On exponential splitting methods for semilinear abstract Cauchy problems

TL;DR

This paper extends exponential splitting methods for semilinear evolution equations by relaxing sectoriality conditions, allowing broader applicability to problems like Navier-Stokes and Ornstein-Uhlenbeck semigroups.

Contribution

It introduces new conditions based on interpolation spaces that generalize existing exponential splitting techniques for semilinear problems.

Findings

Applicable to non-analytic semigroups like Ornstein-Uhlenbeck.

Enables analysis of Navier-Stokes equations around rotating bodies.

Generalizes exponential splitting methods beyond sectorial operators.

Abstract

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 ${\pi}/2$ (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.