# Preprints

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Nummer 2714 Filonov, NikolayShilkin, Timofey On the local boundedness of weak solutions to elliptic equations with divergence-free drifts 16.1.2017 elliptic equationsdivergence-free driftstationary incompressible flow

Abstract:
In this paper we investigate the local boundedness of weak solutions to the equation $-\Delta u + b\cdot\nabla u=0$ describing the diffusion in a stationary incompressible flow. The corresponding theory is well-known in the case of the general (not necessary divergence-free) sufficiently smooth drift (namely, for $b\in L_n$, where $n$ is the dimension of the space). Our main interest is focused on the case of $b$ with limited regularity (namely, $b\in L_2$). In this case the structure assumption $\div b=0$ turns out to be crucial. In our paper (which is partly expository) we recall some known properties of weak solution in the case of the divergence-free drifts $b\in L_2$ and also establish some new results on the local boundedness of weak solutions.

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Verantwortliche Autorin: Anke Meier-Dörnberg

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Fachbereich Mathematik