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7, we could see T has a C 1 map from B1 to B1 (for fixed a, ε, T, α), with quadratic bound T B1 ≤ C b 2B1 , and derivative bound ∂b T L(B1 ≤ C b B1 . 12, [TZ2]). Without regularity of the fixed point of T , we could not use the standard implicit function theorem in Sect. 3. These issues were discussed in detail in Sect. 3 of [TZ2]. 46 B. Texier, K. 13. 18, there exists a function β(a, ε, T, α), bounded from R4+1 to X 1 and C 1 from R4+1 to B1 , such that (a, β(a, ε, T, α), ε, T ) ≡ 0, β X1 + ∂(ε,T ) β L(R2 ,B1 ) ≤ C(|a|2 + |α|), ∂a β L(R2 ,B1 ) ≤ C|a|, ∂α β L(R,B1 ) ≤ C, for |(a, ε, α)| sufficiently small.

SIAM J. Math. Anal. : Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems. Mem. Amer. Math. Soc. : Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, 44. : Mechanism of instabilities in exothermic blunt-body flows. Combus. Sci. Tech. : Stability of undercompressive viscous shock profiles of hyperbolic– parabolic systems. J. Diff. Eqs. : Hopf bifurcation from viscous shock waves. SIAM J. Math. Anal. : On the normal form of the symmetric hyperbolic-parabolic systems associated with the conservation laws.

66), there are two terms in the upper bound for (S + R) +4 . 23) is small enough. 16) the growth assumption on ω. 9. 8). 66)(ii), and θ0 to be small ε ∈ H 2. 2. Reduction. 10. 17) is defined in Sect. 3, is equivalent to b = (Id − S(ε, T ))−1 N˜ (a, b, ε, T ) + ω for ω ∈ Ker(Id − S(ε, T )) ∩ X 1 . Proof. A simple consequence of the definition in the above section of the right inverse (Id − S(ε, T ))−1 . 8 in [TZ2]. Note that, as a consequence of (H4), Sect. 5, the kernel of Id − S(ε, T ) is of dimension one, for all ε, T, generated by (U¯ ε ) .

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