Editing Euler equations (fluid dynamics) (section)

===Properties===
Although Euler first presented these equations in 1755, many fundamental questions or concepts  about them remain unanswered.

In three space dimensions, in certain simplified scenarios, the Euler equations produce singularities.<ref>{{Cite journal |last=Elgindi |first=Tarek M. |date=2021-11-01 |title=Finite-time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$ |url=https://projecteuclid.org/journals/annals-of-mathematics/volume-194/issue-3/Finite-time-singularity-formation-for-C1alpha-solutions-to-the-incompressible/10.4007/annals.2021.194.3.2.full |journal=Annals of Mathematics |volume=194 | arxiv = 1904.04795 |issue=3 |doi=10.4007/annals.2021.194.3.2 |issn=0003-486X}}</ref>

Smooth solutions of the free (in the sense of without source term: g=0) equations satisfy the conservation of specific kinetic energy:
<math display="block">{\partial \over\partial t} \left(\frac{1}{2} u^2 \right) + \nabla \cdot \left(u^2 \mathbf u + w \mathbf u\right) = 0.</math>

In the one-dimensional case without the source term (both pressure gradient and external force), the momentum equation becomes the inviscid [[Burgers' equation]]:
<math display="block">{\partial u \over\partial t}+ u {\partial u \over\partial x} = 0.</math>

This model equation gives many insights into Euler equations.