關於 the generalized Proudman-Johnson equation 的爆炸問題

On the blow-up problems for the generalized Proudman-Johnson equation

黃銘毅(Ming-Yi Huang)

指導教授 ： 卓建宏

國立中正大學/理學院/數學系應用數學研究所/碩士(2016年)

摘要

利用數值的方法來探討 generalized Proudman-Johnson equation 的解的爆炸行為，我們利用了周期邊界條件及Grundy-McLaughlin的邊界條件來做計算。
而目前已知結果中，Hamada利用數值的方法證明出這個方程式在a>1時解會爆炸，並且爆炸速率為1，而我們也利用數值的方法去計算解在a<-3時的行為，我們也發現爆炸解會存在並且爆炸速率為1

關鍵字

爆炸問題

並列摘要

We consider the blow-up problem for the generalized Proudman-Johnson equation. It was proved that, for the periodic boundary condition, the solutions exist globally if $-3leq aleq 1$. For $a>1$, Hamada showed numerically that a blow-up solution might exist. In this paper, we would like to explore the case of $a<-3$.
To this end, we propose a numerical scheme for the computation and use the method given by Cho to determine whether blow-up might occur or not. In addition, we also consider the case of the Grundy-McLaughlin boundary condition. The behavior is different to that of the periodic boundary solution in the case of $a>1$. We check numerically whether this also occur in the case of $a<-3$.
Blow-up rate is computed while blow-up occurs. To verify the validity of our results analytical, we analyze by asymptotic expansion to exclude some possibilities.

並列關鍵字

generalized Proudman-Johnson equation ； blow-up problem

參考文獻

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