Skip to main content Accessibility help
×
Home

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

  • Yasunori Maekawa (a1)

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.

Copyright

References

Hide All
1. Abe, K., On estimates for the Stokes flow in a space of bounded functions, J. Differential Equations 261 (2016), 17561795.
2. Abe, K. and Giga, Y., The L -Stokes semigroup in exterior domains, J. Evol. Equ. 14(1) (2014), 128.
3. Abe, K., Giga, Y. and Hieber, M., Stokes resolvent estimates in spaces of bounded functions, Ann. Sci. Éc. Norm. Supér. (4) 48(3) (2015), 537559.
4. Borchers, W. and Sohr, H., On the equations rot v = g and div u = f with zero boundary conditions, Hokkaido Math. J. 19 (1990), 6787.
5. Borchers, W. and Varnhorn, W., On the boundedness of the Stokes semigroup in two-dimensional exterior domains, Math. Z. 213(2) (1993), 275299.
6. Dan, W. and Shibata, Y., On the L qL r estimates of the Stokes semigroup in a two-dimensional exterior domain, J. Math. Soc. Japan 51 (1999), 181207.
7. 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), 223239.
8. Enomoto, Y. and Shibata, Y., Local energy decay of solutions to the Oseen equations in the exterior domains, Indiana Univ. Math. J. 53 (2004), 12911330.
9. 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), 339367.
10. Finn, R. and Smith, D. R., On the linearized hydrodynamical equations in two dimensions, Arch. Ration. Mech. Anal. 25 (1967), 125.
11. Finn, R. and Smith, D. R., On the stationary solution of the Navier–Stokes equations in two dimensions, Arch. Ration. Mech. Anal. 25 (1967), 2639.
12. Hishida, T., L q -L r estimate of the Oseen flow in plane exterior domains, J. Math. Soc. Japan 68(1) (2016), 295346.
13. 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), 265288.
14. Kobayashi, T. and Shibata, Y., On the Oseen equation in the three dimensional exterior domains, Math. Ann. 310 (1998), 145.
15. 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.
16. Maremonti, P. and Solonnikov, V., On nonstationary Stokes problem in exterior domains, Ann. Sc. Norm. Super Pisa Cl. Sci. 24 (1997), 395449.
17. Miyakawa, T., On nonstationary solutions of the Navier–Stokes equations in an exterior domain, Hiroshima Math. J. 12 (1982), 115140.
18. 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. 135 (World Sci. Publishing, River Edge, 1992).
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed