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We propose a new way of looking at the Navier-Stokes equation (N-S) in dimensions two and three. We consider its regular approximations in which the -P Delta operator is replaced with the fractional power. The 3-D N-S equation is super-critical with respect to the standard L2 a priori estimates; the regular approximating problem in 3-D should contain fractional power with s > 5/4. Using Dan Henry's semigroup approach we construct regular solutions to such approximations. The solutions are unique, smooth and regularized through the equation in time. Solution to 2-D and 3-D N-S equations are obtained next as a limit of the regular solutions of the above approximations., I am sending the final, completed version of the manuscript; New look at the Navier-Stokes equation that is carefully checked and extended with few remarks presented in the final part. I also hope that the manuscript will be soon accepted for publication in the Journal it was submitted early this year. Tomasz Dlotko

Solvability of Cauchy's problem in $\mathbb{R}^2$ for subcritical quasi-geostrophic equation is discussed here in two phase spaces; $L^p(\mathbb{R}^2)$ with $p> \frac{2}{2\alpha-1}$ and $H^s(\mathbb{R}^2)$ with $s>1$. A solution to that equation in critical case is obtained next as a limit of the $H^s$-solutions to subcritical equations when the exponent $\alpha$ of $(-\Delta)^\alpha$ tends to $\frac{1}{2}^+$. Such idea seems to be new in the literature. Existence of the global attractor in subcritical case is discussed in the paper. In section 7 we also discuss solvability of the critical problem with Dirichlet boundary condition in bounded domain $\Omega \subset \mathbb{R}^2$, when $\| \theta_0 \|_{L^\infty(\Omega)}$ is small.

Global solvability and dynamical behaviour of the modified viscous Cahn-Hilliard equation is studied in the Sobolev space $H^1({\mathbb{R}}^N)$. For $\nu \in [0,1]$ we construct $H^1({\mathbb{R}}^N)$ global attractors and show their upper semicontinuity at $\nu = 0$.