## Sharp Sobolev interpolation inequalities for the Stokes operator.(English)Zbl 0892.46030

Let $$\Omega$$ be an open set in $$\mathbb R^n$$ $$(n=2,3)$$ and let $$C_0^\infty (\Omega )$$ denote the set of all smooth solenoidal vector fields with compact support in $$\Omega .$$ The symbols $$\mathcal L^2(\Omega )$$ and $$\mathcal H_0^1(\Omega )$$ are used to denote the closure of $$C_0^\infty (\Omega )$$ in $$L^2$$ and $$H^1,$$ respectively. In the paper the Stokes operator $$\widetilde \Delta$$ is defined as $$\Pi \Delta ,$$ where $$\Delta$$ is the Laplace operator and $$\Pi$$ denotes the orthogonal projection of $$L^2(\Omega )$$ onto $$\mathcal L^2(\Omega ).$$
Using fundamental solutions of generalized Stokes systems, the author proves: If $$u\in \mathcal H^1_0(\mathbb R^2)$$ and $$\widetilde \Delta \mathbf {u}\in \mathcal L^2(\mathbb R^2),$$ then ${\| }\mathbf {u}{\| }_\infty \leq 1/(2\sqrt \pi )\big ({\| }\mathbf {u}{\| }_2 {\| }\widetilde \Delta \mathbf {u}{\| }_2 + {\| }\nabla \mathbf {u}{\| }^2\big )^{1/2}. \tag $$*$$$ The case when equality in ($$*$$) occurs is fully described. This is one of the main results of the paper; the others are its modifications.
Reviewer: B.Opic (Praha)

### MSC:

 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 46E40 Spaces of vector- and operator-valued functions 26D10 Inequalities involving derivatives and differential and integral operators 35E05 Fundamental solutions to PDEs and systems of PDEs with constant coefficients