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On the instantaneous spreading for the Navier-Stokes system in the whole space. (English) Zbl 1080.35063
Summary: We consider the spatial behavior of the velocity field \(u(x, t)\) of a fluid filling the whole space \(\mathbb{R}^n\) (\(n\geq 2\)) for arbitrarily small values of the time variable. We improve previous results on the spatial spreading by deducing the necessary conditions \(\int u_h(x,t)u_k(x,t)\,dx=c(t)\delta_{h,k}\) under more general assumptions on the localization of \(u\). We also give some new examples of solutions which have a stronger spatial localization than in the generic case.

MSC:
35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
76D05 Navier-Stokes equations for incompressible viscous fluids
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
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