On the Cauchy problem for the Hall magnetohydrodynamics
Abstract: In this talk, I will describe a recent series of work with I.-J. Jeong on the Hall MHD equation without resistivity. This PDE, ﬁrst investigated by the applied mathematician M. J. Lighthill, is a one-ﬂuid description of magnetized plasmas with a quadratic second-order correction term (Hall current term), which takes into account the motion of electrons relative to positive ions. Curiously, we demonstrate the ill(!)posedness of the Cauchy problem near the trivial solution, despite the apparent linear stability and conservation of energy. On the other hand, we identify several regimes in which the Cauchy problem is well-posed, which not only includes the original setting that M. J. Lighthill investigated (namely, for initial data close to a uniform magnetic ﬁeld) but also possibly large perturbations thereof. Central to our proofs is the viewpoint that the Hall current term imparts the magnetic ﬁeld equation with a quasilinear dispersive character. With such a viewpoint, the key ill- and well-posedness mechanisms can be understood in terms of the properties of the bicharacteristic ﬂow associated with the appropriate principal symbol.