Viscous quantum hydrodynamics and parameter-elliptic systems

Chen, Li; Dreher, Michael

doi:10.1002/mma.1377

Cited by 9 publications

(6 citation statements)

References 22 publications

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“…In fact, many authors impose periodic boundary conditions [6,9,11,19,26,38], insulating boundary conditions [13], or they consider the whole-space problem [39]. Boundary conditions satisfying the Shapiro-Lopatinskii criterion have been examined in [12]. Furthermore, in [32,35] Dirichlet-type conditions have been employed in the analyzed, but only for the (simpler) one-dimensional equations.…”

Section: Proof First We Multiply (32) By Hmentioning

confidence: 99%

Diffusive semiconductor moment equations using Fermi–Dirac statistics

Jüngel

Krause

Pietra

2010

Z. Angew. Math. Phys.

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Abstract. The global-in-time existence of weak solutions to the barotropic compressible quantum Navier-Stokes equations in a three-dimensional torus for large data is proved. The model consists of the mass conservation equation and a momentum balance equation, including a nonlinear thirdorder differential operator, with the quantum Bohm potential, and a density-dependent viscosity. The system has been derived by Brull and Méhats [10] from a Wigner equation using a moment method and a Chapman-Enskog expansion around the quantum equilibrium. The main idea of the existence analysis is to reformulate the quantum Navier-Stokes equations by means of a so-called effective velocity involving a density gradient, leading to a viscous quantum Euler system. The advantage of the new formulation is that there exists a new energy estimate which implies bounds on the second derivative of the particle density. The global existence of weak solutions to the viscous quantum Euler model is shown by using the Faedo-Galerkin method and weak compactness techniques. As a consequence, we deduce the existence of solutions to the quantum Navier-Stokes system if the viscosity constant is smaller than the scaled Planck constant.Key words. Compressible Navier-Stokes equations, quantum Bohm potential, density-dependent viscosity, global existence of solutions, viscous quantum hydrodynamic equations, third-order derivative, energy estimates. Gardner [20] from the Wigner equation by a moment method. More recently, dissipative quantum fluid models have been proposed. For instance, the moment method applied to the Wigner-Fokker-Planck equation leads to viscous quantum Euler models [21], and a Chapman-Enskog expansion in the Wigner equation leads under certain assumptions to quantum Navier-Stokes equations [10]. In this paper, we will reveal a connection between these two models by introducing an effective velocity variable, first used in capillary Korteweg-type models [5], and we will prove the global existence of weak solutions to the multidimensional initial-value problems for any finite-energy initial data. AMSIn the following, we describe the two dissipative quantum systems studied in this paper. The barotropic quantum Navier-Stokes equations for the particle density n and the particle velocity u read as

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Section: Proof First We Multiply (32) By Hmentioning

confidence: 99%

Diffusive semiconductor moment equations using Fermi–Dirac statistics

Jüngel

Krause

Pietra

2010

Z. Angew. Math. Phys.

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“…REMARK 4.2. Examples of steady states with exponential stability of the linearised problem (with explicit description of the decay rate) have been given in [6,7,14], see also [15].…”

Section: Applicationsmentioning

confidence: 99%

Resolvent estimates for Douglis–Nirenberg systems

Dreher

2009

J. Evol. Equ.

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“…The existence of classical solutions to the one-dimensional stationary model with ε = 0 and with physical boundary conditions was shown in [23]. The transient equations are considered in [5,6,9], and the local-in-time existence and exponential stability of solutions were proved. Global-in-time solutions in one space dimension are obtained if the initial energy is assumed to be sufficiently small [5].…”

Section: Introductionmentioning

confidence: 99%

Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations

Gamba

Jüngel

Vasseur

2009

Journal of Differential Equations

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The existence of global-in-time weak solutions to the one-dimensional viscous quantum hydrodynamic equations is proved. The model consists of the conservation laws for the particle density and particle current density, including quantum corrections from the Bohm potential and viscous stabilizations arising from quantum Fokker-Planck interaction terms in the Wigner equation. The model equations are coupled self-consistently to the Poisson equation for the electric potential and are supplemented with periodic boundary and initial conditions. When a diffusion term linearly proportional to the velocity is introduced in the momentum equation, the positivity of the particle density is proved. This term, which introduces a strong regularizing effect, may be viewed as a classical conservative friction term due to particle interactions with the background temperature. Without this regularizing viscous term, only the nonnegativity of the density can be shown. The existence proof relies on the Faedo-Galerkin method together with a priori estimates from the energy functional.

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Viscous quantum hydrodynamics and parameter-elliptic systems

Cited by 9 publications

References 22 publications

Diffusive semiconductor moment equations using Fermi–Dirac statistics

Diffusive semiconductor moment equations using Fermi–Dirac statistics

Resolvent estimates for Douglis–Nirenberg systems

Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations

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