---+-----+-----
1 | 1 | 4
---+-----+-----
2 |{5} |{8}
---+-----+-----
3 | 9 |{17}
---+-----+-----
4 |{18} |{23}
---+-----+-----
5 | 24 | 27
---+-----+-----
6 | 28 | 34
---+-----+-----
7 |{35} |{41}
---+-----+-----
8 |{42} |{52}
---+-----+-----
This is the equation which governs the flow of fluids such as water and air. However, there is no proof for the most basic questions one can ask: do solutions exist, and are they unique? Why ask for a proof? Because a proof gives not only certitude, but also understanding. (Clay Institute).
By this paper it is derived a sufficient condition keeping energy conservation in terms of the gradient of the velocity for the weak solutions of the compressible Euler equations for both the non-vacuum and the vacuum cases.