Euler–Maxwell systems describe the dynamics of inviscid plasmas. We consider an
incompressible two-dimensional version of such a system and prove the existence and
uniqueness of global weak solutions, uniformly with respect to the speed of light
, for some
threshold value
depending only on the initial data. In particular, the condition
ensures that
the velocity of the plasma nowhere exceeds the speed of light and allows us to analyze the
singular regime
.
The functional setting for the fluid velocity lies in the framework of Yudovich’s
solutions of the two-dimensional Euler equations, whereas the analysis of
the electromagnetic field hinges upon the refined interactions between the damping
and dispersive phenomena in Maxwell’s equations in the whole space. This analysis is
enabled by the new development of a robust abstract method allowing us to incorporate
the damping effect into a variety of existing estimates. The use of this method
is illustrated by the derivation of damped Strichartz estimates (including endpoint
cases) for several dispersive systems (including the wave and Schrödinger equations),
as well as damped maximal regularity estimates for the heat equation. The ensuing
damped Strichartz estimates supersede previously existing results on the same systems.
