On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

Shibata, Yoshihiro; Shimizu, Senjo

doi:10.1515/crelle.2008.013

Cited by 65 publications

(75 citation statements)

References 10 publications

(9 reference statements)

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“…Employing the same argument as in the proof of Theorem 3.8 in [34], by Theorem 2.9 (1) and an interpolation argument we see that for any ðr 0 ;ũ u 0 Þ A _ E E p; q the problem (7.11) admits a unique solution ðr;ũ uÞ with ðr;ũ uÞ A W Finally, we consider the equation: provided that g A ðÀy; g 3 =pÞ, which implies (7.15). Let n 0 be the positive number given in (2.3).…”

Section: ð6:5þmentioning

confidence: 99%

On the $\mathcal{R}$-Sectoriality and the Initial Boundary Value Problem for the Viscous Compressible Fluid Flow

Enomoto

Shibata

2013

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Abstract. In this paper, we prove the R-sectoriality of the resolvent problem for the boundary value problem of the Stokes operator for the compressible viscous fluids in a general domain, which implies the generation of analytic semigroup and the maximal L p -L q regularity for the initial boundary value problem of the Stokes operator. Combining our linear theory with fixed point arguments in the Lagrangian coordinates, we have a local in time unique existence theorem in a general domain and a global in time unique existence theorem for some initial data close to a constant state in a bounded domain for the initial boundary value problem of the Navier-Stokes equations describing the motion of compressible viscous fluids. All the results obtained in this paper were announced in Enomoto-Shibata [12].

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Section: ð6:5þmentioning

confidence: 99%

On the $\mathcal{R}$-Sectoriality and the Initial Boundary Value Problem for the Viscous Compressible Fluid Flow

Enomoto

Shibata

2013

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“…We consider the problem by using the analytic semigroup approach just as Shibata and Shimizu [13,14,15]. One of our main tools to show L p -L q maximal regularity of (4) is R-boundedness and an operator valued Fourier multiplier theorem which have recently been developed by Weis [20] and Denk, Hieber and Prüss [8].…”

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confidence: 99%

Maximal regularity and viscous incompressible flows with free interface

Shimizu

2008

Parabolic and Navier–Stokes Equations

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Abstract. We consider a free interface problem for the Navier-Stokes equations. We obtain local in time unique existence of solutions to this problem for any initial data and external forces, and global in time unique existence of solutions for sufficiently small initial data. Thanks to global in time L p -L q maximal regularity of the linearized problem, we can prove a global in time existence and uniqueness theorem by the contraction mapping principle.1. Introduction. We consider a time dependent problem for the Navier-Stokes equations with free interface in a bounded domain Ω ⊂ R n (n ≥ 2). For example, if a bottle with stopper half filled with water is put under zero gravity, then the air forms a bubble completely surrounded by the water. Let us consider such a model.Let Ω + t ⊂ Ω be occupied by the fluid of viscosity µ + > 0 which is given only at the initial time t = 0, while for t > 0 it is to be determined. Γ t denotes the boundary of Ω +

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“…Concerning the problem without surface tension, the local in time unique solvability for any initial data and the global in time unique solvability for small initial data of the drop problem were proved by Solonnikov [17] in W 2,1 p (n < p < ∞), and by Shibata and Shimizu [11], [12] in W 2,1 q,p (2 < p < ∞ and n < q < ∞). Also the local in time solvability in W 2,1 p (n < p < ∞) was proved by Mucha and Zajaczkowski [6,7] for the drop problem, and Abels [1] for the ocean problem.…”

Section: Introduction and Resultsmentioning

confidence: 97%

Report on a local in time solvability of free surface problems for the Navier–Stokes equations with surface tension

Shibata

Shimizu

2011

Applicable Analysis

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Dedicated to Professor Vsevolod Alekseevich Solonnikovon the occasion of his 75th birthday.Abstract. We consider the free boundary problem of the Navier-Stokes equation with surface tension. Our initial domain Ω is one of a bounded domain, an exterior domain, a perturbed halfspace or a perturbed layer in R n (n ≥ 2). We report a local in time unique existence theorem in the space W 2,1) with some T > 0, 2 < p < ∞ and n < q < ∞ for any initial data which satisfy compatibility condition. Our theorem can be proved by the standard fixed point argument based on the Lp-Lq maximal regularity theorem for the corresponding linearized equations. Our results cover the cases of a drop problem and an ocean problem that were studied by Solonnikov [15,16,18,19], Beale [3] and Tani [21].

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On the Lp-Lq maximal regularity of the Neumann problem for the Stokes equations in a bounded domain

Cited by 65 publications

References 10 publications

On the $\mathcal{R}$-Sectoriality and the Initial Boundary Value Problem for the Viscous Compressible Fluid Flow

On the $\mathcal{R}$-Sectoriality and the Initial Boundary Value Problem for the Viscous Compressible Fluid Flow

Maximal regularity and viscous incompressible flows with free interface

Report on a local in time solvability of free surface problems for the Navier–Stokes equations with surface tension

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