Abstract
We start here by studying the porous media equation problem (1.1) when \(\beta: \mathbb{R} \rightarrow \mathbb{R}\) is monotonically increasing and Lipschitz continuous. The main reason is that general maximal monotone graphs β can be approximated by their Yosida approximations β ε which are Lipschitz continuous and monotonically increasing. So, several estimates proved in this chapter will be exploited later for studying problems with more general β.
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Notes
- 1.
Here \(\mathcal{L}_{2}(H^{-1},L^{2})\) denotes the space of all Hilbert–Schmidt operators from H−1 to L2.
- 2.
\(\mathcal{L}_{1}(H^{-1})\) denotes the space of all symmetric, nonnegative definite, trace-class operators in H −1.
- 3.
\(\vert f\vert _{p} \leq \vert f\vert _{2}^{ \frac{2} {p} }\;\vert f\vert _{\infty }^{\frac{p-2} {p} }\).
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Appendix: Two Analytical Inequalities
Appendix: Two Analytical Inequalities
Let us consider the Laplace operator in \(L^{2}(\mathcal{O}),\;\mathcal{O}\in \mathbb{R}^{d},\) with homogeneous boundary conditions and its orthonormal basis of eigenfunctions, that is
We set f k = α k 1∕2 e k so that {f k } is an orthonormal basis in \(H^{-1}(\mathcal{O})\). We assume that \(\partial \mathcal{O}\) is sufficiently regular (for instance of class C 2) in order to apply [66].
Proposition 1
There exist C 1 > 0 and C 2 > 0 such that
and
Proof
The proof of (2.84) is very simple. In fact for each x ∈ L 2 we have
because by [66] we have \(\vert e_{k}\vert _{\infty }\leq c\alpha _{k}^{\frac{d-1} {2} }\) for all \(k \in \mathbb{N}\).
Let us now consider (2.83). Since H −1 is the dual of H 0 1 we have
But
On the other hand, for all \(k \in \mathbb{N}\)
Since
we have
Therefore,
Now by the Sobolev embedding theorem we have \(H_{0}^{1} \subset L^{ \frac{2d} {d-2} }\) for d > 3, H 0 1 ⊂ ∩ p ≥ 2 L p for d = 1, 2 with continuous embedding. Then, using Hölder in the first term of (2.87) we see that there is a constant c > 0 such that
Now as mentioned earlier we know that
so that, by interpolationFootnote 3
Finally, we find
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Barbu, V., Da Prato, G., Röckner, M. (2016). Equations with Lipschitz Nonlinearities. In: Stochastic Porous Media Equations. Lecture Notes in Mathematics, vol 2163. Springer, Cham. https://doi.org/10.1007/978-3-319-41069-2_2
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DOI: https://doi.org/10.1007/978-3-319-41069-2_2
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