Title:

Numerical study on the regularity of the NavierStokes equations

This thesis is mainly focused on the regularity problem for the threedimensional NavierStokes equations. \\\\ The threedimensional freely decaying NavierStokes and Burgers equations are compared via direct numerical simulations, starting from identical {\it incompressible} initial conditions, with the same kinematic viscosity. From previous work by Kiselev and Ladyzenskaya (1957), the Burgers equations are known to be globally regular thanks to a maximum principle. In this comparison, the Burgers equations are split via Helmholtz decomposition with consequence that the potential part dominates over the solenoidal part. The nonlocal term ${\bm u}\cdot\nabla p$ invalidates the maximum principle in the NavierStokes equations. Its probability distribution function and joint probability distribution functions with both energy and enstrophy are essentially symmetric with random fluctuations, which are temporally correlated in all three cases. We then evaluate nonlinearity depletion quantitatively in the enstrophy growth bound via the exponent $\alpha$ in the powerlaw $\frac{dQ}{dt}+2\nu P\propto(Q aP b) {\alpha}$, where $Q$ is enstrophy, $P$ is palinstrophy and $a$ and $b$ are determined by calculus inequalities. \\\\ CaffarelliKohnNirenberg theory defines a local Reynolds number over parabolic cylinder $Q_r$ as $\delta(r)=1/(\nu r)\int_{Q_r}\nabla {\bm u} 2\,d{\bm x}\,dt$. From this we determine a crossover scale $r_* \propto L\left( \frac{ \overline{\\nabla \bm{u} \ 2_{L 2}} } {\ \nabla \bm{u} \ 2_{L \infty}} \right) {1/3}$, corresponding to the change in scaling behavior of $\delta(r)$. Following the assumption that $E(k)\propto k {q}$ $(1
