Sharp uniform bounds for steady potential fluid-Poisson systems

Abstract

We consider steady potential hydrodynamic-Poisson systems with a dissipation term (viscosity) proportional to a small parametervin a two- or three-dimensional bounded domain. We show here that for any smooth solution of a boundary value problem which satisfies that the speed, denoted by |∇φv|, has an upper coarse bound, uniform in the parameterv, then a sharper, correct uniform bound is obtained: the viscous speed |∇φv| is bounded pointwise, at pointsx0in the interior of the flow domain, by cavitation speed (given by Bernoulli's Law at vacuum states) plus a term ofthat depends on. The exponent is β = 1 for the standard isentropic gas flow model and β = 1/2 for the potential hydrodynamic Poisson system. Both cases are considered to have a γ-pressure law with 1<γ<2 in two space dimensions and 1 < γ< 3/2 in three space dimensions. These systems have cavitation speeds which take not necessarily constant values. In fact, for the potential hydrodynamic-Poisson systems, cavitation speed is a function that depends on the potential flow function and also on the electric potential.In addition, we consider a two-dimensional boundary value problem which has been proved to have a smooth solution whose speed is uniformly bounded. In this case, we show that the pointwise sharper bound can be extended to the section of the boundary ∂Ω\∂3Ω, where ∂3Ω is called the outflow boundary. The exponent β varies between 1 and 1/8 depending on the location ofx0at the boundary and on the curvature of the boundary atx0. In particular, our estimates apply to classical viscous approximation to transonic flow models.