论文标题
长期渐近方程式与没有固定溶液的注入的凝结方程
Long-time asymptotics for coagulation equations with injection that do not have stationary solutions
论文作者
论文摘要
在本文中，我们研究了一类凝血方程，其中包括一个源术语，该术语是在系统大小的系统簇中注入。当$ x $大于$ y $时，凝结核是均匀的，具有均匀性$γ<1 $，因此$ k（x，y）$是$ x^{γ+λ} y^{ - λ} $，当$ x $大于$ y $时。我们将分析限制为$γ+2λ\ geq 1 $。在这一指数范围内，质量向无穷大的运输是由不同大小的颗粒之间的碰撞驱动的。这与$γ+2λ<1 $的情况相反。在这种情况下，质量向无穷大的运输是由于大小相当的颗粒之间的碰撞所致。在$γ+2λ\ geq 1 $的情况下，不同大小的粒子之间的相互作用导致凝血方程中的额外传输项，该方程近似于原始凝结方程与大量注射的溶液。我们证明了一类自相似解决方案的存在，可用于与运输的凝结方程相同的$γ$和$λ$的合适选择。我们证明，对于补充案例，这种自相似解决方案不存在。
In this paper we study a class of coagulation equations including a source term that injects in the system clusters of size of order one. The coagulation kernel is homogeneous, of homogeneity $γ< 1$, such that $K(x,y)$ is approximately $x^{γ+ λ} y^{-λ}$, when $x$ is larger than $y$. We restrict the analysis to the case $γ+ 2 λ\geq 1 $. In this range of exponents, the transport of mass toward infinity is driven by collisions between particles of different sizes. This is in contrast with the case when $γ+ 2 λ<1$. In that case, the transport of mass toward infinity is due to the collision between particles of comparable sizes. In the case $γ+2λ\geq 1$, the interaction between particles of different sizes leads to an additional transport term in the coagulation equation that approximates the solution of the original coagulation equation with injection for large times. We prove the existence of a class of self-similar solutions for suitable choices of $γ$ and $λ$ for this class of coagulation equations with transport. We prove that for the complementary case such self-similar solutions do not exist.
