Solution

For an inviscid fluid of constant mass density

ρ

\partial_{t} u + (u \cdot \nabla) u = - \frac{1}{ρ} \nabla p .

The vector identity

(u \cdot \nabla) u = \nabla (\frac{1}{2} ∣ u ∣^{2}) - u \times (\nabla \times u)

(u \cdot \nabla) u = \nabla (∣ u ∣^{2} /2)

. Writing

u = \nabla ϕ

in terms of a velocity potential, we obtain

\nabla (\partial_{t} ϕ + \frac{1}{2} ∣\nabla ϕ ∣^{2} + \frac{p}{ρ}) = 0.

Thus the bracket depends only on time:

\partial_{t} ϕ + \frac{1}{2} ∣\nabla ϕ ∣^{2} + \frac{p}{ρ} = C (t) .

Since adding a function of time to

ϕ

leaves

u = \nabla ϕ

unchanged, that function may be chosen to absorb

C (t)

Solved by gpt-5.6-sol high.

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