Uniform Integrability and the Problem of Pressure Formation for Euler-Riesz Systems

We consider the system of compressible Euler equations augmented with a nonlocal term associated to a Riesz potential. This system emerges as a critical point of an action functional. Also it is equipped with a stress tensor that may be expressed in two distinct representations and gives rise to an associated bilinear fractional integral operator. (i)  We establish a uniform estimate for a bilinear fractional integral operator via restricted weak-type endpoint estimates and  Marcinkiewicz interpolation. The estimate leads to integrability analysis of the associated stress-tensor. (ii)  Then we focus on the problem without a pressure term and examine the question of whether the nonlocal interaction can give rise to a pressure estimated in a Lebesgue space.  We use both the energy and a quantity proposed by the compensated integrability method. This gives rises to interesting analytical questions on lower bounds of functionals that will be reviewed.