Apprendre MathType

Well, I use the Mathpix tools to grab these formulas from the website Wikiwand about the Euler equations.

Thanks to these vector identities, the incompressible Euler equations with constant and uniform density and without external field can be put in the so-called conservation (or Eulerian) differential form, with vector notation:

$\left\{\begin{aligned}\frac{\partial \mathbf{u}}{\partial t}+\nabla \cdot(\mathbf{u} \otimes \mathbf{u}+w \mathbf{I}) & =\mathbf{0} \\\frac{\partial 0}{\partial t}+\nabla \cdot \mathbf{u} & =0\end{aligned}\right.$

or with Einstein notation:

$\left\{\begin{aligned}\partial_t u_j+\partial_i\left(u_i u_j+w \delta_{i j}\right) & =0 \\\partial_t 0+\partial_j u_j & =0\end{aligned}\right.$

Then incompressible Euler equations with uniform density have conservation variables:

$\mathbf{y}=\binom{\mathbf{u}}{0} ; \quad \mathbf{F}=\binom{\mathbf{u} \otimes \mathbf{u}+w \mathbf{I}}{\mathbf{u}}$

Note that in the second component $u$ is by itself a vector, with length $N$, so $y$ has length $N+1$ and $F$ has size $N(N+1)$. In 3D for example $y$ has length 4 , $I$ has size $3 \times 3$ and $F$ has size $4 \times 3$, so the explicit forms are:

$\mathbf{y}=\left(\begin{array}{c}u_1 \\u_2 \\u_3 \\0\end{array}\right) ; \quad \mathbf{F}=\left(\begin{array}{ccc}u_1^2+w & u_1 u_2 & u_1 u_3 \\u_2 u_1 & u_2^2+w & u_2 u_3 \\u_3 u_1 & u_3 u_2 & u_3^2+w \\u_1 & u_2 & u_3\end{array}\right)$