发表机构
Universidade de Santiago de Compostela（圣地亚哥德孔波斯特拉大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文为 Gelfand 三元组上的线性抛物型发展方程建立 Banach 值侧向 Stieltjes 微积分，处理后验状态和原子跳跃，证明乘积公式、重构定理和能量恒等式，并通过 Galerkin 与 Rothe 格式构造存在性，以带 Stieltjes 时间的热方程为例。
AI 中文摘要
我们发展了一种 Banach 值侧向 Stieltjes 微积分，适用于 Gelfand 三元组 \\[ V\hookrightarrow H\simeq H^*\hookrightarrow V^* \\] 上的线性发展方程。该发展问题以后验状态表述为 \\[ D_g u(t)+A(t) S_g u(t)=f(t),\qquad u(a)=u_0. \\] 这一选择在 Stieltjes 测度的原子处具有决定性作用：若 $\delta=\Delta_g(r)>0$，则方程给出隐式变分预解式 \\[ (I+\delta A(r))u(r^+)=u(r)+\delta f(r). \\] 我们证明了 Banach 值侧向乘积公式、弱-强重构定理，以及 Lions–Magenes 能量恒等式的 Stieltjes 类比。在本文考虑的可分 Hilbert 设定中，证明中所用的稳定投影链通过谱带构造被证明自动存在。后者包含精确的正跳跃贡献 \\[ \frac12\sum_r\\|u(r^+)-u(r)\\|_H^2. \\] 存在性分别通过 Galerkin 逼近和适应 Stieltjes 质量及其原子的 Rothe 格式独立构造。Rothe 构造不使用空间投影。我们还解释了为何现有的关于前向 Stieltjes–Bochner 发展图的 Aubin–Lions 定理不自动适用于后验能量空间，并且在当前线性强制设定中并不需要。作为模型应用，我们包含了一个带 Stieltjes 时间的热方程。这些结论相对于早期的 Stieltjes 抛物型方程、一般测度驱动的发展方程以及算子值测度抛物型系统进行了定位；我们不对测度驱动抛物型理论或一般的 Galerkin 逼近主张优先权。
英文摘要
We develop a Banach-valued lateral Stieltjes calculus adapted to linear evolution equations on a Gelfand triple \[ V\hookrightarrow H\simeq H^*\hookrightarrow V^*. \] The evolution problem is formulated with the posterior state, \[ D_g u(t)+A(t) S_g u(t)=f(t),\qquad u(a)=u_0. \] This choice is decisive at atoms of the Stieltjes measure: if $δ=Δ_g(r)>0$, the equation yields the implicit variational resolvent \[ (I+δA(r))u(r^+)=u(r)+δf(r). \] We prove a Banach-valued lateral product formula, a weak--strong reconstruction theorem, and a Stieltjes analogue of the Lions--Magenes energy identity. In the separable Hilbert setting considered here, the stable projection chain used in the proof is shown to exist automatically by a spectral-band construction. The latter contains the exact positive jump contribution \[ \frac12\sum_r\|u(r^+)-u(r)\|_H^2. \] Existence is constructed independently by Galerkin approximation and by a Rothe scheme adapted to the Stieltjes mass and its atoms. The Rothe construction does not use spatial projections. We also explain why the available Aubin--Lions theorem for anterior Stieltjes--Bochner evolution graphs is not automatically applicable to the posterior energy space and, in the present linear coercive setting, is not needed. A heat equation with Stieltjes time is included as a model application. The claims are positioned relative to earlier Stieltjes parabolic equations, general measure-driven evolution, and operator-valued-measure parabolic systems; no priority is claimed for measure-driven parabolic theory or Galerkin approximation in general.
