×

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
PDFBibTeX XMLCite
Full Text: DOI Numdam EuDML

References:

[1] L. Brandolese , On the Localization of Symmetric and Asymmetric Solutions of the Navier-Stokes Equations dans \(\mathbb{R}^n\) . C. R. Acad. Sci. Paris Sér. I Math 332 ( 2001 ) 125 - 130 . Zbl 0973.35149 · Zbl 0973.35149 · doi:10.1016/S0764-4442(00)01805-X
[2] Y. Dobrokhotov and A.I. Shafarevich , Some integral identities and remarks on the decay at infinity of solutions of the Navier-Stokes Equations . Russian J. Math. Phys. 2 ( 1994 ) 133 - 135 . Zbl 0976.35508 · Zbl 0976.35508
[3] T. Gallay and C.E. Wayne , Long-time asymptotics of the Navier-Stokes and vorticity equations on \(\mathbb{R}^3\) . Preprint. Univ. Orsay ( 2001 ). arXiv
[4] C. He and Z. Xin , On the decay properties of Solutions to the nonstationary Navier-Stokes Equations in \(\mathbb{R}^3\) . Proc. Roy. Soc. Edinburgh Sect. A 131 ( 2001 ) 597 - 619 . Zbl 0982.35083 · Zbl 0982.35083 · doi:10.1017/S0308210500001013
[5] T. Kato , Strong \(L^p\)-Solutions of the Navier-Stokes Equations in \(\mathbb{R}^m\), with applications to weak solutions . Math. Z. 187 ( 1984 ) 471 - 480 . Zbl 0545.35073 · Zbl 0545.35073 · doi:10.1007/BF01174182
[6] O. Ladyzenskaija , The mathematical theory of viscous incompressible flow . Gordon and Breach, New York, English translation, Second Edition ( 1969 ). MR 254401 | Zbl 0184.52603 · Zbl 0184.52603
[7] T. Miyakawa , On space time decay properties of nonstationary incompressible Navier-Stokes flows in \(\mathbb{R}^n\) . Funkcial. Ekvac. 32 ( 2000 ) 541 - 557 . Zbl pre02112739 · Zbl 1142.35545
[8] S. Takahashi , A wheighted equation approach to decay rate estimates for the Navier-Stokes equations . Nonlinear Anal. 37 ( 1999 ) 751 - 789 . Zbl 0941.35066 · Zbl 0941.35066 · doi:10.1016/S0362-546X(98)00070-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.