On gelation for the Smoluchowski equation
Nicolas Fournier
TL;DR
This work analyzes gelation in the Smoluchowski coagulation equation for a broad class of kernels, focusing on $\gamma$-homogeneous kernels with $\gamma>1$ (and certain $\gamma=1$ cases with logarithmic factors). It presents a concise, deterministic proof extending the approach of Escobedo-Mischler-Perthame, showing that under a finite $H$-integral condition $\kappa=\int_{x_0}^{\infty} [H(a)]^{-1/2} da<\infty$ with $H(a)=a\inf\{K(x,y): x,y\in[a,ra]\}$, weak solutions lose mass in finite time with an explicit gelation-time bound $T_{gel}$. The results cover kernels like $K(x,y)=(x\land y)\log^{2+\varepsilon}(e+x\land y)$ and reveal that the optimal logarithmic exponent for gelation depends on kernel shape, including 1-homogeneous kernels where thresholds differ (e.g., $\alpha>2$ vs $\alpha>1$ in representative constructions). Together, these findings sharpen gelation criteria, provide practical bounds on gelation time, and demonstrate that critical exponents are not universal but shaped by kernel structure.
Abstract
Motivated by the recent results of Andreis-Iyer-Magnanini (2023), we provide a short proof, revisiting the one of Escobedo-Mischler-Perthame (2002), that for a large class of coagulation kernels, any weak solution to the Smoluchowski equation looses mass in finite time. The class of kernels we consider is essentially the same as the one of Andreis-Iyer-Magnanini (2023): homogeneous kernels of degree $γ>1$ not vanishing on the diagonal, or homogeneous kernels of degree $γ=1$ not vanishing on the diagonal with some additional logarithmic factor. We also show that when $γ=1$, the power of the logarithmic factor ensuring gelation may depend on the shape of the kernel.