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The existence and stability of collisional kinetic equation, especiallynon-cutoff Boltzmann equation, in bounded domain with physical boundarycondition is longstanding open problem. This work proves the global stabilityof the Landau equation and non-cutoff Boltzmann equation in union of cubes withthe specular reflection boundary condition when an initial datum is nearMaxwellian. Moreover, the solution enjoys exponential large-time decay inbounded domain. Our method is based on that fact that normal derivatives incubes is also derivatives along axis, which allows us to obtain high-orderderivative estimates.

The Landau and Non-cutoff Boltzmann Equation in Union of Cubes