具有源项的非等温Cahn-Hilliard系统的最优温度分布
Optimal temperature distribution for a nonisothermal Cahn-Hilliard system with source term
摘要：
本文研究了Cahn-Hilliard型非等温相场系统的最优控制，该系统是经典Caginalp非等温相变模型的推广，具有守恒序参量.该系统耦合了一个Cahn-Hilliard型方程，其序参量为源项，系统的内能满足普适平衡定律。代替标准的傅立叶形式，热通量的本构律被假定为由Green和Naghdi开发的理论给出的形式，该理论解释了演化的可能的热记忆。这具有这样的结果，即内部能量的平衡定律变成热位移或冻结指数的二阶时间方程，即，温度关于时间的本原。我们的系统的另一个特点是存在的源项在方程的序参数，这需要额外的数学困难，因为质量守恒的序参数，典型的经典的卡恩-希利亚德方程，不再满足。在本文中，我们分析的情况下，驱动相变的演变的双阱势是可微的，无论是（在常规情况下）在整个实数集或（在奇异对数的情况下）在有限的开区间;不可微的情况下，如双障碍潜力被排除在分析之外。我们证明了在适当的巴拿赫空间之间的控制到状态算子的Fréchet可微性，并建立了相应的伴随系统的可解性，以导出最优控制问题的相关一阶必要最优性条件。
英文摘要：
In this note, we study the optimal control of a nonisothermal phase field system of Cahn-Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter, typical of the classic Cahn-Hilliard equation, is no longer satisfied. In this paper, we analyze the case that the double-well potential driving the evolution of the phase transition is differentiable, either (in the regular case) on the whole set of reals or (in the singular logarithmic case) on a finite open interval; nondifferentiable cases like the double obstacle potential are excluded from the analysis. We prove the Fréchet differentiability of the control-to-state operator between suitable Banach spaces for both the regular and the logarithmic cases and establish the solvability of the corresponding adjoint systems in order to derive the associated first-order necessary optimality conditions for the optimal control problem.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
