Reading some articles I seen that we can define the heat semigroup $\{e^{t\Delta}\}_{t\geq 0}$ in weakest spaces as Morrey Spaces. For example: for $1\leq q_1 \leq q_2<\infty$, $0\leq \lambda <n$, $t\geq 0$ $$e^{t\Delta}: \mathcal{M}_{q_1,\lambda}(\mathbb{R}^n) \to \mathcal{M}_{q_2,\lambda}(\mathbb{R}^n)$$ and $\|e^{t\Delta} u \|_{\mathcal{M}_{q_2,\lambda}}\leq Ct^{\frac{1}{2}\left(\frac{n-\lambda}{q_1}-\frac{n-\lambda}{q_2}\right)}\|u\|_{\mathcal{M}_{q_1,\lambda}}$, for all $u \in \mathcal{M}_{q_1,\lambda}(\mathbb{R}^n)$, where $\mathcal{M}_{q,\lambda}=\mathcal{M}_{q,\lambda}(\mathbb{R}^n)$ is the Morrey Space
$$\mathcal{M}_{q,\lambda}=\{f \in L_{loc}^q(\mathbb{R}^n): \ \|f\|_{q,\lambda}<\infty\}$$ with norm given by $$\|f\|_{q,\lambda}=\sup \{R^{-\frac{\lambda}{p}}\|f\|_{L^q(B(x,R))}; \ x \in \mathbb{R}^n, \ R>0\}.$$ A lot of references cite this, one of then is Taylor (1992). My doubty is... What can we say about the Stokes Operator $-P\Delta$ in the case of Navier-Stokes equation?
- Can we define the Stokes Operator on Morrey-Spaces? How to do this?
- The Domain of the Stokes operator will be dense in any commom Morrey Space?
- We define the "heat semigroup" from the Stokes operator or vice versa? What theorem is used (I don't think it's Lumer phillips)? Here the emphasis is devide in Kato (1984) one says that the semigroup associed to the Stokes operator is essensily the Heat semigroup, so i think that from the Stokes operator we define the "heat semigroup".
I thanks any help. I am really deficient in this subject.