具有加性摩擦系数的 Maxwell-Stefan 扩散的全局强解

中文摘要 AI 辅助

我们研究具有加性摩擦系数 $f_{ij}=g_i+g_j$ 的 Maxwell-Stefan 扩散。在质量分数中，系统隔离了受约束的成对摩擦块；在摩尔分数中，它是恒定总摩尔浓度下的经典理想等温/等压 Maxwell-Stefan 系统。加性使得受约束的成对摩擦耗散在物种上呈对角化；反之，在一个内部重心约束空间上的物种对角化强制了成对和表示。在算子层面，正受约束松弛算子是对 $G=diag[g_1,\ldots,g_N]$ 的压缩的标量平移。其标量预解式既产生显式的受约束逆，也产生交错谱根，这些谱根构成开单纯形上的全局实解析坐标，其微分是左特征余向量。在根坐标中，主部是对角的，不出现自平方梯度项，标量比较产生不变矩形和与单纯形边界的分离。对于正则性，我们引入熵稳定的单侧多 EPD 截断：Euler-Poisson-Darboux 熵抵消混合二次产生，而对于 $N\ge4$，截断加权的混合熵修正提供横向强制性。Caccioppoli 和对数估计、收缩和临界质量给出直到 Neumann 边界的 Hölder 连续性。混合熵也使矩系统对称化；冻结的余法向估计给出空间 Lipschitz 界。结合时间 Hölder 控制和短区间极大正则性，这为每个 $N\ge2$、$d\ge2$ 和 $p>d+2$，在有界 $C^{2+\alpha}$ 域（$0<\alpha<1$）上，对所有一致正、相容的初始浓度（在自然迹类中）产生全局强可解性。解在正时间变为经典解，并在相对熵、$L^2$ 和 $C^1(\bar\Omega)$ 中指数收敛到平衡。

英文摘要

We study Maxwell-Stefan diffusion with additive friction coefficients $f_{ij}=g_i+g_j$. In mass fractions, the system isolates the constrained pair-friction block; in mole fractions, it is the classical ideal isothermal/isobaric Maxwell-Stefan system at constant total molar concentration. Additivity makes the constrained pair-friction dissipation species-diagonal; conversely, species-diagonality on one interior barycentric constraint space forces a pair-sum representation. At operator level, the positive constrained relaxation operator is a scalar shift of a compression of $G=diag[g_1,\ldots,g_N]$. Its scalar resolvent yields both an explicit constrained inverse and interlacing spectral roots, which form global real-analytic coordinates on the open simplex and whose differentials are left eigen-covectors. In root coordinates the principal part is diagonal, no self-square gradient term occurs, and scalar comparison yields invariant rectangles and separation from the simplex boundary. For regularity we introduce entropy-stabilized one-sided multi-EPD truncations: Euler-Poisson-Darboux entropies cancel mixed quadratic production, while for $N\ge4$ a truncation-weighted mixing-entropy correction supplies transverse coercivity. Caccioppoli and logarithmic estimates, shrinking, and critical mass yield Hölder continuity up to the Neumann boundary. The mixing entropy also symmetrizes the moment system; frozen conormal estimates give spatial Lipschitz bounds. Together with time Hölder control and short-interval maximal regularity, this yields global strong solvability on bounded $C^{2+α}$ domains with $0<α<1$, for each $N\ge2$, $d\ge2$, and $p>d+2$, for all uniformly positive, compatible initial concentrations in the natural trace class. Solutions become classical for positive times and converge exponentially to equilibrium in relative entropy, $L^2$, and $C^1(\barΩ)$.

发表机构

• Technische Universität Darmstadt（达姆施塔特工业大学）

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