Effective velocity in compressible Navier-Stokes equations with third-order derivatives. (English) Zbl 1211.35217
Summary: A formulation of certain barotropic compressible Navier-Stokes equations with third-order derivatives as a viscous Euler system is proposed by using an effective velocity variable. The equations model, for instance, viscous Korteweg or quantum Navier-Stokes flows. The formulation in the new variable allows for the derivation of an entropy identity, which is known as the BD (Bresch-Desjardins) entropy equation. As a consequence of this estimate, a new global-in-time existence result for the one-dimensional quantum Navier-Stokes equations with strictly positive particle densities is proved.
MSC:
| 35Q30 | Navier-Stokes equations |
| 35Q53 | KdV equations (Korteweg-de Vries equations) |
| 76Y05 | Quantum hydrodynamics and relativistic hydrodynamics |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
Keywords:
Korteweg-type models; quantum hydrodynamic equations; viscous Euler system; energy estimates; BD entropyReferences:
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