We will start with the Critical Threshold Phenomena (CTP). CTP is observed when there is a manifold in the phase space of initial data, dividing it into two sections: a supercritical region and a subcritical region. If the initial data lies completely in the subcritical region, then there is global-in-time smooth solution. We will look into the Euler–Poisson and Euler–Poisson alignment systems from the point of view of CTP.

For CTP to occur, one must have a “good” term to avoid the effect of the “bad” convective term, in the absence of which, we see discontinuities (shocks) forming, which is the case with conservation laws. In 1D spatial variable, it is widely believed that the appropriate space to look for (weak) solutions is the space of BV functions. Since few decades, Front Tracking schemes have grown to be an essential tool in the analysis of such systems. However, practically all existing FT schemes are catered to small data. We will look into a modified FT scheme that handles general data, specifically strong shocks, and prove the convergence thereof, with the a priori assumption of BV bounds.

The talk constitutes joint works with Robin Young (University of Massachusetts Amherst), Hailiang Liu (Iowa State University) and Changhui Tan (University of South Carolina).