Perfect fluid flows on Rd 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 velocity in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as vertical bar x vertical bar(beta) with beta < 1/2. Moreover, for initial velocity with sufficient spatial decay, 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. For initial data in the Schwartz space, we identify the evolution space of the fluid velocity with a certain space of symbols.

Automated summary

This study focuses on perfect fluid flows in weighted Sobolev spaces, showing that under sufficient spatial decay of the initial velocity, the fluid velocity can develop non-vanishing asymptotic terms expansion at infinity analytically dependent on time and initial data.