arXiv 2603.201532026-03math.AP

一维交叉扩散系统中独立对流的全局解

Global solutions to cross-diffusion systems with independent advections in one dimension

St John’s College, University of Oxford（牛津大学圣约翰学院）；Mathematical Institute, University of Oxford（牛津大学数学研究所）

Jakub Skrzeczkowski

备注：: 55 pages + references

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AI总结：

本文研究了一维中考虑两个物种u和v的交叉扩散系统，利用达西定律描述其演化，并探讨了在存在额外对流场时构造解的问题，提出了统一适用于所有压力指数α的方法，解决了α>1时的首次存在性结果。

摘要：

我们考虑描述两个物种u和v演化的交叉扩散系统，根据达西定律运动，压力定律为p(s) = 1/(α-1) s^{α-1}，其中s=u+v。该领域最具有挑战性的问题是在存在额外对流场时构造解，而不对场或初始条件施加任何人工结构。尽管对流自然出现在这些模型中，但它破坏了系统的对称性，阻碍了最近技术的应用。本文提供了一种新的方法，用于在一维空间中构造解，适用于所有压力指数α∈(0,∞)和任意初始数据。特别是，在α>1的范围内，这给出了首次不存在性结果，无需任何结构假设。我们构造了解作为粘性消失近似（u_ε，v_ε）的极限。主要挑战是确定u_ε ∂_x p(s_ε)的极限，其中s_ε = u_ε + v_ε。关键的新见解是可能的u_ε和∂_x p(s_ε)振荡相关，简化了补偿紧致性论证中的Young测度分析，并允许识别极限。令人惊讶的是，与2×2超几何系统理论相比，该论证仅依赖三个熵-熵通量对。这在α>2时特别有用，因为不清楚是否还有额外的熵可用。

英文摘要：

We consider cross-diffusion systems describing evolution of two species $u$ and $v$ moving according to Darcy's law with the pressure law $p(s) = \frac{1}{α-1} s^{α-1}$ where $s=u+v$. One of the most challenging questions in the field is the construction of solutions to the problem in the presence of additional advection fields, without imposing any artificial structure on the fields or the initial conditions. Although advection arises naturally in these models, it breaks the symmetry of the system and prevents application of techniques developed in recent years. Here, we provide a new approach to construct solutions in one space dimension that works in a unified manner for all pressure exponents $α\in (0,\infty)$ and for arbitrary initial data. In~particular, in the regime $α> 1$, this yields the first existence result of its kind, obtained without any structural assumptions. We construct the solutions as a limit of a vanishing viscosity approximation $(u_{\varepsilon}, v_{\varepsilon})$. The main challenge is to identify the limit of $u_{\varepsilon} \, \partial_x p(s_{\varepsilon})$, where $s_{\varepsilon} = u_{\varepsilon} + v_{\varepsilon}$. The key new insight is that possible oscillations of $u_\varepsilon$ and $\partial_x p(s_\varepsilon)$ are correlated, simplifying the Young measure analysis in the compensated compactness argument and allowing identification of the limit. Somewhat surprisingly, in contrast to the theory of $2\times2$ hyperbolic systems, the argument relies on only three entropy-entropy flux pairs. This is particularly useful for $α>2$, where it is unclear whether additional entropies are available.

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