Publishing type：Research paper (scientific journal)   Publisher：Cambridge University Press (CUP)  

We investigate enstrophy variations by collapse of point vortices in an inviscid flow and, in particular, focus on the enstrophy dissipation that is a significant property characterising two-dimensional (2-D) turbulent flows. To reveal the vortex dynamics causing the enstrophy dissipation, we consider the dynamics of point vortices, whose vorticity is concentrated on points and dynamics on the inviscid flow, governed by the point-vortex system. The point-vortex system has self-similar collapsing solutions, which are expected to be a key to understand the enstrophy dissipation, but the collapsing process cannot be described by solutions to the 2-D Euler equations. We thus consider the 2-D filtered-Euler equations, which are a regularised model of the 2-D Euler equations, and their point-vortex solutions. The preceding studies (Gotoda and Sakajo, J. Nonlinear Sci. 2016, vol. 26, pp. 1525–1570, Gotoda and Sakajo, SIAM J. Appl. Math. 2018, vol. 78, 2105–2128) have proven that there exist three point-vortex solutions to the filtered model such that they converge to self-similar collapsing orbits in the three point-vortex system and dissipate the enstrophy at the event of collapse in the zero limit of the filter parameter. In this study, we numerically show that the enstrophy dissipation by the collapse of point vortices could occur for the four and five vortex problems in a filtered model. Moreover, we show the detailed convergence process of the point vortices for gradually decreasing filter parameters, which provides a new insight for the three vortex problem. In addition, numerical computations suggest that the enstrophy dissipation is caused by collapse of separated point vortices with the negative interactive energy.

DOI： 10.1017/jfm.2025.10342

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Publishing type：Research paper (scientific journal)   Publisher：IOP Publishing  

Abstract

We consider energy conservation in a two-dimensional incompressible and inviscid flow through weak solutions of the filtered-Euler equations, which describe a regularized Euler flow based on a spatial filtering. We show that the energy dissipation rate for the filtered weak solution with vorticity in L^p, p > 3/2 converges to zero in the limit of the filter parameter. Although the energy defined in the whole space is not finite in general, we formally extract a time-dependent part, which is well-defined for filtered solutions, from the energy and define the energy dissipation rate as its time-derivative. Moreover, the limit of the filtered weak solution is a weak solution of the Euler equations and it satisfies a local energy balance in the sense of distributions. For the case of p = 3/2, we find the same result as p > 3/2 by assuming Onsager’s critical condition for the family of the filtered solutions.

DOI： 10.1088/1361-6544/ac8715

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Other Link： https://iopscience.iop.org/article/10.1088/1361-6544/ac8715/pdf

Language：English   Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1038/s41598-021-92540-1

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PubMed

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Other Link： http://www.nature.com/articles/s41598-021-92540-1

Publishing type：Research paper (scientific journal)   Publisher：Springer Science and Business Media LLC  

DOI： 10.1007/s10884-020-09867-y

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Other Link： https://link.springer.com/article/10.1007/s10884-020-09867-y/fulltext.html

Publishing type：Research paper (scientific journal)   Publisher：IOP Publishing  

Abstract

In recent years, a mathematical model for collective motions of a self-propelled material have been introduced on the basis of the experiments and numerical analysis for the model have made progress in theoretical understanding of the mechanism of collective motions. On the other hand, there are few mathematically rigorous studies for the mathematical model. In this study, to provide mathematical justification for the mathematical model and its numerical analysis, we show the global existence of a unique solution of the mathematical model for a self-propelled motion on ${\mathbb{R } }$. Moreover, we give sufficient conditions for the existence of a non-trivial travelling solution on ${\mathbb{R } }$, which we call a bimodal travelling wave solution.

DOI： 10.1088/2399-6528/ac0100

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Other Link： https://iopscience.iop.org/article/10.1088/2399-6528/ac0100/pdf

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Language：English   Publisher：Kyoto University  

Thesis / Doctoral Dissertation (Science)
Supervisor: Takashi Sakajo, Professor (Kyoto University)
Vice supervisor: Robert Krasny, Professor (University of Michigan)

DOI： 10.14989/doctor.k20154

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