We provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting wavetrains is close to the uniquely determined exact solution for small wavelengths ( ). Waves reflecting off of fixed noncharacteristic boundaries and off of multidimensional shocks are considered under the assumption that the underlying fixed (respectively, free) boundary problem is uniformly spectrally stable in the sense of Kreiss (respectively, Majda). Our results apply to a general class of problems that includes the compressible Euler equations; as a corollary we rigorously justify the leading term in the geometric optics expansion of highly oscillatory multidimensional shock solutions of the Euler equations. An earlier stability result of this type [21] was obtained by a method that required the construction of high-order approximate solutions. That construction in turn was possible only under a generically valid (absence of) small divisors assumption.Here we are able to remove that assumption and avoid the need for high-order expansions by studying associated singular (because they involve coefficients of order 1 ) fixed and free boundary problems. The analysis applies equally to systems that cannot be written in conservative form.

We generalize the analysis of [12] and develop a singular pseudodifferential calculus. The symbols that we consider do not satisfy the standard decay with respect to the frequency variables. We thus adopt a strategy based on the Calderón-Vaillancourt Theorem. The remainders in the symbolic calculus are bounded operators on L 2 , whose norm is measured with respect to some small parameter. Our main improvement with respect to [12] consists in showing a regularization effect for the remainders. Due to a nonstandard decay in the frequency variables, the regularization takes place in a scale of anisotropic, and singular, Sobolev spaces. Our analysis allows to extend the results of [12] on the existence of highly oscillatory solutions to nonlinear hyperbolic problems by dropping the compact support condition on the data. The results are also used in our companion work [6] to justify nonlinear geometric optics with boundary amplification, which corresponds to a more singular regime than the one considered in [12]. The analysis is carried out with either an additional real or periodic variable in order to cover problems for pulses or wavetrains in geometric optics.Contents 1991 Mathematics Subject Classification. Primary: 35S05; Secondary: 47G30.The norm · 0,γ does not depend on γ and coincides with the usual L 2 -norm on R d . We shall thus write · 0 instead of · 0,γ for the L 2 -norm on R d . For simplicity, we also write · s instead of · s,1 for the standard H s -norm (when the parameter γ equals 1).2.2. Functional spaces on R d × T. We now extend the previous definitions to functions that depend in a periodic way on an additional variable θ. We shall in some sense "interpolate" between Fourier transform and Fourier series. Let us begin with the definition of the Schwartz space. The Schwartz space S (R d × T) is the set of C ∞ functions f on R d × R, that are 1-periodic with respect to the last variable, and with fast decay at infinity in the first variable, that is:

We study highly oscillating solutions to a class of weakly well-posed hyperbolic initial boundary value problems. Weak well-posedness is associated with an amplification phenomenon of oscillating waves on the boundary. In the previous works [CGW14, CW14], we have rigorously justified a weakly nonlinear regime for semilinear problems. In that case, the forcing term on the boundary has amplitude O(ε 2 ) and oscillates at a frequency O(1/ε). The corresponding exact solution, which has been shown to exist on a time interval that is independent of ε ∈ (0, 1], has amplitude O(ε). In this paper, we deal with the exact same scaling, namely O(ε 2 ) forcing term on the boundary and O(ε) solution, for quasilinear problems. In analogy with [CGM03], this corresponds to a strongly nonlinear regime, and our main result proves solvability for the corresponding WKB cascade of equations, which yields existence of approximate solutions on a time interval that is independent of ε ∈ (0, 1]. Existence of exact solutions close to approximate ones is a stability issue which, as shown in [CGM03], highly depends on the hyperbolic system and on the boundary conditions; we do not address that question here.This work encompasses previous formal expansions in the case of weakly stable shock waves [MR83] and two-dimensional compressible vortex sheets [AM87]. In particular, we prove well-posedness for the leading amplitude equation (the "Mach stem equation") of [MR83] and generalize its derivation to a large class of hyperbolic boundary value problems and to periodic forcing terms. The latter case is solved under a crucial nonresonant assumption and a small divisor condition.

We provide a justification with rigorous error estimates showing that the leading term in weakly nonlinear geometric optics expansions of highly oscillatory reflecting pulses is close to the uniquely determined exact solution for small wavelengths ε. Pulses reflecting off fixed noncharacteristic boundaries are considered under the assumption that the underlying boundary problem is uniformly spectrally stable in the sense of Kreiss. There are two respects in which these results make rigorous the formal treatment of pulses in Majda and Artola [16], and Hunter, Majda and Rosales [10]. First, we give a rigorous construction of leading pulse profiles in problems where pulses traveling with many distinct group velocities are, unavoidably, present; and second, we provide a rigorous error analysis which yields a rate of convergence of approximate to exact solutions as ε → 0. Unlike wavetrains, interacting pulses do not produce resonances that affect leading order profiles. However, our error analysis shows the importance of estimating pulse interactions in the construction and estimation of correctors. Our results apply to a general class of systems that includes quasilinear problems like the compressible Euler equations; moreover, the same methods yield a stability result for uniformly stable Euler shocks perturbed by highly oscillatory pulses.