  In this work we aim to determine the atmospheric survivability of the
TRAPPIST-1 planets by modelling the response of the upper atmosphere to
incoming stellar high-energy radiation. Through this case study, we also aim to
learn more about rocky planet atmospheres in the habitable zone around low-mass
M dwarfs. We simulated the upper atmospheres using the Kompot code, a
self-consistent thermo-chemical code. Specifically, we studied the atmospheric
mass loss due to Jeans escape induced by stellar high-energy radiation. This
was achieved through a grid of models that account for the differences in
planetary properties, irradiances, and atmospheric properties, allowing the
exploration of the different factors influencing atmospheric loss. The
present-day irradiance of the TRAPPIST-1 planets would lead to the loss of an
Earth's atmosphere within just some 100 Myr. Taking into account the much more
active early stages of a low-mass M dwarf, the planets undergo a period of even
more extreme mass loss, regardless of planetary mass or atmospheric
composition. This indicates that it is unlikely that any significant atmosphere
could survive for any extended amount of time around any of the TRAPPIST-1
planets. The assumptions used here allow us to generalise the results, and we
conclude that the results tentatively indicate that this conclusion applies to
all Earth-like planets in the habitable zones of low-mass M dwarfs.
