On the role of continuous symmetries in the solution of the 3D Euler fluid equations and related models

Miguel Bustamante, University College Dublin

03/31, 2023 at 1:30pm-2:30pm (HH410)

This work is published in Philosophical Transactions A, as open access:  https://doi.org/10.1098/rsta.2021.0050.

We review the continuous symmetry approach based on Sophus Lie's transformation theory, and apply it to find the solution of the 3D Euler fluid equations in several instances of interest, via the construction of constants of motion and infinitesimal symmetries, without recourse to Noether's theorem, in a setup that allows for unsteady flows as well as unsteady infinitesimal symmetries. Roughly speaking, an infinitesimal symmetry is a vector field that is continuously transported along the flow.

When the flow admits two linearly independent infinitesimal symmetries, we obtain a number of general results:

(i) If these symmetries commute, then we construct a constant of motion for the flow.

(ii) If these symmetries do not commute, then we construct a new infinitesimal symmetry and can repeat the search (i) with a new pair of symmetries, or repeat the search (ii) to find a complete Lie algebra of infinitesimal symmetries.

Another general result, of remarkable geometrical and dynamical importance for both steady and unsteady flows, is that the vorticity field is an infinitesimal symmetry of the flow.

Therefore, if the flow admits another infinitesimal symmetry other than the vorticity, then by points (i) and (ii) above one can construct new constants of motion and/or a Lie algebra of new symmetries. For steady Euler flows this leads directly to the distinction of (non-)Beltrami flows: an example is given where the topology of the spatial manifold determines whether extra infinitesimal symmetries can be constructed.

As for unsteady flows, we study the stagnation-point-type exact solution of the 3D Euler fluid equations introduced by (Gibbon et al., Physica D, vol. 132, 1999, pp. 497-510) along with a one-parameter generalisation of it introduced by (Mulungye et al., J. Fluid Mech., vol. 771, 2015, pp.468-502). Applying the continuous symmetry approach to these models allows for the explicit construction of constants of motion and the subsequent integration of the fields (vorticity, its stretching rate, and the back-to-labels map) along pathlines, revealing a fine structure of blowup, depending on the value of the free parameter and on the initial conditions. A remarkable formula for the blowup time is obtained, which shows how the flow's regularity depends on the initial conditions. We produce explicit blowup exponents and prefactors for a generic type of initial conditions.