On $L^\infty$-BMO estimates for derivatives of the Stokes semigroup

Abstract

We consider the Stokes equations in a class of domains that we will call admissible domains including bounded domains, the half space and exterior domains. We will prove new $L^\infty$ estimates for derivatives of velocity and pressure. The estimates will be given in terms of a BMO-type norm of the initial data.

Appendix: \(L^\infty \)-BMO estimate for the heat semigroup

In this section we will give a simple proof of (1.2) for the reader’s convenience.

Theorem 7.1

For a given $v_0\in BMO (\mathbb {R}^n)$ ($n\ge 1$) the $L^\infty$-norm of $\nabla G_t*v_0$ is estimated as

$$\begin{aligned} t^{1/2}\Vert \nabla G_t*v_0\Vert _\infty \le C_* [v_0]_{ BMO }\quad (t>0) \end{aligned}$$

with a constant $C_*$ depending only on n.

Proof

It suffices to prove that

$$\begin{aligned} t^{1/2}\Vert \nabla G_t*u_0\Vert _{\mathcal {H}^1}\le C\Vert u_0\Vert _1 \end{aligned}$$

(7.1)

for $u_0\in L^1(\mathbb {R}^n)$ since $(\mathcal {H}^1)^*= BMO$ by [13], where $\mathcal {H}^1=\mathcal {H}^1(\mathbb {R}^n)$ denotes the Hardy space consisting of all $f\in L^1(\mathbb {R}^n)$ for which

$$\begin{aligned} \Vert f\Vert _{\mathcal {H}^1}=\Vert \sup _{s>0}|G_s*f|\Vert _1=\int _{\mathbb {R}^n}\sup _{s>0}|G_s*f|(x)\,dx<\infty . \end{aligned}$$

Indeed by duality

$$\begin{aligned} \Vert \partial _j G_t*v_0\Vert _\infty =\sup \left\{ \left| \int _{\mathbb {R}^n}u_0(\partial _j G_t)*v_0\,dx\right| {:}\,\Vert u_0\Vert _1\le 1\right\} \end{aligned}$$

for $j=1,\ldots , n$. Since $(\mathcal {H}^1)^*= BMO$ and by the antisymmetry of $\partial _j G_t$ we observe that

$$\begin{aligned} \left| \int _{\mathbb {R}^n}u_0((\partial _j G_t)*v_0)\,dx\right|&=\left| -\int _{\mathbb {R}^n}((\partial _j G_t)*u_0)v_0\,dx\right| \\&\le \Vert \partial _j G_t*u_0\Vert _{\mathcal {H}^1}[v_0]_{ BMO }. \end{aligned}$$

The estimate (7.1) now yields

$$\begin{aligned} \Vert \partial _j G_t*v_0\Vert _\infty \le C t^{-1/2}[v_0]_{ BMO }. \end{aligned}$$

The proof of (7.1) is given in [7, 17]. We will give it here for the sake of completeness. We first show (7.1) for $t=1$. By definition we observe that

$$\begin{aligned} \Vert \partial _j G_1*u_0\Vert _{\mathcal {H}^1}&= \Vert \sup _{s>0}|G_s*\partial _j G_1*u_0|\Vert _1\\&\le \Vert \sup _{s>0}\left( |\partial _j G_{s+1}|*|u_0|\right) \Vert _1\\&\le \Vert \left( \sup _{s>0}|\partial _j G_{s+1}|\right) *|u_0|\Vert _1\\&\le \Vert \sup _{s>0}|\partial _j G_{s+1}|\Vert _1\Vert u_0\Vert _1. \end{aligned}$$

Since $\partial _j G_t=-x_j G_t/(2t)$, we observe that

$$\begin{aligned} |\partial _j G_t(x)|\le \frac{2|x_j|}{|x|^{n+2}}\frac{1}{\pi ^{n/2}}\varrho ^{\frac{n+2}{2}}\mathrm {e}^{-\varrho } \quad \text{ with }\quad \varrho =\frac{|x|^2}{4t}. \end{aligned}$$

This implies

$$\begin{aligned} |\partial _j G_t(x)|\le \frac{C_0}{|x|^{n+1}} \end{aligned}$$

with $C_0=2\pi ^{-n/2}\sup _{\varrho >0}\varrho ^\frac{n+2}{2}\mathrm {e}^{-\varrho }$. Additionally we can estimate $|\partial _j G_t(x)|\le (4\pi )^{-n/2}$ for $t\ge 1$ and thus observe that

$$\begin{aligned} |\partial _j G_{s+1}(x)|\le \min \left\{ \frac{C_0}{|x|^{n+1}}, \frac{1}{(4\pi )^{n/2}}\right\} =:a(x) \text{ for } s>0, x\in \mathbb {R}^n. \end{aligned}$$

The right hand side is integrable in x and therefore

$$\begin{aligned} \Vert \partial _j G_1*u_0\Vert _{\mathcal {H}^1}\le C_*\Vert u_0\Vert _1, \end{aligned}$$

where $C_*=\int _{\mathbb {R}^n}a(x)\,dx$. We have now proved (7.1) with $t=1$. To obtain (7.1) for general t we apply a scaling transformation. For $\lambda >0$ and a function f in $\mathbb {R}^n$ let $f_\lambda$ be defined by $f_\lambda (x)=\lambda ^n f(\lambda x)$. We notice that

$$\begin{aligned} (\partial _j G_1)* (u_0)_\lambda = \lambda \left( (\partial _j G_{\lambda ^2})* u_0\right) _\lambda . \end{aligned}$$

Since both the $\mathcal {H}^1$-norm and the $L^1$-norm are invariant under this scaling transformation, the estimate (7.1) with $t=1$ yields

$$\begin{aligned} \lambda \Vert \partial _j G_{\lambda ^2}* u_0\Vert _{\mathcal {H}^1} \le C_* \Vert u_0\Vert _1 \end{aligned}$$

which yields (7.1) by taking $\lambda =t^{1/2}$.

Note that our proof implies

$$\begin{aligned} \Vert \partial _j G_1\Vert _{\mathcal {H}^1}\le C_* \text{ and } \Vert \partial _j G_t\Vert _{\mathcal {H}^1}\le C_*t^{-1/2}, \end{aligned}$$

which was established by [7]. $\square$