Perfect fluid flows on $\R^d$ with growth/decay conditions at infinity

We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on $\R^d$ with initial data in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as $|x|^β$ with $β<1/2$. In particular, we show that the solution of the Euler equation generically develops an asymptotic expansion at infinity with non-vanishing asymptotic terms that depend analytically on time and the initial data. We identify the evolution space for initial data in the Schwartz class with a certain space of symbols.

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