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On the Collisions of an \(N\)-Particle System Interacting via the Newtonian Gravitational Potential

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Abstract

This article investigates an \(N\)-particle system with the Newtonian gravitational interaction in dimensions \(n\geq 3\). For the many-particle system without noise term, we prove that there is a finite time \(T\) such that the pairs of particles do collide with each other for any initial condition. When the noise effect is concerned and the initial data are independent and identically distributed (i.i.d.), we prove that pairs of particles do collide with positive probability: the singularity of the drift is indeed visited in finite time, which means that, for the stochastic \(N\)-particle system interacting via the Newtonian gravitational potential, global strong solution dose not exist.

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References

  1. Bertozzi, A.L., Laurent, T.: Finite-time blow up of solutions of an aggregation equation in \(\mathbb{R}^{n}\). Commun. Math. Phys. 274, 717–735 (2007)

    Article  Google Scholar 

  2. Burger, M., Di Francesco, M.: Large time behavior of nonlocal aggregation models with nonlinear diffusion. Netw. Heterog. Media 3(4), 749–785 (2008)

    Article  MathSciNet  Google Scholar 

  3. Cattiaux, P., Pédèches, L.: The 2-D stochastic Keller-Segel particle model: existence and uniqueness. ALEA Lat. Am. J. Probab. Math. Stat. 13, 447–463 (2016)

    Article  MathSciNet  Google Scholar 

  4. Chavanis, P.H.: Critical mass of bacterial populations and critical temperature of self-gravitating Brownian particles in two dimensions. Phys. A, Stat. Mech. Appl. 384, 392–412 (2007)

    Article  Google Scholar 

  5. Chavanis, P.H.: Exact diffusion coefficient of self-gravitating Brownian particles in two dimensions. Eur. Phys. J. B 57(4), 391–409 (2007)

    Article  MathSciNet  Google Scholar 

  6. DiPerna, R.J., Lions, P.-L.: Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98, 511–547 (1989)

    Article  MathSciNet  Google Scholar 

  7. Fournier, N., Jourdain, B.: Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes. Ann. Appl. Probab. 27(5), 2807–2861 (2017)

    Article  MathSciNet  Google Scholar 

  8. Gihman, I.I., Skorohod, A.V.: Stochastic Differential Equations. Springer Berlin Heidelberg, New York (1972)

    Book  Google Scholar 

  9. Godinho, D., Quininao, C.: Propagation of chaos for a sub-critical Keller-Segel model. Ann. Inst. Henri Poincaré Probab. Stat. 51(3), 965–992 (2015)

    Article  MathSciNet  Google Scholar 

  10. Hauray, M., Jabin, P.-E.: \(N\)-particles approximation of the vlasov equations with singular potential. Arch. Ration. Mech. Anal. 183, 489–524 (2007)

    Article  MathSciNet  Google Scholar 

  11. Huang, H., Liu, J.-G.: Error estimate of a random particle blob method for the Keller-Segel equation. Math. Comput. 86(308), 2719–2744 (2017)

    Article  MathSciNet  Google Scholar 

  12. Laurent, T.: Local and global existence for an aggregation equation. Commun. Partial Differ. Equ. 32, 1941–1964 (2007)

    Article  MathSciNet  Google Scholar 

  13. Liu, J.-G., Yang, R.: Propagation of chaos for Keller-Segel equation with a logarithmic cut-off. Methods Appl. Anal. 26(4), 319–348 (2019)

    MathSciNet  MATH  Google Scholar 

  14. Liu, J.-G., Yang, R.: Propagation of chaos for large Brownian particle system with Coulomb interaction. Res. Math. Sci. 3(1), 40 (2016)

    Article  MathSciNet  Google Scholar 

  15. Jabin, P.-E., Hauray, M.: Particles approximations of Vlasov equations with singular forces: propagation of chaos. Ann. Sci. Éc. Norm. Supér. 48(4), 891–940 (2015)

    Article  MathSciNet  Google Scholar 

  16. Øksendal, B.: Stochastic Differential Equations: An Introduction with Applications, 6th edn. Springer, Berlin (2003)

    Book  Google Scholar 

  17. Osada, H.: A stochastic differential equation arising from the vortex problem. Proc. Jpn. Acad., Ser. A, Math. Sci. 61(10), 333–336 (1985)

    Article  MathSciNet  Google Scholar 

  18. Saari, D.G.: Improbability of collisions in newtonian gravitational systems II. Trans. Am. Math. Soc. 181, 351–368 (1973)

    Article  MathSciNet  Google Scholar 

  19. Saari, D.G.: A global existence theorem for the four-body problem of Newtonian mechanics. J. Differ. Equ. 26(1), 80–111 (1977)

    Article  MathSciNet  Google Scholar 

  20. Takanobu, S.: On the existence and uniqueness of SDE describing an \(n\)-particle system interacting via a singular potential. Proc. Jpn. Acad. 61A, 287–290 (1985)

    MathSciNet  MATH  Google Scholar 

  21. Xia, Z.: The existence of noncollision singularities in Newtonian systems. Ann. Math. 135(3), 411–468 (1992)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

The research of R. Yang is supported by National Natural Science Foundation of China No. 11601021. The research of H. Min is supported by the Beijing National Science Foundation No. 1184013 and National Natural Science Foundation of China No. 12001023.

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  1. Faculty of Science, Beijing University of Technology, Ping Le Yuan100, Chaoyang District, Beijing, 100124, People’s Republic of China

    Rong Yang & Hui Min

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  1. Rong Yang
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  2. Hui Min
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Correspondence to Rong Yang.

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Yang, R., Min, H. On the Collisions of an \(N\)-Particle System Interacting via the Newtonian Gravitational Potential. Acta Appl Math 172, 7 (2021). https://doi.org/10.1007/s10440-021-00401-w

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