论文标题
加权$ \ ell^1 $空间中的离散碎片系统
Discrete fragmentation systems in weighted $\ell^1$ spaces
论文作者
论文摘要
我们研究了一个无限的线性系统，该系统的普通微分方程模拟了碎片簇的演变。我们假设每个群集都由相同的单位（单体）组成，并且在每个碎片事件中允许质量丢失，获得或保守。通过将系统的初始价值问题提出为抽象的库奇问题（ACP），以适当的加权$ \ ell^1 $空间提出，然后采用扰动，这是从操作员分离群的理论中产生的，我们证明了物理相关的经典溶液的存在和独特性，用于广泛的初始集群分布。此外，我们确定始终有可能识别片段半群是分析的加权$ \ ell^1 $空间，这立即暗示相应的ACP对属于此特定空间的任何初始分布都很好地构成。我们还研究了溶液的渐近行为，并表明，在对碎片系数的适当限制下，解决方案显示了融合到纯粹单体稳态的预期长期行为。此外，当碎片分析分析时，溶液被证明以明确定义的指数速率向这种稳态衰减。
We investigate an infinite, linear system of ordinary differential equations that models the evolution of fragmenting clusters. We assume that each cluster is composed of identical units (monomers) and we allow mass to be lost, gained or conserved during each fragmentation event. By formulating the initial-value problem for the system as an abstract Cauchy problem (ACP), posed in an appropriate weighted $\ell^1$ space, and then applying perturbation results from the theory of operator semigroups, we prove the existence and uniqueness of physically relevant, classical solutions for a wide class of initial cluster distributions. Additionally, we establish that it is always possible to identify a weighted $\ell^1$ space on which the fragmentation semigroup is analytic, which immediately implies that the corresponding ACP is well posed for any initial distribution belonging to this particular space. We also investigate the asymptotic behaviour of solutions, and show that, under appropriate restrictions on the fragmentation coefficients, solutions display the expected long-term behaviour of converging to a purely monomeric steady state. Moreover, when the fragmentation semigroup is analytic, solutions are shown to decay to this steady state at an explicitly defined exponential rate.
