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 ℝ n. We strive for time independent, a priori estimates in regularity spaces of type L 1(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.
2010 Mathematics Subject Classification: 35K05, 35K10, 35B65.
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Notes
- 1.
Without any support assumption, it is obvious that if s > 0, then we have \(\|\cdot \|_{\dot{B}_{p,r}^{s}({\mathbb{R}}^{n})} \lesssim \|\cdot \|_{B_{p,r}^{s}({\mathbb{R}}^{n})}\) and that, if s < 0, then \(\|\cdot \|_{\dot{B}_{p,r}^{s}({\mathbb{R}}^{n})} \gtrsim \|\cdot \|_{B_{p,r}^{s}({\mathbb{R}}^{n})}.\)
- 2.
Nonhomogeneous Besov spaces on domains may be defined by the same token.
- 3.
Here we just consider the case q < ∞ to shorten the presentation.
- 4.
Below \(\mathcal{M}(X)\) denotes the multiplier space associated to the Banach space X that is the set of those functions f such that fg ∈ X, whenever g is in X, endowed with the norm \(\|f\|_{\mathcal{M}(X)} :=\inf _{g}\|fg\|_{X}\) where the infimum is taken over all g ∈ X with norm 1.
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The second author has been supported by the MN grant IdP2011 000661.
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Danchin, R., Mucha, P.B. (2013). New Maximal Regularity Results for the Heat Equation in Exterior Domains, and Applications. In: Cicognani, M., Colombini, F., Del Santo, D. (eds) Studies in Phase Space Analysis with Applications to PDEs. Progress in Nonlinear Differential Equations and Their Applications, vol 84. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6348-1_6
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