关于迁移方程解的结构性等问题研究

The projection main research: (1) The constitutive theory and stability theory of transport equation solution in cell populations , It is to prove that the semigroups generated by the transport operator is compact in general condition, and the finite order remainder term of transport semigroups is compact, and the spectrum of the transport operators is only consists of finitely isolate eigenvalues with finite algebraic multiplicities in the trip, and the distribution of real isolate eigenvalues and their algebrai index, and the existence of dominant eigenvalue,and the solution of transport equation is complete expansion of generalize eigenfuncyions and so no；(2)the spectral analysis of transport operators with abstract boundary condition, It is to obtain that the spectrum of transport operator is only consists of finitely isolate eigenvalues with finite algebraic multiplicities in the trip and the asymptotic behavior of the transport equation's solution and so on；(3) It is to prove that the finite order remainder term of the singular transport semigroups is compact, and it is to research that the spectral analysis of the transport operators, and it is to prove that the semigroup generated by the Jorgens transport operator is differentiable, and it is to obtain that the solution of transport equation is complete expansion of generalize eigenfuncyions and so no；(2)the spectral analysis of transport operators with abstract boundary condition. These problems research are very dificuty in international transport theory and it is to have important theory and application value .

本项目主要研究:(1)种群细胞中迁移方程解的结构性和稳定性等理论,证明在一般条件下其迁移算子生成半群的紧性、该迁移半群某有限阶余项的紧性、迁移算子的谱在某区域内仅由有限个具有限代数重数的离散本征值组成、实离散本征值的分布及其代数指标和占优本征值的存在性以及该迁移方程的解按广义本征函数完全展开等结果；（2）研究具抽象边界条件的迁移算子的谱分析，获得该迁移算子的谱在某区域内仅有有限个具有限代数重数的离散本征值组成和其迁移方程解的渐近稳定性等；（3）证明最一般奇异迁移算子产生半群的某有限阶余项的紧性，研究该算子的谱分析，特别是证明具Jorgens型迁移算子产生的半群是可微的，并得到其迁移方程的解按广义本征函数完全展开等结果。这些问题都是目前国际迁移理论界研究的热点和难点问题，都具有重要的理论意义和应用价值。

本项目研究的是以种群细胞和粒子(中子、量子等)迁移等为背景的迁移方程解的结构性理论和渐近行为等，其研究的主要内容之一是种群细胞中的Rotenberg模型和L-R模型或具扰动项的L-R模型,在更一般的情况下(即:细胞成熟速度可为0或无限,或细胞遗传周长可为0或无限,或一般边界条件等),得到了相应的迁移算子生成正不可约 半群和其Dyson-Phillips展式的9阶余项的紧性,讨论了该迁移算子的谱分析和相应模型解的渐近行为以及异步生长特性等,获得了该迁移算子的谱在右半平面上仅由有限个具有限代数重数的离散本征值组成;对具光滑边界条件的这类模型,进一步得到其迁移半群的紧性,从而获得了该迁移算子的谱仅由至多可数个具有限代数重数的离散本征值组成, 是唯一可能的聚点以及该模型的解可按广义本征函数展开和完整性等结果;研究的另一个主要内容是粒子(中子、量子等)迁移中具抽象边界条件或奇异迁移方程解的结构性理论和渐近行为等,也得到了相应的迁移算子生成正不可约 半群和其Dyson-Phillips展式的9阶余项的紧性,讨论了该迁移算子的谱分析和相应模型解的渐近行为等,从而获得了该迁移算子的谱在右半平面上仅由有限个具有限代数重数的离散本征值组成等,对奇异的情况,进一步得到其模型的Streaming算子和迁移算子的本质谱型的一致性等结果。