Novosibirsk, Russia, May, 30 – June, 4, 2011

"Modern Problems of Applied Mathematics and Mechanics: Theory, Experiment and Applications", devoted to the 90th anniversary of professor Nikolai N. Yanenko

## Tkachev D.L. Блохин А.М.## The regularity of a solution and the well-posedness of an initial-boundary value problem for an elliptic system with a gradient quadratic nonlinearity## Reporter: Tkachev D.L.In the last time, for the macroscopic description of the charge transport in semiconductors besides well-known drift-diffusion equations [1] and energy transport models[2] one becomes to use new hydrodynamical models [3]. These models are derived from the infinite system of moment equations by a suitable truncation procedure (the moment equations follow from the Boltzman transport equation). For justifying the stabilization method used for funding stationary solutions of the initial-boundary value problem (as a material basis we choose a planar silicon transistor MESFET (Metal Semiconductor Field Effect Transistor)) we have to prove that the obtained “limit” (in our case, elliptic) problem is well-posed. The essential feature of our problem is that the interior equations contain squares of gradients of the unknown functions. In the case when the right-hand side of the elliptic problem satisfies the condition that it grows as the “almost” uniform norm of the solution does (so-called “natural condition” [4]) we obtained the following two results: 1. The bounded solution of the problem has an additional smoothness and belongs to an intersection of Hölder and Sobolev spaces. 2. There exists a solution of the problem and it is unique under an additional assumption. This work was partially supported by RFBR (grant N 10-01-00320-a) and was done in the framework of the programs of the Russian Education Ministry “Russian scientific and educational personnel” 2009-2013 (grant N P1180) and “Development of the scientific potential of the High school” 2009-2011 (grant N 2.1.1/4591). References
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