论文标题
关于半线性抽象库奇问题的指数分裂方法
On exponential splitting methods for semilinear abstract Cauchy problems
论文作者
论文摘要
由于Hochbruck和Ostermann指数分裂的开创性作品，是利用操作员半群理论来处理半线性演化方程的良好数值方法，其主要线性零件涉及一个角度的角度，其角度大于$π/2 $（这意味着本质上是下层半群落的Holomorphy）。本文通过放松部门条件来对这一主题做出贡献，但反过来要求半群操作员始终如一地在插值夫妇（或在Banach空间的规模上）行动。我们的条件（在半群和半线性上）的灵感来自T. Kato对Navier-Stokes方程的局部解决性的方法，其中Stokes Semigroup的$ l^p-l^r $平滑是基本的。当前的抽象操作者理论结果适用于后一种问题（如Ostermann和Hochbruck的结果），或更一般地在Holomorphic semigroups的环境中，但也允许考虑示例，例如非分析性的Ornstein-uhlenbeck semigroups或Navier-Stokes或Navier-Stokes或Navier-Stokes或Navier-Stokes围绕旋转的身体流动。
Due to the seminal works of Hochbruck and Ostermann exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part involves a sectorial operator with angle greater than $π/2$ (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing on the sectoriality condition, but in turn requiring that the semigroup operators act consistently on an interpolation couple (or on a scale of Banach spaces). Our conditions (on the semigroup and on the semilinearity) are inspired by the approach of T. Kato to the local solvability of the Navier-Stokes equation, where the $L^p-L^r$ smoothing of the Stokes semigroup was fundamental. The present abstract operator theoretic result is applicable for this latter problem (as was already the result of Ostermann and Hochbruck), or more generally in the setting of holomorphic semigroups, but also allows the consideration of examples, such as non-analytic Ornstein-Uhlenbeck semigroups or the Navier-Stokes flow around rotating bodies.
