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Maximal regularity for non-autonomous evolution equations

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Abstract

We consider the maximal regularity problem for non-autonomous evolution equations

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Each operator \(A(t)\) is associated with a sesquilinear form \({\mathop {{\varvec{\mathfrak {a}}}}(t; \cdot , \cdot )}\) on a Hilbert space \(H\). We assume that these forms all have the same domain and satisfy some regularity assumption with respect to \(t\) (e.g., piecewise \(\alpha \)-Hölder continuous for some ). We prove maximal \(L_p\)-regularity for all \(u_0 \) in the real-interpolation space . The particular case where \(p = 2\) improves previously known results and gives a positive answer to a question of Lions (Équations différentielles opérationnelles et problèmes aux limites, Springer, Berlin, 1961) on the set of allowed initial data \(u_0\).

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Notes

  1. See Solutions in \(L_{p}\) of abstract parabolic equations in Hilbert spaces. J. Math. Kyoto Univ. 33 (2), 299–314 (1993).

  2. It seems to be difficult to understand the Proof of Lemma 4.3 at the beginning of page 306, the end of the Proof of Proposition 2 at page 307, as well as estimates after formula (5.5) at page 310 in the Proof of Lemma 3.4.

  3. In [14], \(L_2(\mathbb {R}; H)\)-boundedness of \(T_\sigma \) is claimed for symbols \(\sigma : \mathbb {R}{\times } \mathbb {R}\rightarrow \mathcal {B}(H)\) that admit a bounded holomorphic extension to a double sector of \(\mathbb {C}\) in the variable \(\xi \), without any kind of regularity in the variable \(x\).

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Acknowledgments

Both authors wish to thank the anonymous referee for his or her thorough reading of the manuscript and his or her suggestions and improvements. We further wish to thank Dominik Dier for his useful remarks on an earlier version of this paper.

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Correspondence to El Maati Ouhabaz.

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The research of both authors was partially supported by the ANR project HAB, ANR-12-BS01-0013-02.

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Haak, B.H., Ouhabaz, E.M. Maximal regularity for non-autonomous evolution equations. Math. Ann. 363, 1117–1145 (2015). https://doi.org/10.1007/s00208-015-1199-7

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