On L$^1$-Stability of Stationary Navier-Stokes Flows in $\Bbb R^n$
Vol. 4 (1997), No. 1, Page 67--119.
Miyakawa, Tetsuro
On L$^1$-Stability of Stationary Navier-Stokes Flows in $\Bbb R^n$
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Abstract:
Stability of stationary Navier-Stokes flows in $\mbox{\tenopenface R}^n$, $n\geq 3$, is discussed in the function space ${\bm L}^1$ or ${\bm H}^1$ (Hardy space). It is shown that a stationary flow ${\bm w}$ is stable in ${\bm H}^1$\ (resp.\ ${\bm L}^1$)\ if $\sup|x|\cdot|{\bm w}(x)|+\sup|x|^2|\nabla{\bm w}(x)|$ (resp.\ $\|{\bm w}\|_{(n,1)}+\|\nabla{\bm w}\|_{(n/2,1)}$) is small. Explicit decay rates of the form $O(t^{-β/2})$, $0<β\leq 1$, are deduced for perturbations under additional assumptions on ${\bm w}$ and on initial data. The proofs of the results heavily rely on the theory of Hardy spaces ${\bm H}^p$\ $(0
Mathematics Subject Classification (1991): Primary 35Q30; Secondary 76D05
Mathematical Reviews Number: MR1451304
Received: 1996-01-16
