Abstract
This paper exposes several recent mathematical results on an interacting system of continuum field distributed over the d-dimensional Euclidean space Rd, which is often referred to as time-dependent Ginzburg-Landau (TDGL) model in physical literatures, e.g. [12]. The evolution law of the system is defined through certain stochastic partial differential equations (SPDE's) on Rd. In Sect. 1, we explain the model from rather heuristic point of view and introduce SPDE's of GL type. The most part of our discussion is devoted to the model for real-valued continuum field; however, the case where the values of the field range over a manifold is also discussed. The existence and uniqueness theorems for these SPDE's are formulated in Sect. 2. Some results on reversible or stationary measures of the SPDE's are summarized briefly. Then mainly from motives of physics two kinds of problems, namely, the hydrodynamic limit and the low temperature limit for the GL model are investigated in Sect.'s 3 and 4, respectively.
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Funaki, T. (1992). On the stochastic partial differential equations of Ginzburg-Landau type. In: Rozovskii, B.L., Sowers, R.B. (eds) Stochastic Partial Differential Equations and Their Applications. Lecture Notes in Control and Information Sciences, vol 176. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0007326
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DOI: https://doi.org/10.1007/BFb0007326
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