Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski's coagulation equation
Regularity, local behavior and partial uniqueness for self-similar profiles of Smoluchowski's coagulation equation
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Smoluchowski 凝固方程自相似曲线的规律性、局部行为和部分唯一性
DOI：
10.4171/rmi/653
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发表时间：
2011
影响因子：
1.2
通讯作者：
S. Mischler
中科院分区：
文献类型：
--
作者：
J. Cañizo;S. Mischler
我们考虑具有齐次核$a(x，y)=x^\αy^\beta+x^\beta y^\α$的Smoluchowski方程，其中$-1<；\α\leq\beta<；1$和$\lambda：=\α+\beta\in(-1，1)$。我们首先证明了该方程的自相似解是无穷可微的，并证明了当$y=0$时，在$\α<；0$情形下自相似轮廓的行为。我们还给出了一些自相似轮廓的部分唯一性结果：在$\α=0$的情况下，我们证明了具有相同质量和阶矩的两个轮廓必然相等，而在$\α和t；0$的情形下，我们证明了在$y=0$处渐近的两个具有相同阶矩的轮廓是相等的。我们的方法包括凝聚算符的一种新的表示，以及使用分数阶导数来估计其正则性。
We consider Smoluchowski's equation with a homogeneous kernel of the form $a(x,y) = x^\alpha y ^\beta + x^\beta y^\alpha$ with $-1 < \alpha \leq \beta < 1$ and $\lambda := \alpha + \beta \in (-1,1)$. We first show that self-similar solutions of this equation are infinitely differentiable and prove sharp results on the behavior of self-similar profiles at $y = 0$ in the case $\alpha < 0$. We also give some partial uniqueness results for self-similar profiles: in the case $\alpha = 0$ we prove that two profiles with the same mass and moment of order $\lambda$ are necessarily equal, while in the case $\alpha < 0$ we prove that two profiles with the same moments of order $\alpha$ and $\beta$, and which are asymptotic at $y = 0$, are equal. Our methods include a new representation of the coagulation operator, and estimates of its regularity using derivatives of fractional order.