Formal constant-density incompressible Euler system

Let $I\subseteq \mathbb{R}$ be a nonempty open time interval, let $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ be a nonempty open set of Cartesian coordinates, and fix a constant density ${\rho }_0>0$. Let $u:I\times \mathrm{\Omega }\to {\mathbb{R}}^3$ have continuous first time derivatives, first spatial derivatives, and second spatial derivatives, and let $p:I\times \mathrm{\Omega }\to \mathbb{R}$ have continuous first spatial derivatives. The formal constant-density incompressible Euler system is the pointwise system

${\mathrm{\partial }}_{t}u+(u\cdot \mathrm{\nabla })u+{\rho }_0^{-1}\mathrm{\nabla }p=0,\mathrm{div}u=0.$

Use the Cartesian derivative conventions $\mathrm{\nabla }f=({\mathrm{\partial }}_{i}f{)}_{i=1}^3$, $\mathrm{div}u={\sum }_i{\mathrm{\partial }}_i{u}_i$, and $(u\cdot \mathrm{\nabla })u={\sum }_j{u}_j{\mathrm{\partial }}_{j}u$. The velocity has continuous first time, first spatial, and second spatial derivatives, and pressure has continuous first spatial derivatives. For this inviscid branch define the formal pressure stress by ${\sigma }_E=-p{I}_3$. For each fixed $t\in I$, set $p^{[t]}(x)=p(t,x)$ and ${\sigma }_E^{[t]}(x)=-p^{[t]}(x)I_3$. On a compact oriented regular surface patch $S\subset \mathrm{\Omega }$ with unit normal $n_S$, define its formal stress resultant at time $t$ to be ${\int }_S{\sigma }_E^{[t]}{n}_{S}dS=-{\int }_S{p}^{[t]}{n}_{S}dS$. For time-independent fields the resultant is independent of $t$, and the section superscripts may be omitted. A steady solution has ${\mathrm{\partial }}_{t}u=0$. These definitions impose no boundary condition, assert no solution existence, and do not claim that the system describes an observed fluid.

Formal fluid derivative conventions and regularity: Formal parameterized fluid systems and observables.

Iain W. Stewart, MIT 8.09/8.309, Chapter 6, §6.4

David Tong, Fluid Mechanics lecture notes, Chapter 1, §§2.3.2–2.3.3