On the L p - L q decay estimate for the Stokes equations with free boundary conditions in an exterior domain

Abstract

This paper deals with the $L_{p}$ - $L_{q}$ decay estimate of the $C^{0}$ analytic semigroup ${T (t)}_{t ⩾ 0}$ associated with the perturbed Stokes equations with free boundary conditions in an exterior domain. The problem arises in the study of free boundary problem for the Navier–Stokes equations in an exterior domain. We proved that $\begin{matrix} {‖ \nabla^{j} T (t) f ‖}_{L_{p}} ⩽ C_{p, q} t^{- \frac{j}{2} - \frac{N}{2} (\frac{1}{q} - \frac{1}{p})} ‖ f ‖_{L_{q}} (j = 0, 1) \end{matrix}$ provided that $1 < q ⩽ p ⩽ \infty$ and $q \neq \infty$ . Compared with the non-slip boundary condition case, the gradient estimate is better, which is important for the application to proving global well-posedness of free boundary problem for the Navier–Stokes equations. In our proof, it is crucial to prove the uniform estimate of the resolvent operator, the resolvent parameter ranging near zero.

Keywords

Exterior domains Stokes equations free boundary problem without surface tension

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References

M.E.

Bogovskiĭ, Solution of the first boundary value problem for the equation of continuity of an incompressible medium, Dokl. Acad. Nauk. SSSR 248 (1976), 1037–1049. English transl: Soviet Math. Dokl.20 (1976), 1094–1098.

M.E.

Bogovskiĭ, Solution of some vector analysis problems connected with operators div and grad, in: Trudy Seminar S. L. Sobolev, Vol. 80, Akademia Nauk SSSR, Sibirskoe Otdelnie Matematik, Nowosibirsk in Russian, 1980, pp. 5–40.

Dan,

Kobayashi and

Shibata, On the local energy decay approach to some fluid flow in exterior domain, in: Recent Topics on Mathematical Theory of Viscous Incompressible Fluid, Lecture Notes Numer. Appl. Math., Vol. 16, Kinokuniya, Tokyo, 1998, pp. 1–51.

Dan and

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. doi:10.2969/jmsj/05110181.

Dan and

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. doi:10.2140/pjm.1999.189.223.

G.P.

Galdi, An Introduction to the Mathematical Theory of the Navier–Stokes Equations, Vol. I, Linearized Steady Problems, Springer Tracts in Natural Philosophy, Vol. 38, Springer-Verlag, New York, 1994.

Hishida, The nonstationary Stokes and Navier–Stokes flows through an aperture, in: Contributions to Current Challenges in Mathematical Fluid Mechanics, Adv. Math. Fluid Mech., Vol. 3, Birkhäuser, Basel, 2004, pp. 79–123. doi:10.1007/978-3-0348-7877-7_4.

Hishida,

L^{q}

L^{r}

estimate of the Oseen flow in plane exterior domains, J. Math. Soc. Japan 68 (2016), 295–346. doi:10.2969/jmsj/06810295.

Hishida and

Shibata,

L_{p}

L_{q}

estimate of the Stokes operator and Navier–Stokes flows in the exterior of a rotating obstacle, Arch. Rational Mech. Anal. 193 (2009), 339–421. doi:10.1007/s00205-008-0130-8.

10.

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_{p}

spaces, Math. Ann. 285 (1989), 265–288. doi:10.1007/BF01443518.

11.

Kobayashi and

Shibata, On the Oseen equation in the three dimensional exterior domains, Math. Ann. 310 (1998), 1–45. doi:10.1007/s002080050134.

12.

Kozono, Rapid time-decay and net force to the obstacles by the Stokes flow in exterior domains, Ibid. 320 (2001), 709–730.

13.

Kubo and

Shibata, On the Stokes and Navier–Stokes equations in a perturbed half-space, Adv. Diff. Eqns. 10 (2005), 695–720.

14.

Maremonti and

V.A.

Solonnikov, On nonstationary Stokes problem in exterior domains, Ann. Sc. Norm. Sup. Pisa 24 (1997), 395–449.

15.

Prüß and

Simonett, Moving Interfaces and Quasilinear Parabolic Evolution Equations, Birkhauser Monographs in Mathematics, 2016. ISBN:978-3-319-27698-4.

16.

Shibata, Generalized resolvent estimates of the Stokes equations with first order boundary condition in a general domain, J. Math. Fluid Mech. 15(1) (2013), 1–40. doi:10.1007/s00021-012-0130-1.

17.

Shibata, On the

R

-boundedness of solution operators for the Stokes equations with free boundary condition, Diff. Int. Eqns. 27(3-4) (2014), 313–368.

18.

Shibata, On some free boundary problem for the Navier–Stokes equations in an exterior domain, I. Local well-posedness. Submitted.

19.

Shibata, Global wellposedness of free boundary problem for the Navier–Stokes equations in an exterior domain, Fluid Mechanics Research International Journal. To appear.

20.

Shibata and

Shimizu, Decay properties of the Stokes semigroup in exterior domains with Neumann boundary condition, J. Math. Soc. Japan 59(1) (2007), 1–34. doi:10.2969/jmsj/1180135499.