Speaker:
Sauli Lindberg
Title:
Taylor's conjecture on magnetic helicity dissipation in magnetohydrodynamics
Abstract:
'Magnetohydrodynamics (MHD) couples Navier-Stokes equations and Maxwell
equations to study the macroscopic behaviour of plasmas and liquid metals. In
ideal MHD, where fluid viscosity and electrical resistivity are assumed to
vanish, smooth solutions conserve total energy, cross helicity and magnetic
helicity in time.
Simulations point, however, towards anomalous dissipation where in near-ideal
MHD, total energy dissipation rate approaches a non-zero constant when viscosity
and resistivity tend to zero. A similar phenomenon has been observed for total
kinetic energy in Navier-Stokes equations. In order to reconcile the
(deterministic) Navier-Stokes equations with anomalous dissipation, the famous
physicist and chemist Onsager (Nuovo Cimento 1949) conjectured that weak
solutions of Euler equations can dissipate total kinetic energy as long as they
are non-smooth enough. After a series of works starting from Scheffer (J. Geom.
Anal. 1993), Shnirelman (Comm. Pure Appl. Math. 1999) and de Lellis-Székelyhidi
(Ann. Math. 2009), the conjecture was recently proved by Isett (Ann. Math. 2018)
and Buckmaster-de Lellis-Székelyhidi-Vicol (Comm. Pure Appl. Math. 2018). By
modifying the tools of de Lellis-Székelyhidi 2009, it has also been shown that
weak solutions can dissipate numerous classically conserved quantities in fluid
dynamics.
Taylor (Phys. Rev. Lett. 1974) conjectured, however, that magnetic helicity is
approximately conserved in MHD at very small resistivities. Mathematically, the
conjecture says that magnetic helicity is conserved by weak solutions at the
inviscid limit of Leray-Hopf solutions. Daniel Faraco and I recently proved the
mathematical version of the conjecture (https://arxiv.org/abs/1806.09526, to
appear in Comm. Math. Phys.) after prior work of Caflisch-Klapper-Steele (Comm.
Math Phys. 1997) and others.
In the first part of the talk I discuss anomalous dissipation and Onsager's and
Taylor's conjectures in non-technical fashion, and in the second part I present
a proof of Taylor's conjecture.'