Maximal regularity for time-fractional Schrödinger equations and application to nonlinear equations
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| Format: | Preprint |
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2026
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| author | Chorfi, S. E. Et-tahri, F. Maniar, L. Yamamoto, M. |
| author_facet | Chorfi, S. E. Et-tahri, F. Maniar, L. Yamamoto, M. |
| contents | We study the maximal regularity problem for abstract time-fractional Schrödinger equations $\partial_t^α(u-u_0) -\mathrm{i} A u=f$, with a fractional derivative $\partial_t^α$ of order $α\in (0,1)$. We assume that $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$. First, we prove the maximal $L^2$-regularity by leveraging properties of Mittag-Leffler functions with an imaginary argument. Compared to existing results for the subdiffusion equations, our proof avoids using the complete monotonicity of Mittag-Leffler functions, which seems difficult to prove within the setting of an imaginary argument. Then, we prove the maximal $L^p$-regularity for $p\in (1,\infty)$ using the operator-valued version of Mikhlin's multiplier theorem. Finally, we apply the maximal regularity results to prove the local well-posedness of quasilinear and semilinear time-fractional Schrödinger equations. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16726 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Maximal regularity for time-fractional Schrödinger equations and application to nonlinear equations Chorfi, S. E. Et-tahri, F. Maniar, L. Yamamoto, M. Analysis of PDEs Functional Analysis 35R11, 35B65, 35D30, 35Q55, 33E12 We study the maximal regularity problem for abstract time-fractional Schrödinger equations $\partial_t^α(u-u_0) -\mathrm{i} A u=f$, with a fractional derivative $\partial_t^α$ of order $α\in (0,1)$. We assume that $A$ is a self-adjoint operator with compact resolvent on a Hilbert space $H$. First, we prove the maximal $L^2$-regularity by leveraging properties of Mittag-Leffler functions with an imaginary argument. Compared to existing results for the subdiffusion equations, our proof avoids using the complete monotonicity of Mittag-Leffler functions, which seems difficult to prove within the setting of an imaginary argument. Then, we prove the maximal $L^p$-regularity for $p\in (1,\infty)$ using the operator-valued version of Mikhlin's multiplier theorem. Finally, we apply the maximal regularity results to prove the local well-posedness of quasilinear and semilinear time-fractional Schrödinger equations. |
| title | Maximal regularity for time-fractional Schrödinger equations and application to nonlinear equations |
| topic | Analysis of PDEs Functional Analysis 35R11, 35B65, 35D30, 35Q55, 33E12 |
| url | https://arxiv.org/abs/2603.16726 |