# Embedding the SQG equation in a modified Euler equation

The Euler equations for three-dimensional incompressible inviscid fluid flow are

\$latex displaystyle partial_t u + (u cdot nabla) u = – nabla p (1)&fg=000000\$

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

where \$latex {u: {bf R} times {bf R}^3 rightarrow {bf R}^3}&fg=000000\$ is the velocity field, and \$latex {p: {bf R} times {bf R}^3 rightarrow {bf R}}&fg=000000\$ is the pressure field. For the purposes of this post, we will ignore all issues of decay or regularity of the fields in question, assuming that they are as smooth and rapidly decreasing as needed to justify all the formal calculations here; in particular, we will apply inverse operators such as \$latex {(-Delta)^{-1}}&fg=000000\$ or \$latex {|nabla|^{-1} := (-Delta)^{-1/2}}&fg=000000\$ formally, assuming that these inverses are well defined on the functions they are applied to.

Meanwhile, the surface quasi-geostrophic (SQG) equation is given by

\$latex displaystyle partial_t theta + (u cdot nabla) theta = 0…

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