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

TL;DR

This paper proves that the 3D incompressible MHD and Navier-Stokes systems are ill-posed in various function spaces, showing solutions can be discontinuous at initial time despite finite energy initial data.

Contribution

It extends previous ill-posedness results for Navier-Stokes to a broader class of spaces and demonstrates similar phenomena for the MHD system, highlighting the delicate nature of well-posedness.

Findings

Ill-posedness of MHD in a wide range of spaces.

Extension of Navier-Stokes ill-posedness beyond critical spaces.

Existence of initial data with finite energy leading to discontinuous solutions.

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 $L^2$.