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arXiv · 1510.05733

Ill-posedness of the Navier-Stokes and magneto-hydrodynamics systems

Alexey Cheskidov·Mimi Dai
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Abstract

We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed due to the discontinuity of weak solutions in a wide range of spaces. Specifically, we construct initial data which has finite energy and is small in certain spaces, such that any Leray-Hopf type of weak solution to the MHD system starting from this initial data is discontinuous at time $t=0$ in such spaces. An analogous result is also obtained for the Navier-Stokes equation which extends the previous result of ill-posedness in $\dot B^{-1}_{\infty,\infty}$ by Cheskidov and Shvydkoy to spaces that are not necessarily critical. The region of the spaces where the norm inflation occurs almost touches $L^2$.

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Alexey Cheskidov, Mimi Dai. 2015-10-20. Ill-posedness of the Navier-Stokes and magneto-hydrodynamics systems. https://arxiv.org/abs/1510.05733

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