17 research outputs found
Global regularity for a family of 3D models of the axisymmetric Navier-Stokes equations
We consider a family of 3D models for the axi-symmetric incompressible
Navier-Stokes equations. The models are derived by changing the strength of the
convection terms in the axisymmetric Navier-Stokes equations written using a
set of transformed variables. We prove the global regularity of the family of
models in the case that the strength of convection is slightly stronger than
that of the original Navier-Stokes equations, which demonstrates the potential
stabilizing effect of convection
Global Regularity of the 3D Axi-symmetric Navier-Stokes Equations with Anisotropic Data
In this paper, we study the 3D axi-symmetric Navier-Stokes Equations with
swirl. We prove the global regularity of the 3D Navier-Stokes equations for a
family of large anisotropic initial data. Moreover, we obtain a global bound of
the solution in terms of its initial data in some norm. Our results also
reveal some interesting dynamic growth behavior of the solution due to the
interaction between the angular velocity and the angular vorticity fields
Dynamic Stability of the 3D Axi-symmetric Navier-Stokes Equations with Swirl
In this paper, we study the dynamic stability of the 3D axisymmetric
Navier-Stokes Equations with swirl. To this purpose, we propose a new
one-dimensional (1D) model which approximates the Navier-Stokes equations along
the symmetry axis. An important property of this 1D model is that one can
construct from its solutions a family of exact solutions of the 3D
Navier-Stokes equations. The nonlinear structure of the 1D model has some very
interesting properties. On one hand, it can lead to tremendous dynamic growth
of the solution within a short time. On the other hand, it has a surprising
dynamic depletion mechanism that prevents the solution from blowing up in
finite time. By exploiting this special nonlinear structure, we prove the
global regularity of the 3D Navier-Stokes equations for a family of initial
data, whose solutions can lead to large dynamic growth, but yet have global
smooth solutions
Potentially singular solutions of the 3D axisymmetric Euler equations
The question of finite-time blowup of the 3D incompressible Euler equations is numerically investigated in a periodic cylinder with solid boundaries. Using rotational symmetry, the equations are discretized in the (2D) meridian plane on an adaptive (moving) mesh and is integrated in time with adaptively chosen time steps. The vorticity is observed to develop a ring-singularity on the solid boundary with a growth proportional to ∼(t_s − t)^(−2.46), where t_s ∼ 0.0035056 is the estimated singularity time. A local analysis also suggests the existence of a self-similar blowup. The simulations stop at τ_2 = 0.003505 at which time the vorticity amplifies by more than (3 × 10^8)-fold and the maximum mesh resolution exceeds (3 × 10^(12))^2. The vorticity vector is observed to maintain four significant digits throughout the computations