带退化迁移率的粘弹性相分离的耗散解：一种动态输运方法

中文摘要 AI 辅助

我们研究了带退化凹迁移率的四阶粘弹性相分离系统的弱解的极小化运动近似的收敛性。所采用的半隐式变分格式将相场变量的动态输运距离与变换后应力变量的加权L²距离相结合。我们证明了该格式的子序列收敛性，并表明极限解满足能量耗散不等式。为此，我们推导了度量斜率的强制性估计，该估计给出了De Giorgi插值所需的时空正则性，并构造了本构通量，其耗散由度量斜率的平方控制。这些估计使我们能够在离散能量耗散不等式中取极限。相场变量演化方程的弱形式通过流交换技术推导得出。在数据的附加假设下，我们进一步证明极限解满足熵耗散不等式。

英文摘要

We study the convergence of minimising movement approximations to weak solutions of a fourth-order viscoelastic phase separation system with degenerate concave mobility. The underlying semi-implicit variational scheme combines a dynamic transport distance for the phase-field variable with a weighted $L^2$-distance for a transformed stress variable. We establish subsequential convergence of the scheme and show that the limiting solutions satisfy the energy-dissipation inequality. To this end, we derive a coercivity estimate for the metric slope, which yields the required space-time regularity for the De Giorgi interpolant, and construct constitutive fluxes whose dissipation is controlled by the squared metric slope. These estimates allow us to pass to the limit in the discrete energy-dissipation inequality. The weak formulation of the evolution equation for the phase-field variable is derived using the flow interchange technique. Under additional assumptions on the data, we further show that the limiting solutions satisfy an entropy-dissipation inequality

发表机构

• Weierstrass Institute for Applied Analysis and Stochastics（魏尔斯特拉斯应用分析与随机性研究所）

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