On the maximal $L_p$-$L_q$ regularity of the Stokes problem with first order boundary condition; model problems

Shibata, Yoshihiro; Shimizu, Senjo

doi:10.2969/jmsj/06420561

Cited by 52 publications

(87 citation statements)

References 41 publications

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“…On the other hand, considering the Stokes equations with other boundary conditions like non-slip condition, Navier slip condition, Robin condition or pure Neumann condition appearing in the study of the one phase problem, we essentially assume that nth component of the velocity fields vanishes on the boundary, because physically the fluid does not go out or come in through the rigid boundary (cf. [12,13,22,23,[25][26][27]). The appearance of the jump of nth component of the velocity fields across the interface is the unavoidable character of the two-phase problem.…”

Section: Introduction and Resultsmentioning

confidence: 98%

“…Such spaces never appear in the study of the Stokes equations with other boundary conditions like non-slip condition, Navier slip condition, Robin condition or pure Neumann condition appearing in the study of one phase problem (cf. Desch et al, 2001 [12], Farwig and Sohr, 1994 [13], Saal, 2003 [22], Shibata and Shimada, 2007 [23], Shibata and Shimizu 2008 [25], 2009 [26], in press [27]), because the normal component of the velocity fields vanishes at the boundary which is physical requirement that the flow does not go out and come in through the rigid boundary.…”

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confidence: 95%

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Maximal Lp–Lq regularity for the two-phase Stokes equations; Model problems

Shibata

Shimizu

2011

Journal of Differential Equations

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In this paper we prove the generalized resolvent estimate and maximal L p -L q regularity of the Stokes equation with and without surface tension and gravity in the whole space with flat interface.We prove R boundedness of solution operators defined in a sector Σ ,γ 0 = {λ ∈ C \ {0} | |arg λ| π − , |λ| γ 0 } with 0 < < π /2 and γ 0 0, which combined with the Fourier multiplier theorem of S.G. Mihlin and the operator valued Fourier multiplier theorem of L. Weis yields the required generalized resolvent estimate and maximal L p -L q regularity at the same time. One of the character of the paper is to introduce special function spaces E q (Ṙ n , Σ ,γ 0 ) and E p,q,γ 0 (Ṙ n × R) (cf. (1.7) and (1.8)), which is necessary to treat the situation that the normal component of velocity fields jumps across the interface. Such spaces never appear in the study of the Stokes equations with other boundary conditions like non-slip condition, Navier slip condition, Robin condition or pure Neumann condition appearing in the study of one phase problem (cf. Desch et al., 2001 [12], Farwig and Sohr, 1994 [13], Saal, 2003, Shibata and Shimada, 2007 [23], Shimizu 2008 [25], 2009 [26], in press [27]), because the normal component of the velocity fields vanishes at the boundary which is physical requirement that the flow does not go out and come in through the rigid boundary.

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Section: Introduction and Resultsmentioning

confidence: 98%

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confidence: 95%

Maximal Lp–Lq regularity for the two-phase Stokes equations; Model problems

Shibata

Shimizu

2011

Journal of Differential Equations

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“…Recently, there are two papers due to Shibata and Shimizu [15,17], which treat the linearized problem of (1.2) and some resolvent problem. But all the papers do not have any results about decay properties of solutions for the linearized problem of (1.2).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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

Saito

Shibata

2016

J. Math. Soc. Japan

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In this paper, we proved decay properties of solutions to the Stokes equations with surface tension and gravity in the half space R. 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± = ±icwhere cg > 0 is the gravitational acceleration and ξ ′ ∈ R N−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 F −1is the inverse Fourier transform with respect to ξ ′ . However, main task in our approach is to show that the remainder part in the above decomposition decay faster than the residue part.

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“…As an example, we mention the two-phase Stokes equations where the normal component of the velocity jumps across the interface. In the paper [22], Y. Shibata and S. Shimizu have shown maximal L p -L q -regularity for this system, introducing a special function space adapted to the inhomogeneous jump conditions. The proofs in this and many other papers in fluid mechanics (see, e.g., Shibata [21]) are based on partial Fourier transform and careful estimates of the solution operators.…”

Section: Introductionmentioning

confidence: 98%

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity

Denk

Seger

2016

Recent Developments of Mathematical Fluid Mechanics

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Abstract. Uniform a priori estimates for parameter-elliptic boundary value problems are well-known if the underlying basic space equals L p (Ω). However, much less is known for the W s p (Ω)-realization, s > 0, of a parameter-elliptic boundary value problem. We discuss a priori estimates and the generation of analytic semigroups for these realizations in various cases. The Banach scale method can be applied for homogeneous boundary conditions if the righthand side satisfies certain compatibility conditions, while for the general case parameter-dependent norms are used. In particular, we obtain a resolvent estimate for the general situation where no analytic semigroup is generated.

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On the maximal $L_p$-$L_q$ regularity of the Stokes problem with first order boundary condition; model problems

Cited by 52 publications

References 41 publications

Maximal Lp–Lq regularity for the two-phase Stokes equations; Model problems

Maximal Lp–Lq regularity for the two-phase Stokes equations; Model problems

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

Inhomogeneous Boundary Value Problems in Spaces of Higher Regularity

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