Global existence for the continuous nonhomogeneous Smoluchowski equation and asymptotic behaviour of the solutions
Global existence of weak solutions to the continuous Smoluchowski coagulation equation with diffusion is established for coagulation kernels \(a\) satisfying the structure condition \(a(y,y')\leq a(y,y+y') + a(y',y+y')\), \((y,y')\in (0,\infty)^2\). This assumption is a generalization of the monotonicity condition \(a(y,y')\leq a(y,y+y')\) used previously, and includes several coagulation kernels which could not be considered with the latter, particularly Smoluchowski's coagulation kernel \((y^{1/3} + {y'}^{1/3}) (y^{-1/3} +{ y'}^{-1/3})\). The existence proof relies on the fact that, under this structure condition, the \(L^p\)-norms of the solutions are non-increasing functions of time, while the dissipation rate of the \(L^p\)-norm provides a weak compactness estimate in \(L^1\) for the coagulation reaction term.
scientific article; zbMATH DE number 1981263