EXPONENTIAL ASYMPTOTIC STABILITY OF RIEMANN SHOCKS OF HYPERBOLIC SYSTEMS OF BALANCE LAWS

Grégory Faye (1) , L. Miguel Rodrigues (2, 3)

1 IMT - Institut de Mathématiques de Toulouse UMR5219
2 IRMAR - Institut de Recherche Mathématique de Rennes
3 IUF - Institut universitaire de France

For strictly entropic Riemann shock solutions of strictly hyperbolic systems of balance laws, we prove that exponential spectral stability implies large-time asymptotic orbital stability. As a preparation, we also prove similar results for constant solutions of initial value and initial boundary value problems, that seem to be new in this generality. Main key technical ingredients include the design of a nonlinear change of variables providing a hypocoercive Kawashima-type structure with dissipative boundary conditions in the high-frequency regime and the explicit identification of most singular parts of the linearized evolution, both being deduced from the mere spectral assumption.

Mots clés: Riemann shocks; asymptotic stability; hyperbolic systems of balance laws 2010 MSC: 35B35; 35L02; 35L67; 35B40; 35L03; 37L15

Domaines: Equations aux dérivées partielles [math.AP]

Grégory Faye, L. Miguel Rodrigues. EXPONENTIAL ASYMPTOTIC STABILITY OF RIEMANN SHOCKS OF HYPERBOLIC SYSTEMS OF BALANCE LAWS. SIAM Journal on Mathematical Analysis, In press. ⟨hal-03738369⟩