On Exponential Splitting Methods for Semilinear Abstract Cauchy problems
Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) exponential splittings are well established numerical methods utilizing operator semigroup theory for the treatment of semilinear evolution equations whose principal linear part i...
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| Published in | Integral equations and operator theory Vol. 95; no. 2 |
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| Language | English |
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| ISSN | 0378-620X 1420-8989 |
| DOI | 10.1007/s00020-023-02735-6 |
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| Abstract | Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) 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 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 Kato (Math Z 187(4):471–480, 1984) 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 Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann (2005), but also allows the consideration of examples, such as non-analytic Ornstein–Uhlenbeck semigroups or the Navier–Stokes flow around rotating bodies. |
|---|---|
| AbstractList | Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) 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 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 Kato (Math Z 187(4):471–480, 1984) 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 Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann (2005), but also allows the consideration of examples, such as non-analytic Ornstein–Uhlenbeck semigroups or the Navier–Stokes flow around rotating bodies. Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) 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 $$\frac{\pi }{2}$$ π 2 (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing 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 Kato (Math Z 187(4):471–480, 1984) to the local solvability of the Navier–Stokes equation, where the $$\textrm{L}^p$$ L p - $$\textrm{L}^r$$ 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 Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann (2005), but also allows the consideration of examples, such as non-analytic Ornstein–Uhlenbeck semigroups or the Navier–Stokes flow around rotating bodies. Due to the seminal works of Hochbruck and Ostermann (Appl Numer Math 53(2–4):323–339, 2005, Acta Numer 19:209–286, 2010) 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 π2 (meaning essentially the holomorphy of the underlying semigroup). The present paper contributes to this subject by relaxing 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 Kato (Math Z 187(4):471–480, 1984) to the local solvability of the Navier–Stokes equation, where the Lp - Lr-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 Hochbruck and Ostermann), or more generally in the setting of Hochbruck and Ostermann (2005), but also allows the consideration of examples, such as non-analytic Ornstein–Uhlenbeck semigroups or the Navier–Stokes flow around rotating bodies. |
| ArticleNumber | 15 |
| Author | Jacob, Birgit Schmitz, Merlin Farkas, Bálint |
| Author_xml | – sequence: 1 givenname: Bálint surname: Farkas fullname: Farkas, Bálint organization: School of Mathematics and Natural Sciences, IMACM, University of Wuppertal – sequence: 2 givenname: Birgit surname: Jacob fullname: Jacob, Birgit organization: School of Mathematics and Natural Sciences, IMACM, University of Wuppertal – sequence: 3 givenname: Merlin orcidid: 0000-0002-5522-6512 surname: Schmitz fullname: Schmitz, Merlin email: meschmitz@uni-wuppertal.de organization: School of Mathematics and Natural Sciences, IMACM, University of Wuppertal |
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| Title | On Exponential Splitting Methods for Semilinear Abstract Cauchy problems |
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