Publications

Home Schools Mathematics Publications

Title: On illposedness of the Hall and electron magnetohydrodynamic equations without resistivity on the whole space

Abstract: It has been shown in Jeong and Oh (2019 On the Cauchy problem for the Hall and electron magnetohy-drodynamic equations without resistivity I: illposedness near degenerate stationary solutions Ann. PDE 8 15) that the incompressible and irresistive Hall- and electron-magneto-hydrodynamic (MHD) equations are illposed on flat domains M=R-k x T3-k for 0 <= k <= 2. The data and solutions therein were assumed to be independent of one coordinate, which not only significantly simplifies the systems but also allows for a large class of steady states. In this work, we remove the assumption of independence and conclude strong illposedness for compactly supported data in R-3. This is achieved by constructing degenerating wave packets for linearized systems around time-dependent axisymmetric magnetic fields. A few main additional ingredients are: a more systematic application of the generalized energy estimate, use of the Bogovski- operator, and a priori estimates for axisymmetric solutions to the Hall- and electron-MHD systems.

List