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Local and global existence for the coupled Navier-Stokes-Poisson problem. (English) Zbl 1039.35075
The author studies a model of semiconductor device gathering a continuity equation and a balance momentum with viscosity terms, but omitting the energy equation. She thus gets a problem coupling a Navier-Stokes equation for the particle velocity field \(u\), Poisson’s equation for the electron field \(\phi \) and a continuity equation for the electron density \(n\). The pressure \(p\) is taken as \(p=an^{\gamma }\), with \(\gamma >1\). The problem is posed in \(\Omega \times \left( 0,T\right) \) where \(\Omega \) is a smooth bounded and connected domain in \({\mathbb R}^{N}\). Homogeneous Dirichlet boundary conditions are imposed for \(u\) on the boundary \(\partial \Omega \times \left( 0,T\right) \). The quantities \(u\), \(nu\) and \(\phi \) start from initial values given in adequate spaces. In order to solve the Navier-Stokes equation, the author uses the results presented in P. L. Lions [Mathematical topics in fluid mechanics. Vol. 1: Incompressible models. Oxford: Clarendon Press (1996; Zbl 0866.76002); Vol. 2: Compressible models. Oxford: Clarendon Press (1998; Zbl 0908.76004)]. She then uses the compactness properties of the Laplace operator in order to use Schauder’s fixed point argument for \(\phi \). This requires more restrictive conditions on the exponent \(\gamma \), which depend on the dimension \(N\). The author thus proves a short-time existence result for the solution of this problem. She ends the paper with further regularity results concerning this solution.

MSC:
35Q30 Navier-Stokes equations
82D37 Statistical mechanical studies of semiconductors
35D05 Existence of generalized solutions of PDE (MSC2000)
76X05 Ionized gas flow in electromagnetic fields; plasmic flow
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