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Supported by: DFG: ``Kopplung von van Roosbroeck und SchrödingerPoissonSystemen mit Ladungsträgeraustausch'' (Coupling between van Roosbroeck and SchrödingerPoisson systems including exchange of carriers)
Description:
The work of 2002 continued that reported in 2000; our aim was to prove a
theorem on (local)
existence for quasilinear parabolic systems
on an L^{p} space which should enable the treatment of the
van Roosbroeck system
and similar reactiondiffusion systems
in heterogeneous media in an L^{p} context.
The motivation for this is the following:
The natural formulation of balance laws is
(1) 
Apart from that, space discretization of such systems is often performed using finite volume schemes, cf. [4]. In order to compare such a scheme with the original equations these equations have to be valid in the space of integrable functions on the simulation domain.
Mathematically, the main results are as follows:
Theorem 1
Spatially twodimensional quasilinear systems of the type
(2) 
This theorem rests heavily on classical results for abstract quasilinear parabolic equations in a Banach space [7], [10], interpolation results for function spaces which carry a trace condition on part of the boundary [2] and resolvent estimates for elliptic operators in divergence form including discontinuous coefficients and mixed boundary conditions [1].
Having the above theorem at hand, one succeeds in proving:
Theorem 2 The spatially twodimensional van Roosbroeck system admits (locally in time) a classical solution in L^{p}. Hereby a broad class of recombination terms, including nonlocal ones, are admissible.
This theorem is derived from the first one by reformulating van Roosbroeck's system in a rather cumbersome procedure: One expresses the densities in terms of the electrostatic potential and the quasiFermi levels, then eliminates the time derivative of the electrostatic potential and afterwards the electrostatic potential itself via the nonlinear Poisson equation. This way, one ends up with a quasilinear system as above for the quasiFermi levels. The authors believe that the consistent treatment of the equations in much more regular spaces as before is not accidental but caused by the physical expectation that the quasiFermi levels should be the most regular objects within a heterogeneous context.
Finally, we will give a short outlook on the 3D case, which in our opinion should also be treated in a concept which provides as much regularity as possible (strong differentiability of the solution, the divergence of the currents being L^{p} functions....). The cornerstone to carry out the same program for the 3D case would be a result that the operators including nonsmooth coefficients and mixed boundary conditionsprovide isomorphisms between spaces , H^{1,q} for a summability index q greater than 3. Unfortunately, this is known to be false in general (cf. [9]). In a forthcoming paper [8] it will be shown that it is true, however, in case of pure Dirichlet conditions and restrictions on the domain and thediscontinuouscoefficient functions. Further, it is expected that the same is true for smooth domains and the Laplace operator including mixed boundary conditions [3]. The task would be to prove that the latter assertion is really true, further to prove the analogous assertion in case of pure Neumann conditions and discontinuous coefficients and fit all these things together.
References:



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