Stability of undercompressive shock profiles

Abstract

Using a simplified pointwise iteration scheme, we establish nonlinear phase-asymptotic orbital stability of large-amplitude Lax, undercompressive, overcompressive, and mixed underovercompressive type shock profiles of strictly parabolic systems of conservation laws with respect to initial perturbations |u 0 (x)| E 0 (1 + |x|) −3/2 in C 0+α , E 0 sufficiently small, under the necessary conditions of spectral and hyperbolic stability together with transversality of the connecting profile.

Galloping instability of viscous shock waves

“…rectangular) crosssection; "spinning" instabilities for a circular cross-section, with one or more "hot spots", or "combustion heads", moving spirally along the duct. Stability of shocks and detonations may be studied within a unified mathematical framework; see, e.g., [Er1,Er2,Ko1,Ko2,D,BT,FD,LS,T,K,M1,M2,M3,FM,Me,CJLW] in the inviscid case (1.1), (1.5), and [S,Go1,Go2,KM,KMN,MN,L1,L3,GX,SX,GZ,ZH,Br1,Br2,BrZ,BDG,KK,Z1,Z2,Z3,Z4,MaZ2,MaZ3,MaZ4,GMWZ1,GWMZ2,GMWZ3,HZ,BL,LyZ1,LyZ2,JLW,…”

“…For further discussion, see [Z4,HZ]. A related, more subtle point arising also in the Lax shock case is that terms ∼ e −η|y| |K y | appearing in R y bound (3.38) are also not acceptable, since the e −η|y| term is no help in convolution against an L 1 source, and |K y | does not have the sign cancellation of K y .…”

Multidimensional viscous shocks I: Degenerate symmetrizers and long time stability

“…The same Evans assumption has already been shown to imply long time stability of viscous profiles in the 1D case in [KK] for zero mass perturbations and [Z2,MaZ1,MaZ2,MaZ3,MaZ4,MaZ5,Z3,HZ,Ra] for general perturbations, and in the multidimensional case in [Z1, Z3, Z4] (for general perturbations); see also the important groundwork of [GZ, ZH, ZS] and [K1,K2,KS,LZe,H1,H2]. A treatment of the scalar multidimensional case (for which the Evans assumption always holds, by the maximum principle) may be found in [HoZ2,HoZ3].…”

“…The endgame of [Z1] could be described rather as "parabolic": integration on the parabolic contour Γ(ξ ) reveals an additional temporal decay due to diffusion that is essential to the proof of nonlinear stability in dimensions less than or equal to two. (Note: In dimension one, somewhat further care is needed; specifically, translation of the shock must be projected out [Z2,MaZ1,MaZ2,MaZ3,MaZ4,MaZ5,Z3,HZ,Ra]. )…”

Hopf Bifurcation From Viscous Shock Waves

“…It should be possible to establish nonlinear stability using a combination of the approach via pointwise estimates developed by Howard and Zumbrun in [8,21] and our spatial-dynamics technique which can be used to obtain the necessary estimates for the Green's function; this will be pursued elsewhere.…”