TY - JOUR

T1 - On decay properties of solutions to the Stokes equations with surface tension and gravity in the half space

AU - Saito, Hirokazu

AU - Shibata, Yoshihiro

N1 - Funding Information:
The first author was partly supported by Grant-in-Aid for JSPS Fellows (No. 25-5259) and by JSPS Japanese-German Graduate Externship at Waseda University. The second author was partly supported by JST CREST, JSPS Grant-in-aid for Scientific Research (S) (No. 24224004), and JSPS Japanese-Germann Graduate Externship at Waseda University.
Publisher Copyright:
©2016 The Mathematical Society of Japan.

PY - 2016

Y1 - 2016

N2 - In this paper, we proved decay properties of solutions to the Stokes equations with surface tension and gravity in the half space R+N = {(x′, xN) | x′ ∈ RN-1, xN > 0} (N ≥ 2). In order to prove the decay properties, we first show that the zero points λ± of Lopatinskii determinant for some resolvent problem associated with the Stokes equations have the asymptotics: λ± = ±icg1/2|ζ′|1/2-2|ζ′|2+O(|ζ′|5/2) as |ζ′| → 0, where cg > 0 is the gravitational acceleration and ζ′ ∈ RN-1 is the tangential variable in the Fourier space. We next shift the integral path in the representation formula of the Stokes semi-group to the complex left half-plane by Cauchy's integral theorem, and then it is decomposed into closed curves enclosing λ± and the remainder part. We finally see, by the residue theorem, that the low frequency part of the solution to the Stokes equations behaves like the convolution of the (N - 1)-dimensional heat kernel and Fζ′-1[e±icg1/2|ζ′|1/2t](x′) formally, where Fζ′-1 is the inverse Fourier transform with respect to ζ′. However, main task in our approach is to show that the remainder part in the above decomposition decay faster than the residue part.

AB - In this paper, we proved decay properties of solutions to the Stokes equations with surface tension and gravity in the half space R+N = {(x′, xN) | x′ ∈ RN-1, xN > 0} (N ≥ 2). In order to prove the decay properties, we first show that the zero points λ± of Lopatinskii determinant for some resolvent problem associated with the Stokes equations have the asymptotics: λ± = ±icg1/2|ζ′|1/2-2|ζ′|2+O(|ζ′|5/2) as |ζ′| → 0, where cg > 0 is the gravitational acceleration and ζ′ ∈ RN-1 is the tangential variable in the Fourier space. We next shift the integral path in the representation formula of the Stokes semi-group to the complex left half-plane by Cauchy's integral theorem, and then it is decomposed into closed curves enclosing λ± and the remainder part. We finally see, by the residue theorem, that the low frequency part of the solution to the Stokes equations behaves like the convolution of the (N - 1)-dimensional heat kernel and Fζ′-1[e±icg1/2|ζ′|1/2t](x′) formally, where Fζ′-1 is the inverse Fourier transform with respect to ζ′. However, main task in our approach is to show that the remainder part in the above decomposition decay faster than the residue part.

KW - Decay properties

KW - Gravity

KW - Half-space problem

KW - Stokes equations

KW - Surface tension

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U2 - 10.2969/jmsj/06841559

DO - 10.2969/jmsj/06841559

M3 - Article

AN - SCOPUS:84992500429

SN - 0025-5645

VL - 68

SP - 1559

EP - 1614

JO - Journal of the Mathematical Society of Japan

JF - Journal of the Mathematical Society of Japan

IS - 4

ER -