Abstract
This chapter aims to develop a systematic theory of topological phase transitions (TPTs) and explores a few typical examples, including (1) quantum phase transitions (QPTs), (2) galactic spiral structures, (3) electromagnetic eruptions on solar surface, (4) boundary-layer separation of fluid flows, and (5) interior separation of fluid flows.
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Notes
- 1.
The Dirichlet boundary condition for T amounts to saying that there is heat exchange of the system with outside. Hence there is no contradiction between the non-blow-up condition and the blow-up theorem 9.4.2, where the Neumann boundary condition for T is used.
- 2.
We remark here that by the above inequality and the Hölder inequality, we have
$$\displaystyle \begin{aligned}\| \nabla u\|{}_{L^p} \ge \frac{c}{|\varOmega|{}^{(N^\ast-q)/N^\ast q}} \| u\|{}_{L^q} \quad \text{ for } n> p, N^\ast= \frac{np}{n-p}.\end{aligned}$$
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Ma, T., Wang, S. (2019). Topological Phase Transitions. In: Phase Transition Dynamics. Springer, Cham. https://doi.org/10.1007/978-3-030-29260-7_9
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DOI: https://doi.org/10.1007/978-3-030-29260-7_9
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