Analysis Seminar

---

Nearly self-similar blowup of generalized axisymmetric Navier-Stokes equations

Speaker: Thomas Hou, Caltech

Location: Warren Weaver Hall 1302

Date: Thursday, February 20, 2025, 11 a.m.

Synopsis:

We investigate the nearly self-similar blowup of the generalized axisymmetric Navier--Stokes equations. First, we provide a rigorous derivation of the axisymmetric  Navier-Stokes equations with swirl in any integer dimensions, marking the first such derivation for dimensions greater than three. Building on this, we generalize the equations to arbitrary positive real-valued dimensions, preserving many known properties of the 3D  axisymmetric Navier-Stokes equations. To address scaling instability, we dynamically vary the space dimension to balance advection scaling along the radial and the axial directions. We further introduce a novel two-scale dynamic rescaling formulation, leveraging the dimension as an additional degree of freedom. This approach enables us to obtain a one-scale self-similar blowup with solution-dependent viscosity. Although the solution dependent viscosity tends to zero as we approach the blowup time, the vanishing viscosity still has an O(1) effect on the blowup profile and the self-similar profile satisfies the axisymmetric Navier-Stokes equations with constant viscosity. We observe that the effective dimension is approximately 3.188 and appears to converge toward 3 as background viscosity diminishes.