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Existence of global weak solutions to the Navier-Stokes equations in weighted spaces
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abstract
We obtain a global existence result for the three-dimensional Navier-Stokes equations with a large class of data allowing growth at spatial infinity. Namely, we show the global existence of suitable weak solutions when the initial data belongs to the weighted space $\mathring M^{2,2}_{\mathcal C}$ introduced in [Z. Bradshaw and I. Kukavica, Existence of suitable weak solutions to the Navier-Stokes equations for intermittent data, J. Math. Fluid Mech. to appear]. This class is strictly larger than currently available spaces of initial data for global existence and includes all locally square integrable discretely self-similar data. We also identify a sub-class of data for which solutions exhibit eventual regularity on a parabolic set in space-time.
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A remark on very weak suitable solutions and Leray solutions of the Navier-Stokes equations
Very weak suitable Navier-Stokes solutions become Leray solutions once the velocity lies in a local Morrey space M^{p,γ} satisfying γ/p − 3/p + 2/3 < 0.
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