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Large time behaviour of solutions to the Navier-Stokes equations. (English) Zbl 0607.35071
In a classical paper Jean Leray (1934) posed the problem of determining whether or not weak solutions to the Navier-Stokes equations on \({\mathbb{R}}^ 3\) (and \({\mathbb{R}}^ 2)\) tend to zero in \(L^ 2\) as time goes to infinity. This paper makes a significant contribution to this important problem.
It is shown that the rate of decay for solutions with large data in \(L^ p\cap L^ 2\), \(1\leq p<2\), is the same as for solutions to the heat equation, i.e. \[ \| u(t)\|_{L^ 2({\mathbb{R}}^ 3)}\leq C(t+1)^{- 3/4([2/p]-1)}. \] Furthermore, lower bounds are established for the rates of decay. The equivalent problem in \({\mathbb{R}}^ 2\) is also studied. Here, it is shown that the decay is logarithmic.
The author draws attention to the extensive literature on decay results for the Navier-Stokes equations, most of which is, however, for small, smooth initial data. It is worth adding another reference here (in Italian and perhaps not widely known): P. Maremonti, Stabilità asintotica in media per moti fluidi viscosi in domini esterni, Ann. Mat. Pura Appl. 142, 57-75 (1985).
Reviewer: B.Straugham

MSC:
35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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