Singularity formed by the collision of two collapsing solitons in interaction for the 2D Keller-Segel system

Charles Collot, Tej-Eddine Ghoul, Nader Masmoudi, Van Tien Nguyen
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Abstract

It is well-known that the two-dimensional Keller-Segel system admits finite time blowup solutions, which is the case if the initial density has a total mass greater than $8\pi$ and a finite second moment. Several constructive examples of such solutions have been obtained, where for all of them a perturbed stationary state undergoes scale instability and collapses at a point, resulting in a $8\pi$-mass concentration. It was conjectured that singular solutions concentrating simultaneously more than one solitons could exist. We construct rigorously such a new blowup mechanism, where two stationary states are simultaneously collapsing and colliding, resulting in a $16\pi$-mass concentration at a single blowup point, and with a new blowup rate which corresponds to the formal prediction by Seki, Sugiyama and Vel\'azquez. We develop for the first time a robust framework to construct rigorously such blowup solutions involving simultaneously the non-radial collision and concentration of several solitons, which we expect to find applications to other evolution problems.
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二维凯勒-西格尔系统中两个坍缩孤子在相互作用中碰撞形成的奇点
众所周知,二维凯勒-西格尔系统存在有限时间爆炸解,如果初始密度的总质量大于8\pi$且具有有限的第二矩,就会出现这种情况。已经得到了几个这种解的构造性实例,其中所有的扰动静止态都经历了尺度不稳定性,并在一点坍缩,导致质量浓度为8\pi$。有人猜想,同时集中了不止一个孤子的奇异解可能存在。我们严格地构建了这样一种新的炸裂机制,即两个静止态同时发生坍缩和碰撞,从而在单个炸裂点产生一个 16 美元的质量浓度,而且新的炸裂率与 Seki、Sugiyama 和 Vel\'azquez 的正式预测相一致。我们首次建立了一个稳健的框架,以严格地构建这种同时涉及多个孤子的非径向碰撞和浓度的炸裂解,我们期望它能应用于其他演化问题。
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