Title: On the existence and regularity of weakly nonlinear stationary Boltzmann equations : a Fredholm alternative approach Show Chinese title

Title: 弱非线性平稳玻尔兹曼方程的存在性和正则性：Fredholm 替代方法

Abstract: The celebrated Fredholm alternative theorem works for the setting of identity compact operators. This idea has been widely used to solve linear partial differential equations \cite{Evans}. In this article, we demonstrate a generalized Fredholm theory in the setting of identity power compact operators, which was suggested in Cercignani and Palczewski \cite{CP} to solve the existence of the stationary Boltzmann equation in a slab domain. We carry out the detailed analysis based on this generalized Fredholm theory to prove the existence theory of the stationary Boltzmann equation in bounded three-dimensional convex domains. To prove that the integral form of the linearized Boltzmann equation satisfies the identity power compact setting requires the regularizing effect of the solution operators. Once the existence and regularity theories for the linear case are established, with suitable bilinear estimates, the nonlinear existence theory is accomplished. Show Chinese abstract

Abstract: 著名的 Fredholm 择一定理适用于恒等幂紧致算子的设定。该思想已被广泛应用于求解线性偏微分方程 \cite{Evans}。本文，我们证明了一个广义的 Fredholm 理论，该理论在恒等幂紧致算子的设定下成立，该理论由 Cercignani 和 Palczewski \cite{CP} 提出，用于解决平板域中稳态玻尔兹曼方程的存在性。我们基于该广义 Fredholm 理论进行了详细的分析，证明了有界三维凸域中稳态玻尔兹曼方程的存在性。为了证明线性化玻尔兹曼方程的积分形式满足恒等幂紧致设定，需要解算子的正则化效应。一旦建立了线性情形的存在性和正则性理论，并给出了合适的双线性估计，非线性存在性理论就完成了。