Nonlinear geometric optics for hyperbolic boundary problems

Journal Article · Tue Dec 31 04:00:00 EST 1996 · Communications in Partial Differential Equations

OSTI ID:530802

Williams, M ^[1]

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1. Univ. of North Carolina, Chapel Hill, NC (United States)

We give a rigorous justification of geometric optics for a class of Kreiss well-posed semilinear boundary problems where both resonant interactions and glancing modes are present. Errors are o(1) in L{sup 2} as the wavelength tends to zero. We emphasize the features that distinguish boundary problems from hyperbolic problems in free space. These include: (1) the failure of coherence and symmetry hypotheses alone to guarantee existence of the exact solution on a fixed domain independent of the wavelength, (2) inconsistent transport equations for flaning modes connected with the presence of a glancing boundary layer, (3) the need to use (generalized) eigenvectors associated to nonreal eigenvalues in constructing approximate solutions and the related presence of an elliptic boundary layer, (4) and the appearance of unbounded families of projection operators (associated to eigenvalues of high multiplicity) in the profile equations. The difficulties in (1) and (4) are both handled by arguments that make use of the same hypothesis; namely, that the real zeros in {xi} of the principal symbol p({tau},{xi},{eta}) ({xi} is dual to {chi}, where {chi}=0 defines the boundary) have multiplicity at most two. As an essential tool we introduce the class of F{sub {infinity}}, m, s, {gamma} spaces, the natural spaces for obtaining energy estimates uniform with respect to wavelength in highly oscillatory boundary problems. 11 refs., 1 fig.

• OSTI ID: 530802
• Journal Information: Communications in Partial Differential Equations, Journal Name: Communications in Partial Differential Equations Journal Issue: 11-12 Vol. 21; ISSN CPDIDZ; ISSN 0360-5302
• Country of Publication: United States
• Language: English

Related Subjects

66 PHYSICS
99 GENERAL AND MISCELLANEOUS
BOUNDARY-VALUE PROBLEMS
NONLINEAR OPTICS
PARTIAL DIFFERENTIAL EQUATIONS