  The zeroth law is one of the oldest conjecture in turbulence that is still
unproven. Here, we consider weak solutions of one-dimensional compressible
magnetohydrodynamics and demonstrate that the lack of smoothness of the fields
introduces a new dissipative term, named inertial dissipation, into the
expression of energy conservation that is neither viscous nor resistive in
nature. We propose exact solutions assuming that the kinematic viscosity and
the magnetic diffusivity are equal, and we demonstrate that the associated
inertial dissipation is positive and equal on average to the mean viscous
dissipation rate in the limit of small viscosity, proving the conjecture of the
zeroth law of turbulence and the existence of an anomalous dissipation. As an
illustration, we evaluate the shock heating produced by discontinuities
detected by Voyager in the solar wind around 5 AU. We deduce a heating rate of
$\sim 10^{-18}$J m$^{-3}$ s$^{-1}$, which is significantly higher than the
value obtained from the turbulent fluctuations. This suggests that
collisionless shocks can be a dominant source of heating in the outer solar
wind.
