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A simple proof of non-explosion for measure solutions of the Keller-Segel equation

  • *Corresponding author: Nicolas Fournier

    *Corresponding author: Nicolas Fournier 
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  • We give a simple proof, relying on a two-particles moment computation, that there exists a global weak solution to the $ 2 $-dimensional parabolic-elliptic Keller-Segel equation when starting from any initial measure $ f_0 $ such that $ f_0( {\mathbb{R}}^2)< 8 \pi $.

    Mathematics Subject Classification: Primary: 35K57, 35D30; Secondary: 92C17.

    Citation:

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    [3] P. Biler, Singularities of Solutions to Chemotaxis Systems, De Gruyter Series in Mathematics and Life Sciences, 6, De Gruyter, Berlin, 2020. doi: 10.1515/9783110599534.
    [4] P. BilerG. KarchP. Laurençot and T. Nadzieja, The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in a disc, Topol. Methods Nonlinear Anal., 27 (2006), 133-147. 
    [5] P. BilerG. KarchP. Laurençot and T. Nadzieja, The $8\pi$-problem for radially symmetric solutions of a chemotaxis model in the plane, Math. Methods Appl. Sci., 29 (2006), 1563-1583.  doi: 10.1002/mma.743.
    [6] A. Blanchet, J. Dolbeault and B. Perthame, Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Differential Equations, (2006), 32 pp.
    [7] D. BreschP.-E. Jabin and Z. Wang, On mean-field limits and quantitative estimates with a large class of singular kernels: Application to the Patlak-Keller-Segel model, C. R. Math. Acad. Sci. Paris, 357 (2019), 708-720.  doi: 10.1016/j.crma.2019.09.007.
    [8] N. Fournier and B. Jourdain, Stochastic particle approximation of the Keller-Segel equation and two-dimensional generalization of Bessel processes, Ann. Appl. Probab., 27 (2017), 2807-2861.  doi: 10.1214/16-AAP1267.
    [9] W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.  doi: 10.2307/2153966.
    [10] E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability, J. Theoret. Biol., 26 (1970), 399-415.  doi: 10.1016/0022-5193(70)90092-5.
    [11] H. Osada, A stochastic differential equation arising from the vortex problem, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 333-336.  doi: 10.3792/pjaa.61.333.
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    [13] D. Wei, Global well-posedness and blow-up for the 2-D Patlak-Keller-Segel equation, J. Funct. Anal., 274 (2018), 388-401.  doi: 10.1016/j.jfa.2017.10.019.
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