Long time stability of large amplitude n
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Long time stability of large amplitude n
arXiv:0804.1345vl |math.AP| 8 Apr 2008LONG-TIME STABILITY OF LARGE-AMPLITUDE NONCHARACTERISTIC BOUNDARY LAYERS FORHYPERBOLIC-PARABOLIC SYSTEMSTOAN NGU Long time stability of large amplitude nUYEN AND KEVIN ZUMBRUNAbstract. Extending investigations of Yarahmadian and Zumbrun in the strictly parabolic case, we study time-asymptotic stability of arbitrary (possibly large) amplitude noncharacteristic boundary layers of a clast of hyperbolic-parabolic systems including the Navier Stokes equa Long time stability of large amplitude ntions of compressible gits- anti magnetohydrodynamics, establishing that linear and nonlinear stability arc both equivalent to an Evans function, or gLong time stability of large amplitude n
eneralized spectral stability, condition. The latter is readily checkable numerically, and analytically verifiable in certain favorable cases; in partarXiv:0804.1345vl |math.AP| 8 Apr 2008LONG-TIME STABILITY OF LARGE-AMPLITUDE NONCHARACTERISTIC BOUNDARY LAYERS FORHYPERBOLIC-PARABOLIC SYSTEMSTOAN NGU Long time stability of large amplitude nmics, with general adiabiatic index If > 1. Together with these previous results, our results thus give nonlinear stability of large-amplitude isentropic boundary layers, the first such result for Compressive (•shock-type”) layers in other than the nearly-oonstant case. The analysis, as in the stric Long time stability of large amplitude ntly parabolic case, proceeds by derivation of detailed pointwisc Green function Isrunds, with substantial new technical difficulties associated with tLong time stability of large amplitude n
he more singular, hyperbolic behavior in the high- frequency /short time regime.Contents1.Introduction21.1.Equations and assumptions.31.2.Main resultsarXiv:0804.1345vl |math.AP| 8 Apr 2008LONG-TIME STABILITY OF LARGE-AMPLITUDE NONCHARACTERISTIC BOUNDARY LAYERS FORHYPERBOLIC-PARABOLIC SYSTEMSTOAN NGU Long time stability of large amplitude n2.3.High frequency estimates182.4.Low frequency estimates273.Pointwise bounds on Green functionG(x,t;y)294.Energy estimates354.1.Energy estimate I354.2.Energy estimate II50Date: Last Updated: April 5. 2008.This work was supported in part by the National Science Foundation award number DMS-0300487.ht Long time stability of large amplitude ntps://khothuvien.cori!STABILITY OF BOUNDARY LAYERS31.1.Equations and assumptions. We consider the general hyperbolic-parabolic system of conservationLong time stability of large amplitude n
laws (2) in conserved variable u. withử=@’ B=(°i £)’ ^>^>0,ũ € R. and V € Rn~l, where, here and elsewhere, Ơ denotes spectrum of a linearized operatorarXiv:0804.1345vl |math.AP| 8 Apr 2008LONG-TIME STABILITY OF LARGE-AMPLITUDE NONCHARACTERISTIC BOUNDARY LAYERS FORHYPERBOLIC-PARABOLIC SYSTEMSTOAN NGU Long time stability of large amplitude nts of a single scalar equation. As in [MaZ3], the results extend in straightforward fashion to the case ỏ G Rfc, A > 1. with o(An) strictly positive or strictly negative.Following [MaZ4. Z3], we assume that equations (2) can Ih’ written, alternatively, after a triangular change of coordinates-3H’ := Long time stability of large amplitude n Ịỹ(ữ) = fv) ,' '\ w (u, V) Jin the quasilinear, partially symmetric hyperbolic-parabolic form-4ðlỹỂ + Ã1K = (BWX)Z + G,where, defining H'+ := W'((7+Long time stability of large amplitude n
),(Al) Ã(lĩ'+), Ã0,Ô are symmetric, A0 block diagonal, Ã" > #0 > 0.(A2) no eigenvector of Ẩ(À°)~1(Ù,\) lies in the kernel of Ồ(À")“1(H''+),(A3) B = QarXiv:0804.1345vl |math.AP| 8 Apr 2008LONG-TIME STABILITY OF LARGE-AMPLITUDE NONCHARACTERISTIC BOUNDARY LAYERS FORHYPERBOLIC-PARABOLIC SYSTEMSTOAN NGU Long time stability of large amplitude n. A,B,W(-),»(-,-) 6 c1.(Hl) A11 (scalar) Is either strictly positive or strictly negative, that is, either Ã11 > ỚỊ > 0. or A11 < -0\ < 0. (We shall call these cases the inflow case or the outflow case, correspondingly.) Long time stability of large amplitude narXiv:0804.1345vl |math.AP| 8 Apr 2008LONG-TIME STABILITY OF LARGE-AMPLITUDE NONCHARACTERISTIC BOUNDARY LAYERS FORHYPERBOLIC-PARABOLIC SYSTEMSTOAN NGUGọi ngay
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