Rheumatoid arthritis is an autoimmune disease characterized by inflammation in the synovial fluid within the synovial joint connecting two contiguous bony surfaces. The inflammation diffuses into the cartilage adjacent to each of the bony surfaces, resulting in their gradual destruction. The interface between the cartilage and the synovial fluid is an evolving free boundary. In this paper we consider a two-phase free boundary problem based on a simplified model of rheumatoid arthritis. We prove global existence and uniqueness of a solution, and derive properties of the free boundary. In particular it is proved that the free boundary increases in time, and the cartilage shrinks to zero as [Formula: see text], even under treatment by a drug. It is also shown in the reduced one-phased problem, with cartilage alone, that a larger prescribed inflammation function leads to a faster destruction of the cartilage.

中文翻译：

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类风湿关节炎的数学模型分析。

类风湿关节炎是一种自身免疫性疾病，其特征在于连接两个连续的骨表面的滑膜关节内的滑膜液发炎。炎症扩散到每个骨表面附近的软骨中，导致其逐渐破坏。软骨和滑液之间的界面是不断发展的自由边界。在本文中，我们基于类风湿关节炎的简化模型考虑了两阶段自由边界问题。我们证明了一个解的整体存在性和唯一性，并推导了自由边界的性质。特别地，已经证明，即使在药物治疗下，自由边界随时间增加，并且软骨收缩为零，如[公式：见正文]。在减少的一阶段问题中也显示了这一点，仅涉及软骨，