On The Mathematical Study Of The Non-cutoff Boltzmann Equation In A Finite Channel

Boundary effect plays an important role in the mathematical study of the Boltzmann equation.It’s quite difficult to establish the global existence in L_x,v^∞ space due to the complexity of the non-cutoff collision operators and the underlying singularity of the transport operator at the boundary.In this thesis,we construct a global and unique solution in a new function space H _x1¹H_?² for the non-cutoff Boltzmann equation with inflow boundary condition in the finite channel {x｜x（=x₁,（?））,x₁[=-1,1],（?）（=x₂,x₃）=[0,2π]²}.We obtain the large-time behavior)and the propagation of the regularity of the solution by using elementary energy method,and the key point of this thesis is to control the nonlinear term by using Sobolev embedding inequality in the function space L_T²L_v²H _x1¹H_?².