非等温相场系统的广义耗散解：存在性、弱-强唯一性与长时间行为
Generalised dissipative solutions for a non-isothermal phase-field system: existence, weak-strong uniqueness, and long-time behaviour
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中文总结 AI 辅助
该研究针对耦合序参数与逆温度的非等温相场系统，通过完全离散逼近构造广义耗散弱解，证明其弱-强唯一性与平稳ω极限态存在性，为相关热力学系统分析提供理论基础。
中文摘要 AI 辅助
我们研究一个热力学一致的非等温相场系统，该系统通过完全非对角的Onsager迁移率耦合两个序参数与逆温度。该模型描述了质量扩散、热传导和局部相弛豫的相互作用，同时守恒质量与内能并产生熵。我们采用完全离散逼近构造全局广义耗散弱解。该离散格式守恒质量与内能，且满足离散熵不等式。可用性估计结合适配维度的障碍，在固定网格下得到严格正性与一致估计。紧性分析使我们能过渡到连续系统，内能奇异部分的集中由非负缺陷测度描述。熵-可用性结构在无限时间区间上产生有限耗散，并给出平稳ω极限态的存在性。最后，相对熵论证建立了弱解与强解保持在有界热力学状态范围内时的弱-强唯一性。
英文摘要
We study a thermodynamically consistent non-isothermal phase-field system coupling two order parameters and the inverse temperature through a fully non-diagonal Onsager mobility. The model describes the interaction of mass diffusion, heat conduction, and local phase relaxation while conserving mass and internal energy and producing entropy. Global generalised dissipative weak solutions are constructed using a fully discrete approximation. The discrete scheme conserves mass and internal energy and satisfies a discrete entropy inequality. An availability estimate, together with a dimension-adapted barrier, yields strict positivity at fixed mesh and uniform estimates. Compactness then allow passage to the continuous system. Concentration of the singular part of the internal energy is represented by a non-negative defect measure. The entropy-availability structure yields finite dissipation on the infinite time interval and the existence of stationary $ω$-limit states. Finally, a relative-entropy argument establishes weak-strong uniqueness whenever the weak and strong solutions remain in a bounded thermodynamic state range.
