06/09/2016, 11:20 — 12:00 — Amphitheatre Pa2, Mathematics Building
Tomasz Dlotko, University of Silesia at Katowice

Fractional Navier-Stokes Equations

I will present shortly the results concerning fractional generalization of the Navier-Stokes problem reported in two recent papers [1, 2]. The abstracts of that two manuscripts are copied below.

The following reference list contains the most important references; an excellent paper [4], forming a base for the local in time solvability, by Yoshikazu Giga and Tetsuro Miyakawa, and recent publications [6, 5] by Jiahong Wu containing earlier studies of the fractional Navier-Stokes equation.

Abstract 1

We consider the Navier-Stokes equation (N-S) in dimensions two and three as limits of the fractional approximations. In 2-D the N-S problem is critical with respect to the standard $L^2$ a priori estimates and we consider its regular approximations with the fractional power operator $(-P\Delta)^{1+\alpha}$, $\alpha \gt 0$ small, where $P$ is the projector on the space of divergence-free functions. In 3-D different properties of the N-S problem with respect to the standard $L^2$ a priori estimate are obtained and the 3-D regular approximating problem involves fractional power operator $(-P\Delta)^s$ with $s\gt \frac{5}{4}$.

Using Dan Henry's semigroup approach and the Giga-Miyakawa estimates we construct regular solutions to such approximations. The solutions are global in time, 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 such regular solutions of the approximations. Moreover, since the nonlinearity of the N-S equation is of quadratic type, the solutions corresponding to small initial data and small $f$ are shown to be global in time and regular.

Abstract 2

We consider fractional Navier-Stokes equations in a bounded smooth domain $\Omega\in\mathbb{R}^N$, $N\ge 2$. Following the geometric theory of abstract parabolic problems we give the detailed analysis concerning existence, uniqueness, regularization and continuation properties of the solutions. Using these generalizations we construct next global solution of the original Navier-Stokes problem. Focusing finally on the 3-D model with zero external force we show that such solutions regularize after a certain time $T=T(\|u_0\|_{X_2})$.

References

1. J. W. Cholewa, T. Dlotko, Fractional Navier-Stokes equations, submitted, January 2016.
2. T. Dlotko, Navier-Stokes equation and its fractional approximations, Appl. Math. Optim., DOI 10.1007/s00245-016-9368-y (2016).
3. T. Dlotko, M. B. Kania, C. Sun, Quasi-geostrophic equation in $\mathbb{R}^2$, J. Differential Equations 259 (2015), 531-561.
4. Y. Giga, T. Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), 267-281.
5. J. Wu, The quasi-geostrophic equation and its two regularizations, Comm. Partial Differ. Equ. 27 (2002), 1161-1181.
6. J. Wu, Generalized MHD equations, J. Differential Equations 195 (2003), 284-312.