Combustion occurring in porous media has various practical applications, such as in in-situ combustion processes in oil reservoirs, the combustion of biogas in sanitary landfills, and many others. A porous medium where combustion takes place can consist of layers with different physical properties. This study demonstrates that the initial value problem for a combustion model in a multi-layer porous medium has a unique solution, which is continuous with respect to the initial data and parameters in L2(R)n. In summary, it establishes that the initial value problem is well-posed in L2(R)n. The model is governed by a one-dimensional reaction–diffusion–convection nonlinear system, where the unknowns are the temperatures in the layers. Previous studies have addressed the same problem in H2(R)n. However, in this study, we solve the problem in a less restrictive space, namely L2(R)n. The proof employs a novel approach to combustion problems in porous media, utilizing an evolution operator defined from the theory of semigroups in Hilbert space and Kato’s theory for a well-posed associated initial value problem. Finally, we give numerical simulations of the nonlinear problem, where we observe that the characteristics of the solution profiles we find are in accordance with what is physically expected.

中文翻译：

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多层多孔介质中燃烧模型的适定问题


在多孔介质中发生的燃烧具有各种实际应用，例如油藏中的原位燃烧过程、卫生垃圾填埋场中的沼气燃烧等等。发生燃烧的多孔介质可以由具有不同物理性质的层组成。本研究表明，多层多孔介质中燃烧模型的初始值问题具有唯一的解，该解相对于 L2（R）n 中的初始数据和参数是连续的。总之，它确定了初始值问题在 L2（R）n 中是适定的。该模型由一维反应-扩散-对流非线性系统控制，其中未知数是层中的温度。以前的研究已经解决了 H2（R）n 中的相同问题。然而，在这项研究中，我们在一个限制较少的空间，即 L2（R）n 中解决了这个问题。该证明采用了一种新的方法来解决多孔介质中的燃烧问题，利用根据希尔伯特空间中的半群理论和 Kato 理论定义的进化算子来解决适定的关联初始值问题。最后，我们给出了非线性问题的数值模拟，我们观察到我们找到的解曲线的特征与物理预期一致。