多维小应变非线性热粘弹性中的全局可解性与稳定性
Global solvability and stabilization in multi-dimensional small-strain nonlinear thermoviscoelasticity
AI总结：
本文研究了多维小应变非线性热粘弹性模型的全局可解性与稳定性，证明了在无数据大小限制的情况下解的全局存在性，并揭示了温度趋向空间均匀极限的稳定性。
摘要：
尽管过去几十年文献中取得了显著进展，但在连续力学分析中仍存在一个悬而未决的问题，即确定给定的Kelvin-Voigt材料中具有惯性的小应变热粘弹性演化的典型模型在多维情况下是否全局可解。本文在光滑边界$ n $维领域（$ n\ge 2 $）中，针对位移变量$ u $的齐次狄利克雷边界条件和温度$ Θ $的齐次奈曼边界条件，探讨了这一问题。在适当广义的可解性概念下，证明了在数据大小无限制的情况下，解的全局存在性，并且系统实际上比（*）更一般，允许热容量$ κ $依赖于$ Θ $。此外，还推导了关于大时间行为的结果，特别指出温度$ Θ $趋向于空间均匀极限的稳定性。除了与能量守恒和熵产生相关的标准特征外，其核心分析依赖于某些对数改进的经典熵函数的演化性质，据我们所知，在先前文献中未被发现，可能具有独立价值。
英文摘要：
Despite considerable developments in the literature of the past decades, a standing open problem in the analysis of continuum mechanics appears to consist of determining how far the prototypical model for small-strain thermoviscoelastic evolution in Kelvin-Voigt materials with inertia, as given by \[ u_{tt} = μΔu_t + (λ+μ)\nabla\nabla\cdot u_t + \hatμ Δu + (\hatλ+\hatμ) \nabla\nabla\cdot u - B\nablaΘ, \qquad \qquad κΘ_t = DΔΘ+ μ|\nabla u_t|^2 + (λ+μ) |{\rm div} \, u_t|^2 - BΘ{\rm div} \, u_t, \qquad \qquad \qquad (\star) \] is globally solvable in multi-dimensional settings and for initial data of arbitrary size. The present manuscript addresses this in the context of an initial value problem in smoothly bounded $n$-dimensional domains with $n\ge 2$, posed under homogeneous boundary conditions of Dirichlet type for the displacement variable $u$, and of Neumann type for the temperature $Θ$. Within suitably generalized concepts of solvability, global existence of solutions is shown without any size restrictions on the data, and for a system actually more general than ($\star$) by, inter alia, allowing the heat capacity $κ$ to depend on $Θ$. Apart from that, results on large time behavior are derived which particularly assert stabilization of $Θ$ toward a spatially homogeneous limit. Besides on standard features related to energy conservation and entropy production, in its core parts the analysis relies on an evolution property of certain logarithmic refinements of classical entropy functionals, to the best of our knowledge undiscovered in precedent literature and possibly of independent interest.
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