具有流出边界的非定域凝聚和初始问题解的性质
Properties of the solutions of delocalised coagulation and inception problems with outflow boundaries
摘要：
适定性建立了一个家庭的方程建模粒子种群经历离域凝结，平流，流入和流出在一个外部指定的速度场。当空间域是$d$维空间的一个有界区域，其中每个点恰好位于与速度场相关的一个流线上时，允许非常一般的粒子类型。该问题被表述为粒子位置和型空间上有界测度的巴拿赫空间中的半线性常微分方程。一个本地Lipschitz属性建立在全变差规范的传播（广义半群）与问题，并用于构建一个皮卡德迭代，建立任何初始条件的局部存在性和全局唯一性。唯一的弱解进一步证明是一个可微的或至少有界的变化强解光滑假设下的参数的凝聚相互作用。在一个空间维度的情况下，强可微性的建立，即使凝固参数在空间中具有特定的有界变化结构。这个一维扩展建立了[Patterson，Stoch. Anal. 2013年10月31日]，一个独特的和可微的限制。
英文摘要：
Well posedness is established for a family of equations modelling particle populations undergoing delocalised coagulation, advection, inflow and outflow in a externally specified velocity field. Very general particle types are allowed while the spatial domain is a bounded region of $d$-dimensional space for which every point lies on exactly one streamline associated with the velocity field. The problem is formulated as a semi-linear ODE in the Banach space of bounded measures on particle position and type space. A local Lipschitz property is established in total variation norm for the propagators (generalised semi-groups) associated with the problem and used to construct a Picard iteration that establishes local existence and global uniqueness for any initial condition. The unique weak solution is shown further to be a differentiable or at least bounded variation strong solution under smoothness assumptions on the parameters of the coagulation interaction. In the case of one spatial dimension strong differentiability is established even for coagulation parameters with a particular bounded variation structure in space. This one dimensional extension establishes the convergence of the simulation processes studied in [Patterson, Stoch. Anal. Appl. 31, 2013] to a unique and differentiable limit.
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