AI 中文总结
研究具有完全单调记忆的扩散方程，通过引入扩展状态、利用算子伴随性使增强生成器\(m\)-耗散，得到唯一温和解等，证明了相关收敛性，为记忆主导扩散的离散化提供连续稳定性目标。
AI 中文摘要
我们研究具有完全单调记忆的扩散方程，此时瞬时扩散形式仅为非负，可能失去强制性。对于其伯恩斯坦表示测度具有有限总质量\(M_{0}=\nu([0,\infty))\)的核，我们引入由物理变量及其内部变量连续体组成的扩展状态。聚集和常数嵌入算子关于记忆能量是伴随的，由此产生的交叉项抵消使增强生成器\(m\)-耗散。这产生了唯一的温和解、对数据的利普希茨依赖性以及对瞬时形式无正下界的收缩估计。零预历史轨迹形成一个记忆图空间，其中问题在哈达玛意义下是适定的。此外，如果第一伯恩斯坦矩\(M_{1}=\int_{[0,\infty)}\lambda\,\diff\nu(\lambda)\)有限，记忆势和第一矩场具有将半群解与编码弱形式识别并获得显式稳定性界所需的正则性。我们还证明了当强制瞬时贡献消失时的一致范数预解式收敛和相关半群的收敛。在对极限解的额外\(L^{2}(0,\Tend;V)\)正则性假设下，记忆图范数中的收敛速率为\(O(\varepsilon^{1/2})\)。这些结果为记忆主导扩散的结构保持和认证离散化提供了连续稳定性目标。
英文摘要
We study diffusion equations with completely monotone memory when the instantaneous diffusion form is merely non-negative and may therefore lose coercivity. For a kernel whose Bernstein representing measure has finite total mass $M_{0}=ν([0,\infty))$, we introduce an extended state consisting of the physical variable and its continuum of internal variables. The aggregation and constant-embedding operators are adjoint with respect to the memory energy, and the resulting cross-term cancellation makes the augmented generator $m$-dissipative. This yields a unique mild solution, Lipschitz dependence on the data, and a contraction estimate that contains no positive lower bound for the instantaneous form. The zero-prehistory trajectories form a memory graph space, in which the problem is well posed in the sense of Hadamard. If, in addition, the first Bernstein moment $M_{1}=\int_{[0,\infty)}λ\,\diffν(λ)$ is finite, the memory potential and first-moment field possess the regularity needed to identify the semigroup solution with an encoded weak formulation and to obtain explicit stability bounds. We further prove uniform norm-resolvent convergence and convergence of the associated semigroups when a coercive instantaneous contribution vanishes. Under an additional $L^{2}(0,\Tend;V)$-regularity assumption on the limiting solution, the convergence rate in the memory graph norm is $O(\varepsilon^{1/2})$. These results provide a continuous stability target for structure-preserving and certified discretisations of memory-dominated diffusion.
