Dissipation and Semigroup on Weighted Sobolev Space: Linearized Boltzmann Operator


Student thesis: Doctoral Thesis

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Award date20 Apr 2021


The non-cutoff Boltzmann equation for the long range interaction potentials is a fundamental mathematical models in collisional kinetic theory, which describe the dynamics of a non-equilibrium rarefied gas. A well-establishded framework in which to study global well-posedness is to look for solutions near global Maxwellian equilibrium with the help of linearized Boltzmann operator.

In this thesis, we find that the linearized Boltzmann operator L of the non-cutoff Boltzmann equation with soft potential generates a strongly continuous semigroup on Hkn , with k, nR. In the theory of Boltzmann equation without angular cutoff, the weighted Sobolev space plays a fundamental role. On the other hand, we split the linearized Boltzmann operator into L=-A+K, where A is a positive operator and K is a compact operator. The proof is based on pseudo-differential calculus and in general, the L2 dissipation of A implies Hkn dissipation. This kind of estimate is also known as the Gårding's inequality. With these results established, we further give the result on global existence and regularity of the non-cutoff Boltzmann equation in weighted Sobolev space.

In order to obtain the Gårding's inequality over the weighted Sobolev space, we consider pseudo-differential symbol <v>n<η>k. The main difficulty is to establish the estimate on commutator between A and (<v>n<η>k)w. Our proof is also valid for a general class of Weyl quantization.