Research on the Kinetic Type Equations with Relevant Numerical Methods


Student thesis: Doctoral Thesis

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Award date29 Apr 2021


In the first part of the thesis, we study the well-posedness theory of the solution to the initial value problem of the kinetic type equation, mainly the inelastic Boltzmann equation without Grad's angular cutoff assumption, in a space of probability measure defined by Cannone-Karch (Comm. Pure. Appl. Math. 63: 747-778, 2010) via Fourier transform, where the infinite energy cases are not priori excluded. In showing the well-definedness of the integral operator, a new geometric relation of the inelastic collision mechanism is introduced to handle the strong singularity of the non-cutoff collision kernel. Moreover, with the help of asymptotic stability result, we also show the self-similar profile of the inelastic Boltzmann equation in possibly infinite energy case by a constructive approach, which is proved to be the large-time asymptotic steady solution in a certain sense.

In the second part, we introduce the relevant numerical method for the spatially homogeneous Boltzmann equation, mainly the Fourier spectral method with our new accelerated algorithm. By overcoming the difficulty brought by the non-integrable singularity in the collision kernels, we demonstrate that the general framework of the fast Fourier spectral method can be extended to handle the non-cutoff kernels, achieving the accuracy and efficiency comparable to the cutoff case. We also show through several numerical examples that the solution to the non-cutoff Boltzmann equation enjoys the smoothing effect, a striking property absent in the cutoff case.

In the last part, we provide a new proof to illustrate the spectral accuracy and stability result for our fast Fourier spectral method proposed in the second part above, based on a careful and technical $ L^{2} $ estimate of the negative part of the numerical solution. We discuss the applicability of the result to various initial data as well, including both continuous and discontinuous functions.