Discontinuity of weak solutions to the 3D NSE and MHD equations in critical and supercritical spaces

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 (B) over dot(infinity,infinity)(-1) 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. (C) 2019 Elsevier Inc. All rights reserved.