arXiv subjects

Tomasz Dlotko

Publications and source records attributed to Tomasz Dlotko.

2 recordsLinked to original sources

arXiv2015

New look at the Navier-Stokes equation

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.

math-ph↗

arXiv2014

Quasi-geostrophic equation in $\mathbb{R}^2$

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 $p> \frac{2}{2α-1}$ 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 $α$ of $(-Δ)^α$ 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 $Ω\subset \mathbb{R}^2$, when $\| θ_0 \|_{L^\infty(Ω)}$ is small.

math-ph↗