Lower Bound on the Blow-up Rate of the Axisymmetric Navier-Stokes Equations

By Chiun-Chuan Chen, Robert M. Strain, Tai-Peng Tsai, Horng-Tzer Yau. 2007

84 citations recorded by the source corpus.

Consider axisymmetric strong solutions of the incompressible Navier-Stokes equations in with nontrivial swirl. Such solutions are not known to be globally defined, but it is shown in ([1], Partial regularity of suitable weak solutions of the Navier-Stokes equations. Communications on Pure and Applied Mathematics, 35 (1982), 771-831) that they could only blow up on the axis of symmetry. Let z denote the axis of symmetry and r measure the distance to the z-axis. Suppose the solution satisfies the pointwise scale invariant bound |v(x, t)| ⩽ C * (r 2 − t) −1/2 for −T 0 ⩽ t < 0 and 0 < C * < ∞ allowed to be large, we then prove that v is regular at time zero.

Subfields: Applied Mathematics, Computational Mechanics, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/ac89c5f4b81335aeb327808d975f36e1a9d76106

Citations in the corpus

• Criticality of the Axially Symmetric Navier-Stokes Equations (2015, unknown): “3) is that smooth solutions of the axi-symmetric Navier-Stokes equations satisfy the following maximum principle (see, for instance, [2, 7, 3, 20, 21]): sup t ‖Γ(t, ·)‖L∞ ≤ ‖Γ0‖L∞ .”
• On axially symmetric incompressible magnetohydrodynamics in three dimensions (2012, unknown): “It is well-known that the axially symmetric Navier-Stokes equations (in the case of B ≡ 0) are (see, for instance, [14])    ∂tu r + u∂ru r + u∂zu r + ∂rp = ( ∆− 1 r(2) ) u + (u )(2) r , ∂tu z + u∂ru z + u∂zu z + ∂zp = ∆u , ∂tu θ + u∂ru θ + u∂zu θ = ( ∆− 1 r(2) ) u − uu r .”