Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-19T01:59:35.240Z Has data issue: false hasContentIssue false
  • English
  • Français

ON LOCAL ENERGY DECAY ESTIMATE OF THE OSEEN SEMIGROUP IN TWO DIMENSIONS AND ITS APPLICATION

Part of: Partial differential equations Incompressible viscous fluids

Published online by Cambridge University Press:  01 July 2019

Yasunori Maekawa
Yasunori Maekawa*
Affiliation:
Department of Mathematics, Kyoto University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto606-8502, Japan ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the temporal decay estimate of the Oseen semigroup in a two-dimensional exterior domain. We establish the local energy decay estimate with a suitable dependence on the small translation speed, which is a significant improvement of Hishida’s result in 2016. As an application, we prove the $L^{q}$-$L^{r}$ estimates of the Oseen semigroup uniformly in the small translation speed.

Keywords

Oseen semigrouptwo-dimensional flows past an obstaclelocal energy decayresolvent analysis

MSC classification

Primary: 76D07: Stokes and related (Oseen, etc.) flows
Secondary: 35B35: Stability
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 20 , Issue 3 , May 2021 , pp. 859 - 891
Check for updates
Copyright
© Cambridge University Press 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abe, K., On estimates for the Stokes flow in a space of bounded functions, J. Differential Equations 261 (2016), 1756–1795.CrossRefGoogle Scholar
Abe, K. and Giga, Y., The L ∞-Stokes semigroup in exterior domains, J. Evol. Equ. 14(1) (2014), 1–28.CrossRefGoogle Scholar
Abe, K., Giga, Y. and Hieber, M., Stokes resolvent estimates in spaces of bounded functions, Ann. Sci. Éc. Norm. Supér. (4) 48(3) (2015), 537–559.CrossRefGoogle Scholar
Borchers, W. and Sohr, H., On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J. 19 (1990), 67–87.CrossRefGoogle Scholar
Borchers, W. and Varnhorn, W., On the boundedness of the Stokes semigroup in two-dimensional exterior domains, Math. Z. 213(2) (1993), 275–299.CrossRefGoogle Scholar
Dan, W. and Shibata, Y., 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.CrossRefGoogle Scholar
Dan, W. and Shibata, Y., Remark on the L q-L ∞ estimate of the Stokes semigroup in a 2-dimensional exterior domain, Pacific J. Math. 189 (1999), 223–239.CrossRefGoogle Scholar
Enomoto, Y. and Shibata, Y., Local energy decay of solutions to the Oseen equations in the exterior domains, Indiana Univ. Math. J. 53 (2004), 1291–1330.CrossRefGoogle Scholar
Enomoto, Y. and Shibata, Y., On the rate of decay of the Oseen semigroup in exterior domains and its application to Navier–Stokes equation, J. Math. Fluid Mech. 7 (2005), 339–367.CrossRefGoogle Scholar
Finn, R. and Smith, D. R., On the linearized hydrodynamical equations in two dimensions, Arch. Ration. Mech. Anal. 25 (1967), 1–25.CrossRefGoogle Scholar
Finn, R. and Smith, D. R., On the stationary solution of the Navier–Stokes equations in two dimensions, Arch. Ration. Mech. Anal. 25 (1967), 26–39.CrossRefGoogle Scholar
Hishida, T., L q -L r estimate of the Oseen flow in plane exterior domains, J. Math. Soc. Japan 68(1) (2016), 295–346.CrossRefGoogle Scholar
Iwashita, H., 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(2) (1989), 265–288.CrossRefGoogle Scholar
Kobayashi, T. and Shibata, Y., On the Oseen equation in the three dimensional exterior domains, Math. Ann. 310 (1998), 1–45.CrossRefGoogle Scholar
Maekawa, Y., On stability of physically reasonable solutions to the two-dimensional Navier–Stokes equations, J. Inst. Math. Jussieu (accepted). Published online: 16 May 2019.CrossRefGoogle Scholar
Maremonti, P. and Solonnikov, V., On nonstationary Stokes problem in exterior domains, Ann. Sc. Norm. Super Pisa Cl. Sci. 24 (1997), 395–449.Google Scholar
Miyakawa, T., On nonstationary solutions of the Navier–Stokes equations in an exterior domain, Hiroshima Math. J. 12 (1982), 115–140.Google Scholar
Simader, C. G. and Sohr, H., A new approach to the Helmholtz decomposition and the Neumann problem in L q-spaces for bounded and exterior domains, in Mathematical Problems Relating to the Navier–Stokes Equations, Ser. Adv. Math. Appl. Sci., 11, pp. 1–35 (World Sci. Publishing, River Edge, 1992).Google Scholar