Abstract
In this paper we study maximal Lp-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 Lp-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 Lp(Lq)-theory for such equations for p,qin (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.
Highlights
In this paper we study maximal Lp-regularity of the Cauchy problem: u (t) + A(t)u(t) = f (t), t ∈ (0, T ) (1.1)
In this paper we focus on maximal Lp-regularity as this usually requires the least regularity of the data in PDEs
In the present paper we develop a functional analytic approach to maximal Lp-regularity in the case t → A(t) is only measurable
Summary
In this paper we study maximal Lp-regularity of the Cauchy problem:. u(0) = x. When the time-dependence is just measurable, an operator-theoretic condition for maximal Lp-regularity is known only in the Hilbert space setting for p = 2 (see [65, 66] and [88, Section 5.5]). For instance Theorems 1.2 and 5.4 we will present a weighted Lp(Lq )-maximal regularity result in the case A is a 2m-th order elliptic operator, assuming only measurability in the time variable and continuity in the space variable. The proof of Theorem 1.2 is given at the end of Section 5 It will be derived from Theorem 5.4 which is a maximal regularity result with weights in time and space.
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