Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations

Lower bounds on the blow-up rate of the axisymmetric Navier-Stokes equations

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Robert Strain, Princeton University
Fine Hall 110

Consider axisymmetric strong solutions, v, of the incompressible Navier-Stokes equations in $\mathbb{R}^3$ with non-trivial swirl. Leray, in his 1934 Acta. Math paper, has shown that at a blow-up time, $T$, such solutions would have to satisfy for some fixed $\epsilon_1>0$ $$ \liminf_{t\uparrow T} \sqrt{T-t} \sup_x |v(x,t)| \ge \epsilon_1$ $$ We will discuss our recent proof which rules out this scale invariant blow-up rate: $$ |v(x,t)| \le C_*/\sqrt{T-t}. $$ Above $C_*$ is allowed to be large. This is joint work with Tsai, Chen and Yau.