A remark on the global existence of weak solutions to the compressible quantum Navier–Stokes equations

Tang, Tong; Zhang, Zujin

doi:10.1016/j.nonrwa.2018.07.009

Cited by 8 publications

(4 citation statements)

References 17 publications

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“…Also, Haspot proved the existence of local and global strong solutions with initial density

\ln ρ_{0}

belonging to the Besov space

B_{2 \infty}^{\frac{d}{2}}

, which implies in particular that some classes of discontinuous initial density generate strong solutions, and as a corollary, he gets the existence of global strong solution for any large initial data (under an irrotational condition on the initial velocity), provided that the Mach number is sufficiently large. Last, we would like to emphasise that there are some recent progresses on the global existence of finite energy weak solutions for the quantum Navier‐Stokes system, for example, we refer to and Hsieh and Wang, Huang et al, and Korteweg …”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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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“…Also, Haspot proved the existence of local and global strong solutions with initial density

\ln ρ_{0}

belonging to the Besov space

B_{2 \infty}^{\frac{d}{2}}

Section: Introduction and Main Resultsmentioning

confidence: 99%

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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“…A similar analysis to Refs. 5 and 29 is done by Tang–Zhang 43 and Dong, 10 respectively, for the case that the viscosity constant is equal to the Plank constant. One can refer to Refs.…”

Section: Introductionmentioning

confidence: 99%

Global existence and the time decay estimates of solutions to the compressible quantum Navier–Stokes–Maxwell system in R3$\mathbb {R}^3$

Tong,

Luo

2023

Stud Appl Math

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We consider the Cauchy problem of the compressible quantum Navier–Stokes–Maxwell equations with the linear damping in the isentropic case under the small perturbation of the constant equilibrium state in three dimensions. Based on the refined energy method, we establish the classical solution globally in time in Sobolev space. By the combination of the energy estimates with the interpolation between the positive Sobolev norms and the negative Sobolev norms with , we also obtain the algebraic decay rates of the classical solution. What is more, the –L2 types of the time decay rates of the solution are obtained without small assumption on the initial data in L1 norm.

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“…Recently, Antonelli and Spirito [3] further improved the result of [32] to the cases of ε < ν, γ > 1 if d = 2, and ε < ν < 3 √ 2 4 ε, 1 < γ < 3 if d = 3, where the authors use a new regular approximating system. We also refer to [4,42,46] for some improvements of [3]. Lacroix-Violet and Vasseur [37] construct the global finite energy weak solutions which are uniform with respect to the Plank constant ε for the compressible quantum Navier-Stokes equations in the torus T d with d = 2, 3.…”

Section: Introductionmentioning

confidence: 99%

“…In [20,37], the authors studied the semiclassical limit of global weak solutions of the quantum Navier-Stokes equations to the global weak solutions of the compressible Navier-Stokes equations in the torus T d with d = 2, 3. Note that the initial data in the above references [3][4][5]17,20,27,32,36,37,42,46,47,49] concerning the weak solution of the compressible quantum Navier-Stokes equations can be arbitrarily large.…”

Section: Introductionmentioning

confidence: 99%

Global Strong/Classical Solutions to the One-dimensional Compressible Quantum Navier-Stokes Equations with Large Initial Data

Chen¹,

Zhao²

2022

Preprint

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We are concerned with the global existence and singular limit of strong/classical solutions to the Cauchy problem of the one-dimensional barotropic compressible quantum Navier-Stokes equations, which consists of the compressible Navier-Stokes equations with a linearly densitydependent viscosity and a nonlinear third-order differential operator known as the quantum Bohm potential. The pressure p(ρ) = ρ γ is considered with γ ≥ 1 being a constant. We focus on the case when the viscosity constant ν and the Planck constant ε are not equal. Under some suitable assumptions on ν, ε, γ, and the initial data, we proved the global existence and large-time behavior of strong and classical solutions away from vacuum to the compressible quantum Navier-Stokes equations with large initial data. This extends the previous results on the construction of global strong large-amplitude solutions of the compressible quantum Navier-Stokes equations to the case ν = ε. Moreover, the vanishing dispersion limit for the classical solutions of the quantum Navier-Stokes equations is also established with certain convergence rates. Some new mathematical techniques and useful estimates are developed to deduce the uniform-in-time positive lower and upper bounds on the specific volume.

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A remark on the global existence of weak solutions to the compressible quantum Navier–Stokes equations

Cited by 8 publications

References 17 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 the time decay estimates of solutions to the compressible quantum Navier–Stokes–Maxwell system in R3$\mathbb {R}^3$

Global Strong/Classical Solutions to the One-dimensional Compressible Quantum Navier-Stokes Equations with Large Initial Data

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