## 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

### Keywords:

Navier-Stokes equations; space-decay; symmetries
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### References:

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