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_{ L∞}ds , improving the previously known results. We also prove the persistence of the sub-analytic Gevrey-class regularity for the Euler equations in a half space, and obtain an explicit rate of decay of the radius of Gevrey-class regularity.

$ u_{t}-u_{x x}+f(x,u,u_x)=0$

we prove that the global attractor can be parametrized by a finite number of Fourier modes and that the number of modes is algebraic in parameters. This improves our earlier result [15], where the number of required modes is exponential. The method extends to equations of order higher than two.

sup_{ $|x-x_0|+\sqrt{t-t_0} < r_0$}
sup_{ $r\in(0,r_0)$}
$ \frac{1}{r^{n+2-q}}
\int_{t-r^2}^{t+r^2}\ \ \ \int_{|y-x|\le r}
|u(y,s)|^{q}\,dy\,ds
\le \epsilon $

then the solution $u$ is regular in a neighborhood of $(x_0,t_0)$. There is no assumption on the integrability of the pressure or the vorticity.

$ \|\| \nabla u_0\|\|_{L^2(\Omega)} \le \frac{1}{C(L_1,L_2)\epsilon^{1/6}} $

then there exists a unique global smooth solution with the initial datum $u_0$.

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