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| A NEW REGULARITY CLASS FOR THE NAVIER-STOKES EQUATIONS IN Rn |
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Citation： |
H. Beiro da Veiga.A NEW REGULARITY CLASS FOR THE NAVIER-STOKES EQUATIONS IN Rn[J].Chinese Annals of Mathematics B,1995,16(4):407~412 |
| Page view： 1345
Net amount： 979 |
Authors： |
H. Beiro da Veiga; |
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| Abstract： |
Consider the Navier-Stokes equations in $\R^n\times (0, T),$ for
$n\ge 3.$ Let $1<\al\le \min\{2, n/(n-2)\}$ and define $\be$ by
$(2/\al)+(n/\be)=2.$ Set $\al'=\al/(\al-1)$. It is proved that
$Dv$ belongs to $C(0, T; L^{\al'})\cap L^{\al'}(0, T; L^{2\be/(n-2)})$
whenever $Dv\in L^{\al}(0, T;
L^{\be}).$ In particular, $v$ is a regular solution. This results
is the natural extension to $\al\in (1,2]$ of the classical
sufficient condition that establishes that $L^{\al}
(0,T;L^{\ga})$ is a regularity class if $(2/\al)+(n/\ga)=1$. Even
the borderline case $\al=2$ is significant. In fact, this result
states that $L^2(0,T; W^{1,n})$ is a regularity class if $n\le
4.$ Since $W^{1,n}\hookrightarrow L^{\infty}$ is false, this result does not
follow from the classical one that states that $L^2(0,T;
L^{\infty})$ is a regularity class. |
Keywords： |
Navies-Stokes equation, Regularity of solution, Extension |
Classification： |
35B65, 35K55, 76D05 |
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