论文标题
涉及微尺度游离边界的矿物溶解和降水模型的均质化
Homogenization of a mineral dissolution and precipitation model involving free boundaries at the micro scale
论文作者
论文摘要
在这项工作中，我们介绍了包括不断发展的微观结构的反应扩散模型的均质化。这种类型的问题模型，例如，在多孔培养基中的矿物溶解和降水。因此，我们正在处理一个多尺度问题，并在毛孔尺度上进行自由边界。在初始状态下，微观几何形状由周期性穿孔的结构域（包括球形固体晶粒）给出。每种谷物的半径是$ε$的，并且取决于其表面未知（溶质浓度）。因此，谷物的半径会随时间变化，导致非线性，自由边界问题。第一步，我们将不断发展的微型域转换为固定的，定期的域。使用Rothe方法，我们证明存在弱解决方案，并获得相对于$ε$均匀的先验估计。最后，让$ε\至0 $，我们得出了一个宏观模型，其解决方案近似于微尺度解决方案。为此，我们使用两尺度收敛的方法，并获得强大的紧凑性结果，使得可以在非线性项中传递到极限。
In this work we present the homogenization of a reaction-diffusion model that includes an evolving microstructure. Such type of problems model, for example, mineral dissolution and precipitation in a porous medium. Hence, we are dealing with a multi-scale problem with free boundaries on the pore scale. In the initial state the microscopic geometry is given by a periodically perforated domain, including spherical solid grains. The radius of each grain is of order $ε$ and depends on the unknown (the solute concentration) at its surface. Therefore the radii of the grains change in time, leading to a nonlinear, free boundary problem. In a first step, we transform the evolving micro domain to a fixed, periodically domain. Using the Rothe-method we prove the existence of a weak solution and obtain a priori estimates that are uniform with respect to $ε$. Finally, letting $ε\to 0$, we derive a macroscopic model, the solution of which approximates the micro-scale solution. For this, we use the method of two-scale convergence, and obtain strong compactness results enabling to pass to the limit in the nonlinear terms.
