in fact comes from the condensation-evaporation problem and other problems related to the kinetic behavior of the gas near a wall. An interesting feature of this problem is that not all Dirichlet data are admissible and the number of admissible conditions changes with the Mach number of the far field Maxwellian. This has been shown for the linear case by many researchers mainly in the context of the classical Milne and Kramers problems, also for the discrete velocity and the full nonlinear models.On the other hand, except for the hard sphere model, for most of the other molecule interaction potentials such as the inverse power laws, the cross section has a nonintegrable angular singularity. This implies that the collision operator is a singular integral operator on the microscopic velocity variable. It is believed that this kind of singularity has smoothing effect on solutions, that is, the solutions have higher regularity than the initial data which should be similar to the case when one replaces the collision operator in the Boltzmann equation by the Laplacian to some fractional power. The mathematical work on this kind of regularization problems for the Boltzmann equation can be traced back to 1970s and it is still an active topic because there is still no complete mathematical theory for the space inhomogeneous nonlinear problem.The main purpose of this project is to study the boundary layer problems without angular cutoff assumption. It can be viewed as a continuation and combination of what the researchers have been working in the field of kinetic theory for the Boltzmann equation. The researchers expect that the research results in this direction can bring new aspects and understandings to this area so that they will not only enrich its mathematical theories but also justify some physical phenomena in mathematical settings. To their knowledge, there is no mathematical theories so far for the boundary layer problems without angular cutoff.