Díaz Palencia José Luis
Escuela Politécnica Superior, Universidad Francisco de Vitoria, Ctra. Pozuelo-Majadahonda Km 1,800, Pozuelo de Alarcón, 28223, Madrid, Spain.
Sci Rep. 2022 Mar 28;12(1):5228. doi: 10.1038/s41598-022-08623-0.
The presented analysis has the aim of introducing general properties of solutions to an Extended Darcy-Forchheimer flow. The Extended Darcy-Forchheimer set of equations are introduced based on mathematical principles. Firstly, the diffusion is formulated with a non-homogeneous operator, and is supported by the addition of a non-linear advection together with a non-uniform reaction term. The involved analysis is given in generalized Hilbert-Sobolev spaces to account for regularity, existence and uniqueness of solutions supported by the semi-group theory. Afterwards, oscillating patterns of Travelling wave solutions are analyzed inspired by a set of Lemmas focused on solutions instability. Based on this, the Geometric Perturbation Theory provides linearized flows for which the eigenvalues are provided in an homotopy representation, and hence, any exponential bundles of solutions by direct linear combination. In addition, a numerical exploration is developed to find exact Travelling waves profiles and to study zones where solutions are positive. It is shown that, in general, solutions are oscillating in the proximity of the null critical state. In addition, an inner region (inner as a contrast to an outer region where solutions oscillate) of positive solutions is shown to hold locally in time.
所呈现的分析旨在介绍扩展达西 - 福希海默流动解的一般性质。基于数学原理引入了扩展达西 - 福希海默方程组。首先，扩散是用一个非齐次算子来表述的，并通过添加一个非线性平流项以及一个非均匀反应项来支持。所涉及的分析在广义希尔伯特 - 索伯列夫空间中进行，以考虑由半群理论支持的解的正则性、存在性和唯一性。之后，受一组关注解的不稳定性的引理启发，分析了行波解的振荡模式。基于此，几何微扰理论提供了线性化流动，其特征值以同伦表示给出，因此，通过直接线性组合得到解的任何指数束。此外，开展了数值探索以找到精确的行波剖面并研究解为正的区域。结果表明，一般来说，解在零临界状态附近振荡。此外，正解的一个内部区域（与解振荡的外部区域相对而言的内部）在局部时间内被证明是成立的。
