By et al Pierluigi Colli (Editor)
Section transition phenomena come up in numerous proper actual international events, resembling melting and freezing in a solid–liquid process, evaporation, solid–solid section transitions suit reminiscence alloys, combustion, crystal progress, harm in elastic fabrics, glass formation, section transitions in polymers, and plasticity. the sensible curiosity of such phenomenology is obvious and has deeply encouraged the technological improvement of our society, stimulating severe mathematical learn during this quarter. This e-book analyzes and approximates a few types and similar partial differential equation difficulties that contain section transitions in numerous contexts and contain dissipation results.
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Extra resources for Dissipative Phase Transitions (Series on Advances in Mathematics for Applied Sciences)
Example text
Proof of Theorem 5 Applying well-known results in the theory of parabolic equations, we infer that there exists a unique solution to ut + Au = R, u(0) = 0, (40) with u € W1,oo(0, +oo; H) n H^Q, +co; V) D L°°(0, +oo; F 2 (fi)). (41) Note that (41) entails u S ^(Qoo). (42) Hence, we choose the constant M in (38) to be M = exp(\\u\\LOO(Qoo)+u*), where u* = max{|KH L o o ( n ) ,log6» c + ||5|| L oo Woo) }. (43) Let us now take the difference of (22) and (40), test it by (u — u*)+, where (•) + stands for the positive part and u := u — u.
E. in Q\. Subsequently, from (44) we deduce \ ||(S-u')+(i)|& + \\y(u-u*)\\lHo,t]H) < 0. e. e. in Qt. e. in Qoo, and the proof is complete. For the sake of completeness, let us recall the following proposition (for the proof refer to [6]) from which uniqueness of the solution follows. e. e. e. in x i < X2, (51) QT- It is worth to note that the above result can be applied to prove uniqueness of the solution on the whole interval (0,+oo). Moreover, the property of the solutions stated by Theorem 9 is interesting from a thermomechanical point of view.
Aiki, Phase field model including a hysteresis operator. To appear in: Proceedings of the WCNA 2004. [3] T. Aiki, A model of 3D shape memory alloy materials. J. Math. Soc. Japan 57, 903-933 (2005). [4] T. -H. Hoffmann and T. Okazaki, Wellposedness for a new mathematical model for magnetostrictive thin film multilayers. Adv. Math. Sci. Appl. 14, 417-442 (2004). [5] T. Aiki, A. Kadoya and S. Yoshikawa, One-dimensional shape memory alloy problem with small viscosity. To appear in: Mathematical Approach to Nonlinear Phenomena; Modelling, Analysis and Simulations, Gakkotosho, Tokyo.