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

Xie, W.

Differ. Integral Equ. 10, No. 2, 393-399 (1997).

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

Keywords:

Stokes operator; Sobolev interpolation inequality; fundamental solutions