Ellipsoidal BGK model for polyatomic molecules near Maxwellians: A dichotomy in the dissipation estimate

We consider the global existence and asymptotic behavior of classical solutions to the ellipsoidal BGK model for polyatomic molecules when the initial data starts sufficiently close to a global polyatomic Maxwellian. We observe that the linearized relaxation operator is decomposed into a trully polyatomic part and a essentially monatomic part, leading to a dichotomy in the dissipative property in the sense that the degeneracy of the dissipation shows an abrupt jump as the relaxation parameter $\theta$ reaches zero. Accordingly, we employ two different sets of micro-macro system to derive the full coercivity and close the energy estimate.