Abstract
In this paper we study maximal L p-regularity for evolution equations with time-dependent operators A. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the L p-boundedness of a class of vector-valued singular integrals which does not rely on Hörmander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of m-th order elliptic operators A with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an L p(L q)-theory for such equations for \(p,q\in (1, \infty )\). In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.
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The first author is supported by Vrije Competitie subsidy 613.001.206 and the second author by the Vidi subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO)
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Gallarati, C., Veraar, M. Maximal Regularity for Non-autonomous Equations with Measurable Dependence on Time. Potential Anal 46, 527–567 (2017). https://doi.org/10.1007/s11118-016-9593-7
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DOI: https://doi.org/10.1007/s11118-016-9593-7
Keywords
- Singular integrals
- Maximal L p-regularity
- Evolution equations
- Functional calculus
- Elliptic operators
- A p -weights
- \(\mathcal {R}\)-boundedness
- Extrapolation
- Quasi-linear PDE