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 在本論文中我們完成下列三項工作:一. 矩量空間中梯度流系統概念之建立。二. 解析能量消散速率控制某型態之梯度流系統解的矩量速度。三. 架構矩量空間中梯度流系統的伽瑪收斂理論。
 In this manuscript, we introduce the systems of gradient flows on metric spaces and then we investigate an upper control for one form of velocity of solutions by its dissipation rate of energy functional.This paper essentially presents the abstract structure of ”Gamma-convergence of gradient flow systems on metric spaces”.The approach is based on the notion initiated by Sylvia Serfaty.
 1 Introduction2 Preliminaries3 Main results
 [1] L. Ambrosio. Minimizing movements. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5)(1995) 191-246.[2] L. Ambrosio, N. Gigli, and G. Savar´e. Gradient flows in metric spaces and in the space of probability measures. Lectures in Mathematics ETH Z¨urich. Birkh¨auser, 2008.[3] W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander. Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics, vol. 96, Birkh¨auser, Basel 2001.[4] Philippe Cl´ement. An Introduction to Gradient Flows in Metric Spaces. Lecture Note, June 2009.[5] M. Degiovanni, A. Marino, and M. Tosques. Evolution equations with lack of convexity. Nonlinear Anal., 9(1985) 1401-1443.[6] E. De Giorgi, A. Marino, and M. Tosques. Problems of evolution in metric spaces and maximal decreasing curve. Att Accad Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8), 68, (1980), 180-187.[7] N. Q. Le. A Gamma-convergence approach to the Cahn-Hilliard equation. Calc. Var. Partial Differential Equations 32 (2008), no.4, 499-522.[8] N. Q. Le. On the convergence of the Ohta-Kawasaki equation to motion by nonlocal Mullins-Sekerka law. SIAM J. Math. Analysis 42 (2010), no.4, 1602-1638.[9] A. Marino, C. Saccon, and M. Tosques. Curves of maximal slope and parabolic variational inequalities on nonconvex constraints. Ann Scuola Norm. Sup. Pisa Cl. Sci. (4), 16(1989), 281-330.[10] S. Serfaty. Gamma-convergence of gradient flows on Hilbert and metric spaces and applications. Disc. Cont. Dyn. Systems, A, 31, No 4 (2011), 1427-1451.[11] E. Sandier and S. Serfaty. Gamma-convergence of gradient flows with applications to Ginzburg-Landau. Comm. Pure Appl. Math, 57,No 12, (2004), 1627-1672.[12] R.L. Wheeden and A. Zygmund Measure and Integral. (1977).
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