## Journal of the Mathematical Society of Japan

### The Stokes and Navier-Stokes equations in an aperture domain

Takayuki KUBO

#### Abstract

We consider the nonstationary Stokes and Navier-Stokes equations in an aperture domain $\Omega \subset \bm{R}^n$, $n\ge 2$. For this purpose, we prove $L^p$-$L^q$ type estimate of the Stokes semigroup in the aperture domain. Our proof is based on the local energy decay estimate obtained by investigation of the asymptotic behavior of the resolvent of the Stokes operator near the origin. We apply them to the Navier-Stokes initial value problem in the aperture domain. As a result, we can prove the global existence of a unique solution to the Navier-Stokes problem with the vanishing flux condition and some decay properties as $t\to \infty$, when the initial velocity is sufficiently small in the $L^n$ space. Moreover we can prove the time-local existence of a unique solution to the Navier-Stokes problem with the non-trivial flux condition.

#### Article information

Source
J. Math. Soc. Japan, Volume 59, Number 3 (2007), 837-859.

Dates
First available in Project Euclid: 5 October 2007

https://projecteuclid.org/euclid.jmsj/1191591861

Digital Object Identifier
doi:10.2969/jmsj/05930837

Mathematical Reviews number (MathSciNet)
MR2344831

Zentralblatt MATH identifier
1171.35097

Subjects
Primary: 35J55
Secondary: 76D07: Stokes and related (Oseen, etc.) flows

#### Citation

KUBO, Takayuki. The Stokes and Navier-Stokes equations in an aperture domain. J. Math. Soc. Japan 59 (2007), no. 3, 837--859. doi:10.2969/jmsj/05930837. https://projecteuclid.org/euclid.jmsj/1191591861

#### References

• T. Abe and Y. Shibata, On a resolvent estimate of the Stokes equation on an infinite layer, Part 2 $\lambda=0$ case, J. Math. Fluid Mech., 5 (2003), 245–274.
• H. Abels, $L^q$-$L^r$ estimates for the non-stationary Stokes equations in an aperture domain, Z. Anal. Anwendungen, 21 (2002), 159–178.
• H. Abels, Stokes equations in asymptotically flat domains and the motion of a free surface, Doctor Thesis, Technischen Univ. Darmstadt, Shaker Verlag, Aachen, 2003.
• M. E. Bogovskiĭ, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Dokl. Akad. Nauk SSSR, 248 (1979), 1037–1040, English Transl.: Soviet Math. Dokl., 20 (1979), 1094–1098.
• W. Borchers and T. Miyakawa, $L^2$ decay for the Navier-Stokes flow in halfspaces, Math. Ann., 282 (1988), 139–155.
• W. Borchers and H. Sohr, On the equations $rot\ v =g$ and div $u=f$ with zero boundary conditions, Hokkaido Math. J., 19 (1990), 67–87.
• W. Borchers and W. Varnhorn, On the boundedness of the Stokes semigroup in two dimensional exterior domains, Math. Z., 213 (1993), 275–299.
• Z. M. Chen, Solutions of the stationary and nonstationary Navier-Stokes equations in exterior domains, Pacific J. Math., 159 (1993), 227–240.
• W. Dan and Y. Shibata, On the $L_q$-$L_r$ estimates of the Stokes semigroup in a two dimensional exterior domain, J. Math. Soc. Japan, 51 (1999), 181–207.
• W. Dan and Y. Shibata, Remark on the $L_q$-$L_\infty$ estimate of the Stokes semigroup in a 2-dimensional exterior domain, Pacific J. Math., 189 (1999), 223–240.
• W. Dan, T. Kobayashi and Y. Shibata, On the local energy decay approach to some fluid flow in exterior domain, Recent Topics on Mathematical Theory of Viscous Incompressible Fluid, Lecture Notes Numer. Appl. Math., 16, Kinokuniya, Tokyo, 1998, pp.,1–51.
• W. Desch, M. Hieber and J. Prüss, $L^p$ theory of the Stokes equation in a half space, J. Evol. Equations, 1 (2001), 115–142.
• Y. Enomoto and Y. Shibata, Local energy decay of solutions to the Ossen equation in the $n$-dimension exterior domains, Indiana Univ. Math. J., 53 (2004), 1291–1330.
• Y. Enomoto and Y. Shibata, On the rate of decay of the Ossen semigroup in exterior domains and its application to Navier-Stokes equation, J. Math. Fluid Mech., 7 (2005), 339–367.
• R. Farwig, Note on the flux condition and pressure drop in the resolvent problem of the Stokes system, Manuscripta Math., 89 (1996), 139–158.
• 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.
• R. Farwig and H. Sohr, Helmholtz decomposition and Stokes resolvent system for aperture domains in $L^q$-spaces, Analysis, 16 (1996), 1–26.
• M. Franzke, Strong solutions of the Navier-Stokes equations in aperture domains, Ann. Univ. Ferrara Sez. VII. Sc. Mat., 46 (2000), 161–173.
• G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I: Linearized Steady Problems, Vol. II: Nonlinear Steady Problems, Springer-Verlag, New York, 1994.
• Y. Giga, Domains of fractional powers of the Stokes operator in $L^r$ spaces, Arch. Rational Mech. Anal., 89 (1985), 251–265.
• Y. Giga and T. Miyakawa, Solutions in $L^r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., 89 (1985), 267–281.
• T. Hishida, The nonstationary Stokes and Navier-Stokes flows through an aperture, Adv. Math. Fluid Mech., 3 (2004), 79–123.
• J. G. Heywood, On uniqueness questions in the theory of viscous flow, Acta Math., 136 (1976), 61–102.
• H. Iwashita, $L_q$-$L_r$ estimates for solutions of the nonstationary Stokes equations in an exterior domain and the Navier-Stokes initial value problems in $L^q$ spaces, Math. Ann., 285 (1989), 265–288.
• T. Kato, Strong $L^p$-solutions of the Navier-Stokes equation in $\mathbf{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471–480.
• H. Kozono, Global $L^n$-solution and its decay property for the Navier-Stokes equations in half-space $\mathbf{R}^n_+$, J. Differential Equations, 79 (1989), 79–88.
• H. Kozono and T. Ogawa, Two-dimensional Navier-Stokes flow in unbounded domains, Math. Ann., 297 (1993), 1–31.
• H. Kozono and T. Ogawa, Decay properties of strong solutions for the Navier-Stokes equations in two-dimensional unbounded domains, Arch. Rational Mech. Anal., 122 (1993), 1–17.
• T. Kubo and Y. Shibata, On some properties of solutions to the Stokes equation in the half-space and perturbed half-space, Quad. Mat., 15 (2004), 149–220.
• T. Kubo and Y. Shibata, On the Stokes and Navier-Stokes equations in a perturbed half-space, Adv. Differential Equations, 10 (2005), 695–720.
• P. Maremonti and V. A. Solonnikov, On nonstationary Stokes problem in exterior domains, Ann. Sc. Norm. Super Pisa, 24 (1997), 395–449.
• M. McCracken, The resolvent problem for the Stokes equation on halfspaces in $L^p$, SIAM J. Math. Anal., 12 (1981), 201–228.
• T. Miyakawa, The Helmholtz decomposition of vector fields in some unbounded domains, Math. J. Toyama Univ., 17 (1994), 115–149.
• T. Muramatsu, On Besov spaces and Sobolev spaces of generalized functions defined in a general region, Publ. RIMS, Kyto Univ., 9 (1974), 325–396.
• Y. Shibata, On the global existence of classical solutions of second order fully nonlinear hyperbolic equations with first order dissipation in the exterior domain, Tsukuba J. Math., 7 (1983), 1–68.
• Y. Shibata, On an exterior initial boundary value problem for Navier-Stokes equations, Quart. Appl. Math., LVII (1999), 117–155.
• Y. Shibata and S. Shimizu, A decay property of the Fourier transform and its applications to the Stokes problem, J. Math. Fluid Mech., 3 (2001), 213–230.
• H. Tanabe, Equations of evolution, Pitman, London, 1979.
• S. Ukai, A solution formula for the Stokes equation in $\mathbf{R}^n_+$, Comm. Pure Appl. Math., 40 (1987), 611–621.
• M. Wiegner, Decay estimates for strong solutions of the Navier-Stokes equations in exterior domain, Ann. Univ. Ferrara Sez., 46 (2000), 61–79.