Spectral Stability of Shock Profiles for Hyperbolically Regularized Systems of Conservation Laws

Abstract We report a proof that under natural assumptions shock profiles viewed as heteroclinic travelling wave solutions to a hyperbolically regularized system of conservation laws are spectrally stable if the shock amplitude is sufficiently small. This means that an associated Evans function $$\mathcal {E}:\Lambda \rightarrow \mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>E</mml:mi> <mml:mo>:</mml:mo> <mml:mi>Λ</mml:mi> <mml:mo>→</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> with $$\Lambda \subset \mathbb {C}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Λ</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>C</mml:mi> </mml:mrow> </mml:math> an open superset of the closed right half plane $$\mathbb {H}^+\equiv \{\kappa \in \mathbb {C}:\text {Re}\,\kappa \geqq 0\}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> <mml:mo>≡</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mi>κ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>C</mml:mi> <mml:mo>:</mml:mo> <mml:mtext>Re</mml:mtext> <mml:mspace/> <mml:mi>κ</mml:mi> <mml:mo>≧</mml:mo> <mml:mn>0</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math> has only one zero, namely, a simple zero at 0. The result is analogous to the one obtained in Freistühler and Szmolyan (Arch Ration Mech Anal 164:287–309, 2002) and Plaza and Zumbrun (Discrete Contin Dyn Syst 10(4):885–924, 2004) for parabolically regularized systems of conservation laws, and also distinctly extends findings on hyperbolic relaxation systems in Mascia and Zumbrun (Partial Differ Equ 34(1–3):119–136, 2009), Plaza and Zumbrun (2004) and Ueda (Math Methods Appl Sci 32(4):419–434, 2009).