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arXiv · 1711.04486

Local energy weak solutions for the Navier-Stokes equations in the half-space

Yasunori Maekawa·Hideyuki Miura·Christophe Prange
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Abstract

The purpose of this paper is to prove the existence of global in time local energy weak solutions to the Navier-Stokes equations in the half-space $\mathbb R^3_+$. Such solutions are sometimes called Lemari\'e-Rieusset solutions in the whole space $\mathbb R^3$. The main tool in our work is an explicit representation formula for the pressure, which is decomposed into a Helmholtz-Leray part and a harmonic part due to the boundary. We also explain how our result enables to reprove the blow-up of the scale-critical $L^3(\mathbb R^3_+)$ norm obtained by Barker and Seregin for solutions developing a singularity in finite time.

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Yasunori Maekawa, Hideyuki Miura, Christophe Prange. 2017-11-13. Local energy weak solutions for the Navier-Stokes equations in the half-space. https://arxiv.org/abs/1711.04486

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