发表机构
University of Montenegro; University of Zagreb（黑山大学; 萨格勒布大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该文研究黎曼流形上带活性修正的Cahn-Hilliard方程，通过单调性论证建立弱解理论，并证明陡峭分类器族的符号图极限收敛性。
AI 中文摘要
我们在紧致黎曼流形（可能带边界）上建立了一个非变分Cahn-Hilliard演化的弱解理论和奇异本构极限。该模型将经典双阱化学势与集中在扩散过渡层中且对Laplace-Beltrami算子符号敏感的活性修正相结合，并带有向指定参考态的单调锚定机制。活性修正打破了被动梯度流结构，引入了对二阶导数的非线性依赖，这种依赖无法通过自然的弱紧性估计来识别。对于平方可积初始数据和足够可积的参考数据，我们在任意维度构造了弱解。利用分类器和锚定定律的单调性以及双调和强制性，Galerkin层次上的单边比较论证得出了近似Laplacian的强收敛性，从而获得了强二阶紧性。这在普通弱表述中识别了非线性活性项，没有二阶导数缺陷。在二维情形下，我们证明了唯一性和连续依赖性。这些估计仅依赖于分类器的有界性和单调性，而不依赖于其斜率，因此对越来越陡的反正切分类器是一致的。它们的奇异极限由作用于Laplace-Beltrami算子的极大单调符号图控制。在任意维度中，我们获得了对所得微分包含的弱解的子序列强二阶收敛性。在二维情形下，极限状态是唯一的，因此整个陡峭分类器族收敛；在零Laplacian集合上本构乘子的唯一性未被断言。这是一个在固定扩散界面厚度下的本构陡度极限，而非尖锐界面极限。
英文摘要
We establish a weak solution theory and a singular constitutive limit for a non variational Cahn Hilliard evolution on compact Riemannian manifolds, possibly with boundary. The model combines the classical double well chemical potential with an active correction concentrated in diffuse transition layers and sensitive to the sign of the Laplace Beltrami operator, together with a monotone anchoring mechanism toward a prescribed reference state. The active correction breaks the passive gradient flow structure and introduces a nonlinear dependence on second derivatives not identified by the natural weak compactness estimates. For square integrable initial data and sufficiently integrable reference data, we construct weak solutions in arbitrary dimension. A Galerkin level one sided comparison argument, using monotonicity of the classifier and anchoring law together with biharmonic coercivity, yields strong convergence of the approximate Laplacians and hence strong second order compactness. This identifies the nonlinear active term in an ordinary weak formulation, without a second derivative defect. In two dimensions we prove uniqueness and continuous dependence. The estimates depend only on the bound and monotonicity of the classifier, not on its slope, and are therefore uniform for increasingly steep arctangent classifiers. Their singular limit is governed by the maximal monotone sign graph acting on the Laplace Beltrami operator. In arbitrary dimension we obtain subsequential strong second order convergence to a weak solution of the resulting differential inclusion. In two dimensions the limiting state is unique, so the entire steep classifier family converges; uniqueness of the constitutive multiplier on the zero Laplacian set is not asserted. This is a constitutive steepness limit at fixed diffuse interface thickness, rather than a sharp interface limit.
