Existence of strong solutions for nonisothermal Korteweg system

Haspot, Boris

doi:10.5802/ambp.274

Cited by 53 publications

(22 citation statements)

References 14 publications

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“…More precisely they prove that if ρ 0 − 1 ∈Ḃ 2,1 ), near a stable constant state (P ′ (1) > 0) they obtain a global solution. Let us also mention the works of Haspot (see [18,19,20,21]) where the Korteweg system is studied in general settings, or with minimal assumptions on the initial data (or Besov indices) in the case of particular viscosity or capillarity coefficients.…”

Section: Presentation Of the Modelmentioning

confidence: 99%

Lagrangian Methods for a General Inhomogeneous Incompressible Navier--Stokes--Korteweg System with Variable Capillarity and Viscosity Coefficients

Burtea¹,

Charve²

2017

SIAM J. Math. Anal.

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We study the inhomogeneous incompressible Navier-Stokes system endowed with a general capillary term. Thanks to recent methods based on Lagrangian change of variables, we obtain local well-posedness in critical Besov spaces (even if the integration index p = 2) and for variable viscosity and capillary terms. In the case of constant coefficients and for initial data that are perturbations of a constant state, we are able to prove that the lifespan goes to infinity as the capillary coefficient goes to zero, connecting our result to the global existence result obtained by Danchin and Mucha for the incompressible Navier-Stokes system with constant coefficients.

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Section: Presentation Of the Modelmentioning

confidence: 99%

Lagrangian Methods for a General Inhomogeneous Incompressible Navier--Stokes--Korteweg System with Variable Capillarity and Viscosity Coefficients

Burtea¹,

Charve²

2017

SIAM J. Math. Anal.

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“…The three-dimensional non-isothermal motion was also discussed in [16]. Recently, the non-isothermal Korteweg system was also studied in [15].…”

Section: 1mentioning

confidence: 99%

Local well‐posedness of the viscous rotating shallow water equations with a term of capillarity

Wu¹,

Zhang

2010

Z Angew Math Mech

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Making use of the Fourier localization method, we prove the local-in-time existence and uniqueness of solutions to the viscous rotating shallow water equations with a term of capillarity under both the low regularity assumption on the initial data and the assumption that the initial height is bounded away from zero. Viscous rotating shallow water equations Remark 1.1. Throughout this paper, we will denoteḂ s 2,1 by B s . For the definition of B s1,s2 , see Definition 2.3 below. Remark 1.2. Because of the presence of the Coriolis force in system (1.1), it seems that we have to adapt the functional framework based on L 2 (not general L p ) to sufficiently make use of the cancelation of the rotating term.To prove this theorem, we first perform a variable substitution and make use of the Hodge decomposition to separate the vector field into a compressible part and an incompressible part. We then obtain a coupled system due to the rotating effect of the Coriolis force. Secondly, with the help of the Fourier localization method, we obtain a priori estimates for the corresponding linear system. Unlike [9], because of the appearance of the Coriolis frequency, we must perform different estimates for the high and low frequencies, respectively. Finally, we use a classical iterate method to construct an approximate solution to the problem and then prove the local-in-time existence and uniqueness of solutions.

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“…The motion of a one-dimensional compressible nonisothermal viscous fluid with internal capillarity can be described by the Korteweg-type model (see [1,2,3,7,9]): Here τ and y represent the time variable and the spatial variable, respectively. The Korteweg tensor K and the interstitial work flux W are given by…”

Section: Introductionmentioning

confidence: 99%

“…The global existence and large-time behavior of solutions to the compressible fluid models of Korteweg type have been studied by many authors. For small initial data, we refer to [6,7] for the global existence of smooth solutions around constant states in Sobolev space, [10,11,12,13,17,18,20,19] for the large-time behavior of smooth solutions in Sobolev space, and [5,9] for the global existence and uniqueness of strong solutions in Besov space.…”

Section: Introductionmentioning

confidence: 99%

Global smooth solutions to the nonisothermal compressible fluid models of Korteweg type with large initial data

Chen

Zhao

2017

Z. Angew. Math. Phys.

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The global solutions with large initial data for the isothermal compressible fluid models of Korteweg type have been studied by many authors in recent years. However, little is known of global large solutions to the nonisothermal compressible fluid models of Korteweg type up to now. This paper is devoted to this problem, and we are concerned with the global existence of smooth and non-vacuum solutions with large initial data to the Cauchy problem of the one-dimensional nonisothermal compressible fluid models of Korteweg type. The case when the viscosity coefficient µ(ρ) = ρ α , the capillarity coefficient κ(ρ) = ρ β , and the heat-conductivity coefficientα(θ) = θ λ for some parameters α, β, λ ∈ R is considered. Under some assumptions on α, β and λ, we prove the global existence and time-asymptotic behavior of large solutions around constant states. The proofs are given by the elementary energy method combined with the technique developed by Y. Kanel' [23] and the maximum principle.

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Existence of strong solutions for nonisothermal Korteweg system

Cited by 53 publications

References 14 publications

Lagrangian Methods for a General Inhomogeneous Incompressible Navier--Stokes--Korteweg System with Variable Capillarity and Viscosity Coefficients

Lagrangian Methods for a General Inhomogeneous Incompressible Navier--Stokes--Korteweg System with Variable Capillarity and Viscosity Coefficients

Local well‐posedness of the viscous rotating shallow water equations with a term of capillarity

Global smooth solutions to the nonisothermal compressible fluid models of Korteweg type with large initial data

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