Euler subsolution with prescribed energy

Let $e\geq0$ be locally integrable on a space-time cylinder in dimension $n\geq2$. An Euler subsolution with energy bound $e$ is a triple $(v,S,q)$, with $v\in L^2_{\rm loc}$, $S\in L^1_{\rm loc}$ symmetric and trace free, and $q$ a distribution, such that

The equations use distributional derivatives; the inequality says that $(2e/n)I-v\otimes v+S$ is positive semidefinite almost everywhere. A smooth strict subsolution has smooth $v,S,q$, continuous $e$, and a positive definite gap everywhere.

The products are outer products, the coefficients have the indicated local integrability, and the pressure is a distribution. This formulation relaxes Euler while keeping its divergence constraints.