## Journal of the Mathematical Society of Japan

### Weak Neumann implies $H^\infty$ for Stokes

#### Abstract

Let $\Omega\subset {\mathbb R}^n$ be a domain with uniform $C^3$ boundary and assume that the Helmholtz decomposition exists in ${\mathbb L}^q(\Omega):=L^q(\Omega)^n$ for some $q\in(1,\infty)$. We show that a suitable translate of the Stokes operator admits a bounded ${\cal H}^\infty$-calculus in ${\mathbb L}_\sigma^p(\Omega)$ for $p\in(\min\{q,q'\},\max\{q,q'\})$. For the proof we use a recent maximal regularity result for the Stokes operator on such domains ([GHHS12]) and an abstract result for the ${\cal H}^\infty$-calculus in complemented subspaces ([KKW06], [KW13]).

#### Article information

Source
J. Math. Soc. Japan Volume 67, Number 1 (2015), 183-193.

Dates
First available in Project Euclid: 22 January 2015

http://projecteuclid.org/euclid.jmsj/1421936549

Digital Object Identifier
doi:10.2969/jmsj/06710183

Mathematical Reviews number (MathSciNet)
MR3304018

Zentralblatt MATH identifier
1317.35173

#### Citation

GEIßERT, Matthias; KUNSTMANN, Peer Christian. Weak Neumann implies $H^\infty$ for Stokes. J. Math. Soc. Japan 67 (2015), no. 1, 183--193. doi:10.2969/jmsj/06710183. http://projecteuclid.org/euclid.jmsj/1421936549.

#### References

• H. Abels, Bounded imaginary powers and ${H}_\infty$-calculus of the Stokes operator in unbounded domains, In: Nonlinear Elliptic and Parabolic Problems, Zurich, 2004, (eds. M. Chipot and J. Escher), Progr. Nonlinear Differential Equations Appl., 64, Birkhäuser Verlag, Basel, 2005, pp.,1–15.
• H. Abels, Bounded imaginary powers and $H_\infty$-calculus of the Stokes operator in two-dimensional exterior domains, Math. Z., 251 (2005), 589–605.
• H. Abels, Reduced and generalized Stokes resolvent equations in asymptotically flat layers. II. $H_\infty$-calculus, J. Math. Fluid Mech., 7 (2005), 223–260.
• H. Abels, Nonstationary Stokes system with variable viscosity in bounded and unbounded domains, Discrete Contin. Dyn. Syst. Ser. S, 3 (2010), 141–157.
• H. Abels and Y. Terasawa, On Stokes operators with variable viscosity in bounded and unbounded domains, Math. Ann., 344 (2009), 381–429.
• M. E. Bogovskiĭ, Decomposition of $L_p(\Omega;{\bf R}^n)$ into a direct sum of subspaces of solenoidal and potential vector fields, Dokl. Akad. Nauk SSSR, 286 (1986), 781–786.
• R. Farwig, H. Kozono and H. Sohr, An $L^q$-approach to Stokes and Navier-Stokes equations in general domains, Acta Math., 195 (2005), 21–53.
• R. Farwig, H. Kozono and H. Sohr, On the Helmholtz decomposition in general unbounded domains, Arch. Math. (Basel), 88 (2007), 239–248.
• R. Farwig, H. Kozono and H. Sohr, On the Stokes operator in general unbounded domains, Hokkaido Math. J., 38 (2009), 111–136.
• R. Farwig and M.-H. Ri, The resolvent problem and $H^\infty$-calculus of the Stokes operator in unbounded cylinders with several exits to infinity, J. Evol. Equ., 7 (2007), 497–528.
• R. Farwig and H. Sohr, Generalized resolvent estimates for the Stokes system in bounded and unbounded domains, J. Math. Soc. Japan, 46 (1994), 607–643.
• G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol.,I, Linearized Steady Problems, Springer Tracts Nat. Philos., 38, Springer-Verlag, New York, 1994.
• M. Geissert, H. Heck, M. Hieber and O. Sawada, Weak Neumann implies Stokes, J. Reine Angew. Math., 669 (2012), 75–100.
• N. Kalton, P. Kunstmann and L. Weis, Perturbation and interpolation theorems for the $H^\infty$-calculus with applications to differential operators, Math. Ann., 336 (2006), 747–801.
• N. Kalton and M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Amer. Math. Soc., 350 (1998), 3903–3922.
• P. C. Kunstmann, $H^\infty$-calculus for the Stokes operator on unbounded domains, Arch. Math. (Basel), 91 (2008), 178–186.
• P. C. Kunstmann, Navier-Stokes equations on unbounded domains with rough initial data, Czechoslovak Math. J., 60(135) (2010), 297–313.
• P. C. Kunstmann and L. Weis, Erratum to: Perturbation and interpolation theorems for the $H^\infty$-calculus with applications to differential operators, Math. Ann., 357 (2013), 801–804.
• A. Noll and J. Saal, $H\sp \infty$-calculus for the Stokes operator on $L_q$-spaces, Math. Z., 244 (2003), 651–688.
• H. Sohr, The Navier-Stokes Equations, An Elementary Functional Analytic Approach, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001.