Existence of global weak solutions for Navier-Stokes-Poisson equations with quantum effect and convergence to incompressible Navier-Stokes equations. (English) Zbl 1375.35413
Summary: We consider a three dimensional quantum Navier-Stokes-Poisson equations. Existence of global weak solutions is obtained, and convergence toward the classical solution of the incompressible Navier-Stokes equation is rigorously proven for well prepared initial data. Furthermore, the associated convergence rates are also obtained.
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 35D30 | Weak solutions to PDEs |
| 76D03 | Existence, uniqueness, and regularity theory for incompressible viscous fluids |
| 82C10 | Quantum dynamics and nonequilibrium statistical mechanics (general) |
| 35B25 | Singular perturbations in context of PDEs |
Keywords:
quantum Navier-Stokes-Poisson equations; weak solutions; quasi-neutral limit; vanishing damping; incompressible Navier-Stokes equations subclassReferences:
| [1] | LoffredoM, MoratoL. On the creation of quantum vortex lines in rotating He II. Il nouvo cimento1993; 108B:205-215. |
| [2] | FerryD, ZhouJR. Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling. Physical Review B1993; 48:7944-7950. |
| [3] | GCimatti, and GProdi. Existence results for a nonlinear elliptic system modelling a temperature dependent electrical resistor. Annali Di Matematica Pura Ed Applicata1988; 152:227-237. · Zbl 0675.35039 |
| [4] | HXie, and WAllegretto. \( \mathbb{C}^\alpha( \overline{\Omega})\) solutions of a class of nonlinear degenerate elliptic systems arising in the thermistor problem. SIAM Journal on Mathematical Analysis1990; 22:1491-1499. · Zbl 0744.35016 |
| [5] | JGrant. Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations. Journal of Physics A: Mathematical, Nuclear and General1973; 6:L151-L153. · Zbl 0268.76005 |
| [6] | PAntonelli, and PMarcati. On the finite energy weak solutions to a system in Quantum Fluid Dynamics. Communications in Mathematical PHysics2009; 287:657-686. · Zbl 1177.82127 |
| [7] | PCaussignac, JDescloux, and AYamnahakki. Simulation of some quantum models for semiconductors. Mathematical Models and Methods in Applied Sciences2002; 12:1049-1074. · Zbl 1162.35444 |
| [8] | LChen, and MDreher. The viscous model of quantum hydrodynamics in several dimensions. Mathematical Models and Methods in Applied Sciences2007; 17:1065-1093. · Zbl 1144.35012 |
| [9] | PDegond, FMéhats, and CRinghofer. Quantum energy‐transport and drift‐diffusion models. Journal of Statistical Physics2005; 118:625-667. · Zbl 1126.82314 |
| [10] | IGamba, AJüngel, and AVasseur. Global existence of solutions to one‐dimensional viscous quantum hydrodynamic equations. Journal of Differential Equations2009; 247:3117-3135. · Zbl 1181.35209 |
| [11] | CGardner. The quantum hydrodynamic model for semiconductor devices. SIAM Journal on Applied Mathematics1994; 54:409-427. · Zbl 0815.35111 |
| [12] | MPGualdani, and AJüngel. Analysis of the viscous quantum hydrodynamic equations for semiconductors. European Journal of Applied Mathematics2004; 15:577-595. · Zbl 1076.82049 |
| [13] | HLLi, and PMarcati. Existence and asymptotic behavior of multi‐dimensional quantum hydrodynamic model for semiconductors. Communications in Mathematical Physics2004; 245:215-247. · Zbl 1075.82019 |
| [14] | BZhang, and JJerome. On a steady‐state quantum hydrodynamic model for semiconductors. Nonlinear Analysis1996; 26:845-856. · Zbl 0882.76105 |
| [15] | SBrull, and FMéhats. Derivation of viscous correction terms for the isothermal quantum Euler model. Zeitschrift für Angewandte Mathematik und Mechanik2010; 90:219-230. · Zbl 1355.82018 |
| [16] | DBresch, BDesjardins, and CKLin. On some compressible fluid models: Korteweg, lubrication and shallow water systems. Communications Partial Differential Equations2003; 28:1009-1037. |
| [17] | JDong. A note on barotropic compressible quantum Navier‐Stokes equations. Nonlinear Analysis2010; 73:854-856. · Zbl 1193.35128 |
| [18] | FJiang. A remark on weak solutions to the baratropic compressible quantum Navier‐Stokes equations. Nonlinear Analysis: Real World Applications2011; 12:1733-1735. · Zbl 1216.35089 |
| [19] | AJüngel. Global weak solutions to compressible Navier‐Stokes equations for quantum fluids. SIAM Journal on Mathematical Analysis2010; 42:1025-1045. · Zbl 1228.35083 |
| [20] | DBresch, and BDesjardins. Existence of global weak solutions for a 2D viscous shallow water equations and convergence to the quasigeostrophique model. Communications in Mathematical PHysics2003; 238:211-223. · Zbl 1037.76012 |
| [21] | DBresch, BDesjardins, and BDucomet. Quasi‐neutral limit for a viscous capillary model of plasma. Ann. I. H. Poincaré‐AN2005; 22:1-9. · Zbl 1062.35061 |
| [22] | HLLi, and CKLin. Zero Debye length asymptotic of the quantum hydrodynamic model for semiconductors. Communications in Mathematical Physics2005; 256:195-212. · Zbl 1071.82058 |
| [23] | FCLi. Quasineutral limit of the viscous quantum hydrodynamic model for semiconductors. Journal of Mathematical Analysis and Applications2009; 352:620-628. · Zbl 1160.82359 |
| [24] | AJüngel, CKLin, and KCWu. An asymptotic limit of a Navier‐Stokes system with capilllary effects. Communications in Mathematical Physics2014; 329:725-744. · Zbl 1297.35169 |
| [25] | FJMcGrath. Nonstationary plane flow of viscous and ideal fluids. Archive for Rational Mechanics and Analysis1968; 27:229-348. · Zbl 0187.49508 |
| [26] | TKato. Nonstationary flow of viscous and ideal fluids in \(\mathbb{R}^3\). Journal of Functional Analysis1972; 9:296-305. · Zbl 0229.76018 |
| [27] | QCJu, FCLi, and SWang. Convergence of the Navier‐Stokes‐Poisson system to the incompressible Navier‐Stokes equations. Journal of Mathematical Physics2008; 49:073515, 8. · Zbl 1152.76356 |
| [28] | AJüngel, and DMatthes. The Derrida‐Lebowitz‐Speer‐Spohn equation: existence, nonuniqueness, and decay rates of the solutions. SIAM Journal on Mathematical Analysis2008; 39:1996-2015. · Zbl 1160.35428 |
| [29] | LionsPL. Mathematical Topics in Fluid Mechanics, Compressible Models Vol. 2, Clarendon, Oxford/Oxford Science, Oxford, 1998. · Zbl 0908.76004 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
