By adding a spatially localized external force with a time profile depending on initial data, solutions to the Navier-Stokes equations can be constructed with rapid space-time decay beyond standard rates.
Brandolese,Space-time decay of Navier–Stokes flows invariant under rotations, Math
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Solutions of the Navier-Stokes equations with forced rapid space-time decay
By adding a spatially localized external force with a time profile depending on initial data, solutions to the Navier-Stokes equations can be constructed with rapid space-time decay beyond standard rates.