Abstract.
In the paper we study the asymptotic dynamics of strong global solutions of the Navier Stokes equations. We are concerned with the question whether or not a strong global solution w can pass through arbitrarily large fast decays. Avoiding results on higher regularity of w used in other papers we prove as the main result that for the case of homogeneous Navier–Stokes equations the answer is negative: If \(\varepsilon \in\) [0, 1/4) and δ0 > 0, then the quotient \(\|A^{1+\varepsilon}w(t)\|/\|w(t+\delta)\|\) remains bounded for all t ≥ 0 and δ∈[0, δ0]. This result is not valid for the non-homogeneous case. We present an example of a strong global solution w of the non-homogeneous Navier–Stokes equations, where the exterior force f decreases very quickly to zero for \(t \rightarrow \infty\) while w passes infinitely often through stages of arbitrarily large fast decays. Nevertheless, we show that for the non-homogeneous case arbitrarily large fast decays (measured in the norm \(\|A^{\beta}.\|)\) cannot occur at the time t in which the norm \(\|A^{\beta}w(t)\|\) is greater than a given positive number.
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Communicated by I. Straškraba
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Skalák, Z. Fast Decays of Strong Global Solutions of the Navier–Stokes Equations. J. math. fluid mech. 9, 565–587 (2007). https://doi.org/10.1007/s00021-006-0222-x
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DOI: https://doi.org/10.1007/s00021-006-0222-x