利用极大正则性求解一维燃烧型自由边界问题
Solutions to a One-Dimensional Combustion-Type Free Boundary Problem via Maximal Regularity
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中文总结 AI 辅助
该研究针对燃烧理论中的一维自由边界问题，以极大$L^p$-$L^q$正则性为工具，建立了两种情形下解的存在唯一性、正则性及自由边界演化规律。
中文摘要 AI 辅助
我们研究燃烧理论中出现的一维自由边界问题，其中界面的运动由给定的诺伊曼边界通量和零狄利克雷边界条件控制，我们对半直线情形和有界区间情形均进行了研究。对于这两种情形，我们采用极大$L^p$-$L^q$正则性作为主要分析工具。在半直线情形中，即使空间导数属于$L^q(\boldsymbol{R}_+)$，解也不必在无穷远处衰减。为处理自由边界的演化规律，我们引入了一种避免二阶边界迹线的导数公式。通过结合极大$L^p$-$L^q$正则性与绍德尔估计，我们建立了解的局部时间存在性、唯一性、正则性，以及自由边界的演化规律。
英文摘要
We study a one-dimensional free boundary problem arising in combustion theory, where the motion of the interface is governed by a prescribed Neumann boundary flux and a zero Dirichlet boundary condition. We treat both the half-line case and the bounded interval case. For both settings, we employ maximal $L^p$-$L^q$ regularity as our main analytical tool. In the half-line case, the solutions need not decay at infinity, even though the spatial derivatives belong to $L^q(\mathbb{R}_+)$. To handle the evolution law of the free boundary, we introduce a derivative formulation that avoids second-order boundary traces. By combining maximal $L^p$-$L^q$ regularity and Schauder estimates, we establish the local-in-time existence, uniqueness, and regularity of solutions, as well as the evolution law of the free boundary.
