New Maximal Regularity Results for the Heat Equation in Exterior Domains, and Applications
This paper is dedicated to the proof of new maximal regularity results involving Besov spaces for the heat equation in the half-space or in bounded or exterior domains of ℝ n . We strive for time independent, a priori estimates in regularity spaces of typ
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New Maximal Regularity Results for the Heat Equation in Exterior Domains, and Applications Raphaël Danchin and Piotr Bogusław Mucha
Abstract This paper is dedicated to the proof of new maximal regularity results involving Besov spaces for the heat equation in the half-space or in bounded or exterior domains of Rn . We strive for time independent, a priori estimates in regularity spaces of type L1 (0, T ; X) where X stands for some homogeneous Besov space. In the case of bounded domains, the results that we get are similar to those of the whole space or of the half-space. For exterior domains, we need to use mixed Besov norms in order to get a control on the low frequencies. Those estimates are crucial for proving global-in-time results for nonlinear heat equations in a critical functional framework. Key words: Besov spaces, Exterior domain, Heat equation, Maximal regularity, L1 regularity 2010 Mathematics Subject Classification: 35K05, 35K10, 35B65.
6.1 Introduction We are concerned with the proof of maximal regularity estimates for the heat equation with Dirichlet boundary conditions, namely,
R. Danchin () LAMA, UMR 8050, Université Paris-Est et Institut Universitaire de France, 61, avenue du Général de Gaulle, F-94010 Créteil Cedex, France e-mail: [email protected] P.B. Mucha Instytut Matematyki Stosowanej i Mechaniki, Uniwersytet Warszawski, ul. Banacha 2, 02-097 Warszawa, Poland e-mail: [email protected] M. Cicognani et al. (eds.), Studies in Phase Space Analysis with Applications to PDEs, Progress in Nonlinear Differential Equations and Their Applications 84, DOI 10.1007/978-1-4614-6348-1__6, © Springer Science+Business Media New York 2013
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ut − νΔ u = f u=0 u = u0
in (0, T ) × Ω , at (0, T ) × ∂ Ω , on Ω
(6.1)
in various domains Ω of Rn (n ≥ 2). We are interested in L1 -in-time estimates for the solutions to (6.1) with a gain of two full spatial derivatives with respect to the data, that is, ut , ν ∇2 uL1 (0,T ;X) ≤ C u0X + f L1 (0,T ;X) (6.2) with a constant C independent of T. Such time-independent estimates are of importance not only for the heat semigroup theory but also in the applications. Typically, they are crucial for proving global existence and uniqueness statements for nonlinear heat equations with small data in a critical functional framework. Moreover, the fact that two full derivatives may be gained with respect to the source term allows to consider not only the −Δ operator but also small perturbations of it. In addition, we shall see below that it is possible to choose X in such a way that the constructed solution u is L1 in time with values in the set of Lipschitz functions. Hence, if the considered nonlinear heat equation determines the velocity field of some fluid, then this velocity field admits a unique Lipschitzian flow for all time. The model may thus be reformulated equivalently in Lagrangian variables (see, e.g., our recent work [4] in the slightly different context of incompressible flows). This is obviously of interes
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