Conserved quantities for the Euler equations

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The Euler equations for incompressible inviscid fluids may be written as

$latex displaystyle partial_t u + (u cdot nabla) u = -nabla p&fg=000000$

$latex displaystyle nabla cdot u = 0&fg=000000$

where $latex {u: [0,T] times {bf R}^n rightarrow {bf R}^n}&fg=000000$ is the velocity field, and $latex {p: [0,T] times {bf R}^n rightarrow {bf R}}&fg=000000$ is the pressure field. To avoid technicalities we will assume that both fields are smooth, and that $latex {u}&fg=000000$ is bounded. We will take the dimension $latex {n}&fg=000000$ to be at least two, with the three-dimensional case $latex {n=3}&fg=000000$ being of course especially interesting.

The Euler equations are the inviscid limit of the Navier-Stokes equations; as discussed in my previous post, one potential route to establishing finite time blowup for the latter equations when $latex {n=3}&fg=000000$ is to be able to construct “computers” solving the Euler equations, which generate smaller replicas of themselves in a…

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