Large viscous boundary layers for noncharacteristic nonlinear hyperbolic problems

A Note on Complete Hyperbolic Operators with log-Zygmund Coefficients

“…In particular, we will recall some basic facts on LittlewoodPaley decomposition and paradifferential calculus, both in its classical version (see e.g. [1], [2] and [16]) and in the one depending on a parameter (see [15] and [17]). We will also recall some basic facts on functions with low regularity modulus of continuity.…”

“…For reasons which will appear clear in the sequel, we have to resort not to the classical version, as introduced by Bony in [2], but to the one depending on a parameter γ ≥ 1 (see e.g. [17] and [15]). Let us recall here the main ideas of the construction.…”

“…where we have omitted ψ γ because the definition is independent of it, up to lower order terms (see [17]). …”

“…For the details see [17,Appendix B]. What is important to retain is that, once we fix γ ≥ 1 (for example, to obtain positivity of paradifferential operators involved in our computations), all the previous construction is equivalent to the classical one; in particular, the space H s+α log γ coincides with H s+α log , the respective norms are equivalent and the characterization given by Proposition 3.3 still holds true.…”

Derivation of the Zakharov Equations

“…The next propositions are classical results of para-differential calculus (see for instance Appendix 2 of [17]), formulated in our C ∞ M m framework. In the following statements, C denotes non-decreasing functions, and d 0 is a real number such that…”