We consider multidimensional hyperbolic systems of conservation laws
with relaxation, together with their associated limit systems.
A strong stability condition for such asymptotics has been introduced by
Chen, Levermore, Liu in Comm. Pure Appl. Math. 47, 787-830, namely
the existence of an entropy extension.
We propose here a new stability condition, the reduced stability condition,
which is weaker than the previous one, but still has the property to imply
the subcharacteristic or interlacing conditions, and the dissipativity
of the leading term in the Chapman-Enskog expansion.
This reduced stability condition has the advantage to involve only
the submanifold of equilibria, or maxwellians, so that it is much
easier to check than the entropy extension condition.
Our condition generalizes the one introduced by the author in
the case of kinetic, i.e. diagonal semilinear relaxation.
We provide an adapted stability analysis in the context of
approximate Riemann solvers obtained via relaxation systems.
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