We study coagulation equations under non-equilibrium conditions which are induced by the addition of a source term for small cluster sizes. We consider both discrete and continuous coagulation equations, and allow for a large class of coagulation rate kernels, with the main restriction being boundedness from above and below by certain weight functions. The weight functions depend on two power law parameters, and the assumptions cover, in particular, the commonly used free molecular and diffusion limited aggregation coagulation kernels. Our main result shows that the two weight function parameters already determine whether there exists a stationary solution under the presence of a source term. In particular, we find that the diffusive kernel allows for the existence of stationary solutions while there cannot be any such solutions for the free molecular kernel. The argument to prove the non-existence of solutions relies on a novel power law lower bound, valid in the appropriate parameter regime, for the decay of stationary solutions with a constant flux. We obtain optimal lower and upper estimates of the solutions for large cluster sizes, and prove that the solutions of the discrete model behave asymptotically as solutions of the continuous model.

我们研究了在非平衡条件下的凝结方程，这是由增加簇大小的源项引起的。我们同时考虑离散和连续的混凝方程，并考虑了大范围的混凝速率核，其主要限制是通过一定的权重函数从上方和下方有界。权重函数取决于两个幂定律参数，并且这些假设尤其涵盖了常用的自由分子和扩散受限的聚集凝结内核。我们的主要结果表明，两个权重函数参数已经确定了在存在源项的情况下是否存在平稳解。特别是，我们发现，扩散核允许存在固定解，而自由分子核则没有任何此类解。证明解不存在的论点依赖于一种新颖的幂定律下界，该定律在适当的参数范围内对具有恒定通量的固定解的衰减是有效的。我们获得了大簇尺寸解的最优上下估计，并证明了离散模型的解与连续模型的解一样渐近地表现。