On some free boundary problem for a compressible barotropic viscous fluid flow

Enomoto, Yuko; Below, Lorenz von; Shibata, Yoshihiro

doi:10.1007/s11565-013-0194-8

Cited by 18 publications

(47 citation statements)

References 14 publications

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“…The numbers A i appearing in (3)(4)(5)(6)(7)(8)(9) are defined by (3)(4)(5)(6)(7)(8)(9)(10) and P 1 , P 2 , P 3 are constants which will determined later by the boundary conditions. Inserting (3)(4)(5)(6)(7)(8)(9) into the boundary conditions (3-8), we get a linear equation system for the coefficients P i :…”

Section: Definition 12 a Domain ω Is Called A Uniform Cmentioning

confidence: 99%

Maximal regularity for the thermoelastic plate equations with free boundary conditions

Denk

Shibata

2016

J. Evol. Equ.

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We consider the linear thermoelastic plate equations with free boundary conditions in the Lp in time and Lq in space setting. We obtain unique solvability with optimal regularity for the inhomogeneous problem in a uniform C 4 -domain, which includes the cases of a bounded domain and of an exterior domain with C 4 -boundary. Moreover, we prove uniform a priori-estimates for the solution. The proof is based on the existence of R-bounded solution operators of the corresponding generalized resolvent problem which is shown with the help of an operator-valued Fourier multiplier theorem due to Weis.

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Section: Definition 12 a Domain ω Is Called A Uniform Cmentioning

confidence: 99%

Maximal regularity for the thermoelastic plate equations with free boundary conditions

Denk

Shibata

2016

J. Evol. Equ.

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“…In [6] we prove the existence of R-bounded solution operator pertaining to (1.4) when Ω is a uniform W 2−1/r r (N < r < ∞) domain in R N . Thus, as was mentioned above, in [6] we have estimates (1.7) and (1.8) automatically with the help of the Weis operator valued Fourier multiplier theorem [24], and moreover we prove a local in time existence theorem for (1.1). In our strategy, the result in this paper plays an essential role in proving maximal L p -L q results for (1.3) and a local in time existence theorem for nonlinear problem (1.1).…”

Section: And Y Shibata / R-boundedness Of the Solution Operatorsmentioning

confidence: 99%

“…In [8] the existence of the R-bounded solution operators was used only in the model problem like [15], but Murata [9] proved the existence of R-bounded solution operators in a uniform W 3−1/r r (N < r < ∞) domain and as a result she proved the maximal L p -L q regularity for evolution problem. The notion of R-sectoriality of the resolvent operator introduced by Clément and Prüß [1] was extended to the non-homogeneous boundary condition case as R-bounded solution operator in [6,9,13], because the non-homogeneous boundary condition should be taken into account to solve the Navier-Stokes equations in the Navier slip condition case and the free boundary problem case.…”

Section: And Y Shibata / R-boundedness Of the Solution Operatorsmentioning

confidence: 99%

On the ℛ-boundedness of the solution operators in the study of the compressible viscous fluid flow with free boundary conditions

Götz

Shibata

2014

Asymptotic Analysis

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In this paper, we consider a generalized resolvent problem for the linearization system of the Navier-Stokes equations describing some free boundary problem of a compressible barotropic viscous fluid flow without taking the surface tension into account. We prove the existence of the R-bounded solution operators, which drives not only the generation of analytic semigroup but also the maximal Lp-Lq regularity by means of Weis' operator valued Fourier multiplier theorem for the corresponding time dependent problem that enable us to prove a local in time existence theorem of the free boundary problem for a compressible barotropic viscous fluid flow in the Lp in time and Lq space setting (cf. Annali dell Universita di Ferrara 60 (2014), 55-89). The results in this paper were given in the PhD thesis [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] by the first author under supervision of the second author. Here we present a slightly different method of deriving a concrete form of solutions to the model problem. In this paper, one of the essential points is to show the invertibility of a 2 × 2 Lopatinski matrix function. The corresponding system in [Three topics in fluid dynamics: viscoelastic, generalized Newtonian, and compressible fluids, 2012, TU Darmstadt] is a 3 × 3 matrix, so that the method presented here is slightly simpler. of the solution operators following system of equations:

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“…This was possible thanks to postulate of the so-called Bresch-Desjardins condition for the viscosity coefficients, which provides an extra estimate of the density gradient and a special form of the pressure. The last restriction was recently removed by Xi and Xie [52].The global well-posedness in the framework of strong solutions for the compressible Navier-Stokes(-Fourier) system under smallness assuptions on the data is already well investigated, see among others [28] in L 2 framework, [48] in L p setting with slip boundary condition or [38,13] for a free boudary problem. However, for the system coupled with reaction-diffusion equations admitting cross-diffusion the issue of global well posedness of initial-boundary value problems has remained open.The purpose of this work is to prove the global in time existence of strong solutions to the system (1.1).…”

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confidence: 99%

On Strong Dynamics of Compressible Two-Component Mixture Flow

Piasecki

Shibata²,

Zatorska

2019

SIAM J. Math. Anal.

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We investigate a system describing the flow of a compressible two-component mixture. The system is composed of the compressible Navier-Stokes equations coupled with non-symmetric reactiondiffusion equations describing the evolution of fractional masses. We show the local existence and, under certain smallness assumptions, also the global existence of unique strong solutions in Lp − Lq framework. Our approach is based on so called entropic variables which enable to rewrite the system in a symmetric form. Then, applying Lagrangian coordinates, we show the local existence of solutions applying the Lp-Lq maximal regularity estimate. Next, applying exponential decay estimate we show that the solution exists globally in time provided the initial data is sufficiently close to some constants. The nonlinear estimates impose restrictions 2 < p < ∞, 3 < q < ∞. However, for the purpose of generality we show the linear estimates for wider range of p and q. MSC Classification: 76N10, 35Q30Under the assumption (1.6), global in time strong (unique) solutions around the constant equilibrium for the Cauchy problem was proven by Giovangigli in [19]. He introduced the entropic and normal variables to symmetrize the system (1.1) and applied the Kawashima and Shizuta theory [24,25] for symmetric hyperbolic-parabolic systems of conservation laws. For the local in time existence result to the species mass balances equations in the isobaric, isothermal case we refer to [3], see also [21]. Later on, Jüngel and Stelzer generalized this result and combined it with the entropy dissipation method to prove the global in time existence of weak solutions [23], still in the case of constant pressure and temperature. The detailed description of the method and its applicability for a range of models we refer to [22]. For the qualitative and quantitative analysis of the ternary gaseous system together with numerical simulations we refer to [7]. One should note that constant pressure assumption in (1.5) not only significantly simplifies the cross-diffusion equations but basically decouples the fluid and the reactiondiffusion parts of the system (1.1). Stationary problems for compressible mixtures were considered in [49] under the assumption of Fick law and later in [20,35,36] with cross diffusion, however for different molar masses. Existence of weak solutions for the mixture of non-newtonian fluids has been shown in [8]. Let us also mention results on multi-phase systems [16,26] and incompressible mixtures [27,9,5]. We would also like to mention the theoretical results for the systems describing the compressible reacting electrolytes [11], where the authors prove the existence of global in time weak solutions to the Nernst-Planck-Poisson model originating from the modelling approach developed by Bothe and Dreyer in the previous paper [4]. The classical mixture models in the sense of [19] were studied in the series of papers [50,51,30,31,32], where the global in time existence of weak solutions was proved without any simplification of (1.7). This was...

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On some free boundary problem for a compressible barotropic viscous fluid flow

Cited by 18 publications

References 14 publications

Maximal regularity for the thermoelastic plate equations with free boundary conditions

Maximal regularity for the thermoelastic plate equations with free boundary conditions

On the ℛ-boundedness of the solution operators in the study of the compressible viscous fluid flow with free boundary conditions

On Strong Dynamics of Compressible Two-Component Mixture Flow

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