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Pointwise Green Function Bounds for Shock Profiles of Systems with Real Viscosity

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Abstract.

Following the pointwise semigroup approach of [ZH,MZ.1], we establish sharp pointwise Green function bounds and consequent linearized stability for viscous shock profiles of general hyperbolic-parabolic systems of conservation laws of dissipative type, under the necessary assumptions ([Z.1,Z.3,Z.4]) of spectral stability, i.e., stable point spectrum of the linearized operator about the wave; transversality of the connecting profile; and hyperbolic stability of the corresponding ideal shock of the associated inviscid system, with no additional assumptions on the structure or strength of the shock. These bounds are used in a companion paper [MZ.2] to establish nonlinear stability of small-amplitude Lax shocks of symmetrizable hyperbolic-parabolic systems.

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Authors and Affiliations

  1. Dipartimento di Matematica ``G. Gastelnuovo'', Università di Roma ``La Sapienza'', P.le Aldo Moro, 2-00185, Roma, Italy

    Corrado Mascia

  2. Department of Mathematics, Indiana University, Bloomington, In 47405-4301

    Kevin Zumbrun

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  1. Corrado Mascia
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  2. Kevin Zumbrun
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Correspondence to Corrado Mascia.

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Communicated by A. Bressan

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Mascia, C., Zumbrun, K. Pointwise Green Function Bounds for Shock Profiles of Systems with Real Viscosity. Arch. Rational Mech. Anal. 169, 177–263 (2003). https://doi.org/10.1007/s00205-003-0258-5

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