2019/06/10 by M. Williams, Williams, Mark
Engineering · Mathematics · #35L50 #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.1906.04074
openalex publication_date 2019/06/10 · openalex created_date 2022/07/23 · openalex updated_date 2026/07/28
We study weakly stable hyperbolic boundary problems with highly oscillatory\ncoefficients that are large, O(1), compared to the small wavelength eps of\noscillations. Such problems arise, for example, in the study of classical\nquestions concerning the stability of Mach stems and compressible vortex\nsheets. For such applications one seeks to prove energy estimates that are in\nan appropriate sense "uniform" with respect to the small wavelength eps, but\nthe large oscillatory coefficients are a formidable obstacle to obtaining such\nestimates. In this paper we analyze a simplified form of the linearized\nproblems that are relevant to the above stability questions, and obtain results\nthat are both positive and negative. On the one hand we identify favorable\nstructural conditions under which it is possible to prove uniform estimates,\nand then do so by a new approach. We also construct examples showing that large\noscillatory coefficients can give rise to an instantaneous \multiple\namplification of the amplitude of solutions relative to data; for example,\nboundary data of a given amplitude O(1) can \immediately give rise to a\nsolution of amplitude O(\(1)/( epsK)), where K>1. footnoteExamples of\nfirst-order amplification, where K=1, are well-known citeMA,CG. We use\nthe examples of multiple amplification to confirm the optimality of our uniform\nestimates when the favorable structural conditions hold. When those conditions\ndo not hold, we explain how multiple amplification of infinite order may rule\nout useful estimates.\n