On the compressible Navier-Stokes-Korteweg equations

Tang, Tong; Gao, Hongjun

doi:10.3934/dcdsb.2016071

Cited by 10 publications

(5 citation statements)

References 28 publications

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“…On the other hand, for the barotropic NSK system with the viscosity and capillary coefficients depending on density, Bresch‐Desjardins‐Lin showed the global existence of a weak solution with large initial data in a periodic domain when μ = νρ ( ν >0) and λ =0, based on the new entropy estimate in the case that p ′ ( ρ ) ≥ − c 0 , where c 0 >0 is a small constant. Recently, Tang‐Gao obtained a result similar to for the system with friction in the case that

\frac{1}{a} ρ^{γ - 1} - b \leq p^{'} false(ρ false) \leq a ρ^{γ - 1} + b

, where a >0, b >0 and γ >1 is the adiabatic constant. On the other hand, Haspot proved the existence of global weak solution in two dimension for initial data in the energy space, close to a stable equilibrium and with specific choices on the capillary coefficients, and then, he showed the existence of global weak solution in dimension one for a specific type of capillary coefficients with large initial data in the energy space.…”

Section: Introduction and Main Resultsmentioning

confidence: 74%

Existence of smooth solutions for the compressible barotropic Navier‐Stokes‐Korteweg system without increasing pressure law

Huang

Hong

Shi

2020

Math Methods in App Sciences

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This paper is concerned with a barotropic model of capillary compressible fluids describing the dynamics of a liquid-vapor mixture with diffuse interphase, in the case of general pressure law including Van der Waals gas. In the Sobolev spaces as close as possible to the physical energy spaces, we first prove local in time existence and uniqueness of the smooth solutions to the Cauchy problem in R d (d = 2, 3), based on the estimates of a linearized system and the contraction mapping principle. Next, we show that there exists a global unique solution for the initial boundary value problem with periodic conditions in torus T d (d = 2, 3), by using a continuation argument of local solution. Notice that it is one of the main difficulties that pressure p is not increasing function of density . KEYWORDS compressible Navier-Stokes-Korteweg system, existence, uniqueness, Van der Waals gas MSC CLASSIFICATION 35Q30; 35B35; 35L65; 76D33; 74J40 and the stress tensor S is given by S = 2 D(u) + (divu)I − pI.Here p = p( , ) and D(u) = 1 2 (∇u+(∇u) T ) denote pressure and strain tensor, respectively. E = e+ u 2 2 is total energy where e = e( , ) is the inertial energy, and the viscosity coefficients , satisfy the usual physical conditions > 0, + 2 d ≥ 0. Also, and represent capillary coefficient and heat conduction, respectively. I denotes the unit matrix. In addition, f(t, x) is a given external force.

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\frac{1}{a} ρ^{γ - 1} - b \leq p^{'} false(ρ false) \leq a ρ^{γ - 1} + b

Section: Introduction and Main Resultsmentioning

confidence: 74%

Existence of smooth solutions for the compressible barotropic Navier‐Stokes‐Korteweg system without increasing pressure law

Huang

Hong

Shi

2020

Math Methods in App Sciences

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“…Kotschote [37] considered the initial boundary value problem of (1.1) in bounded domain and proved the local existence and uniqueness of strong solutions without assuming increasing pressure laws. Tang and Gao [41] proved the stability of weak solution in the periodic domain T d (d = 2, 3) under a non-monotone pressure law. Chikami and Kobayashi [11] established global existence and decay of strong solution in the L 2 critical Besov spaces.…”

Section: D(u)mentioning

confidence: 99%

Global existence and analyticity of $L^p$ solutions to the compressible fluid model of Korteweg type

Zhang¹,

Xu²

2022

Preprint

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We are concerned with a system of equations in R d (d ≥ 2) governing the evolution of isothermal, viscous and compressible fluids of Korteweg type, that can be used as a phase transition model. In the case of zero sound speed P ′ (ρ * ) = 0, it is found that the linearized system admits the purely parabolic structure, which enables us to establish the global-in-time existence and Gevrey analyticity of strong solutions in hybrid Besov spaces of L p -type. Precisely, if the full viscosity coefficient and capillary coefficient satisfy ν2 ≥ 4κ, then the acoustic waves are not available in compressible fluids. Consequently, the prior L 2 bounds on the low frequencies of density and velocity could be improved to the general L p version with 1 ≤ p ≤ d if d ≥ 2. The proof mainly relies on new nonlinear Besov (-Gevrey) estimates for product and composition of functions.

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“…For results on non-local capillary terms and convergence to various models, we refer to the works by F. Charve and B. Haspot [4][5][6]. In [18], T. Tang and H. Gao obtain the global solution for System (1.1) in T d under a non-monotone pressure law.…”

Section: Compressible Navier-stokes-korteweg Systemmentioning

confidence: 99%

Global Well-Posedness and Time-Decay Estimates of the Compressible Navier–Stokes–Korteweg System in Critical Besov Spaces

Chikami

Kobayashi

2019

J. Math. Fluid Mech.

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We consider the compressible Navier-Stokes-Korteweg system describing the dynamics of a liquid-vapor mixture with diffuse interphase. The global solutions are established under linear stability conditions in critical Besov spaces. In particular, the sound speed may be greater than or equal to zero. By fully exploiting the parabolic property of the linearized system for all frequencies, we see that there is no loss of derivative usually induced by the pressure for the standard isentropic compressible Navier-Stokes system. This enables us to apply Banach's fixed point theorem to show the existence of global solution. Furthermore, we obtain the optimal decay rates of the global solutions in the L 2 (R d ) -framework.

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On the compressible Navier-Stokes-Korteweg equations

Cited by 10 publications

References 28 publications

Existence of smooth solutions for the compressible barotropic Navier‐Stokes‐Korteweg system without increasing pressure law

Existence of smooth solutions for the compressible barotropic Navier‐Stokes‐Korteweg system without increasing pressure law

Global existence and analyticity of $L^p$ solutions to the compressible fluid model of Korteweg type

Global Well-Posedness and Time-Decay Estimates of the Compressible Navier–Stokes–Korteweg System in Critical Besov Spaces

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