Fast Decays of Strong Global Solutions of the Navier–Stokes Equations

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.