| Abstract: |
| The collective motion of camphor boats in the water channel exhibits both a homogeneous and an inhomogeneous state, depending on the number of boats. In order to analyze those phenomena, we have developed the center manifold theorem proposed by S.-I. Ei et al. (2002) and derived a reduced system from the original model in our previous works (K. Ikeda et al., 2019, K. Ikeda and S.-I. Ei, 2020). As shown in those works, the existence of delta functions in our original system raises two difficulties. One is that our theory is established in $(H^1)^*$-framework, where $(H^1)^*$ is the dual space of $H^1$. Recall that S.-I. Ei et al. has developed their theory in $L^2$-framework in their previous work. The other is that the delta functions cause the lack of regularity of functions and then make it impossible to derive higher order terms.
To avoid the difficulties mentioned above, we introduce a function which can exclude the step function in the original system. In this talk, we develop a center manifold theory in a new system without delta functions in $L^2$-framework. |
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