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Crystalline flow starting from a general polygon

  • * Corresponding author: Yoshikazu Giga

    * Corresponding author: Yoshikazu Giga 

The work of the second author was partly supported by the Japan Society for the Promotion of Science (JSPS) through the grants KAKENHI No. 19H00639, No. 18H05323, No. 17H01091 and by Arithmer Inc. through collaborative grant

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  • This paper solves a singular initial value problem for a system of ordinary differential equations describing a polygonal flow called a crystalline flow. Such a problem corresponds to a crystalline flow starting from a general polygon not necessarily admissible in the sense that the corresponding initial value problem is singular. To solve the problem, a self-similar expanding solution constructed by the first two authors with H. Hontani (2006) is effectively used.

    Mathematics Subject Classification: Primary: 35K67; Secondary: 34A12, 53E10, 35C06, 74E15.


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  • Figure 1.  Each arrow indicates the positive direction

    Figure 2.  A given Wulff shape and initial condition

    Figure 3.  Self-similar solution

    Figure 4.  Wulff shape and newly created facets

    Figure 5.  $ i $-th, $ (i-1) $-th and $ (i+1) $-th facet

    Figure 6.  Distance function $ d_i^s(t) $

    Figure 7.  New self-similar solution

    Figure 8.  Wulff shape

    Figure 9.  Compare with a self-similar solution

    Figure 10.  An example of the numerical calculation

    Figure 11.  Another example of the numerical calculation (enlarged)

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