arXiv subjects

Ruo Li

Publications and source records attributed to Ruo Li.

93 records · Page 6Linked to original sources

arXiv2006

Level Set Dynamics and the Non-blowup of the 2D Quasi-geostrophic Equation

In this article we apply the technique proposed in Deng-Hou-Yu (Comm. PDE, 2005) to study the level set dynamics of the 2D quasi-geostrophic equation. Under certain assumptions on the local geometric regularity of the level sets of $θ$, we obtain global regularity results with improved growth estimate on $| \nabla^{\bot} θ|$. We further perform numerical simulations to study the local geometric properties of the level sets near the region of maximum $| \nabla^{\bot} θ|$. The numerical results indicate that the assumptions on the local geometric regularity of the level sets of $θ$ in our theorems are satisfied. Therefore these theorems provide a good explanation of the double exponential growth of $| \nabla^{\bot} θ|$ observed in this and past numerical simulations.

math.AP↗

arXiv2006

Nonexistence of Local Self-Similar Blow-up for the 3D Incompressible Navier-Stokes Equations

We prove the nonexistence of local self-similar solutions of the three dimensional incompressible Navier-Stokes equations. The local self-similar solutions we consider here are different from the global self-similar solutions. The self-similar scaling is only valid in an inner core region which shrinks to a point dynamically as the time, $t$, approaches the singularity time, $T$. The solution outside the inner core region is assumed to be regular. Under the assumption that the local self-similar velocity profile converges to a limiting profile as $t \to T$ in $L^p$ for some $p \in (3,\infty)$, we prove that such local self-similar blow-up is not possible for any finite time.

math.AP↗

arXiv2006

Dynamic Depletion of Vortex Stretching and Non-Blowup of the 3-D Incompressible Euler Equations

We study the interplay between the local geometric properties and the non-blowup of the 3D incompressible Euler equations. We consider the interaction of two perturbed antiparallel vortex tubes using Kerr's initial condition \cite{Kerr93}[Phys. Fluids {\bf 5} (1993), 1725]. We use a pseudo-spectral method with resolution up to $1536\times 1024 \times 3072$ to resolve the nearly singular behavior of the Euler equations. Our numerical results demonstrate that the maximum vorticity does not grow faster than double exponential in time, up to $t=19$, beyond the singularity time $t=18.7$ predicted by Kerr's computations \cite{Kerr93,Kerr04}. The velocity, the enstrophy and enstrophy production rate remain bounded throughout the computations. As the flow evolves, the vortex tubes are flattened severely and turned into thin vortex sheets, which roll up subsequently. The vortex lines near the region of the maximum vorticity are relatively straight. This local geometric regularity of vortex lines seems to be responsible for the dynamic depletion of vortex stretching.

math-ph↗