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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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