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Pointwise semigroup methods and stability of viscous shock waves

Kevin Zumbrun, Peter Howard

1998
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Indiana University Mathematics Journal
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Considered as rest points of ODE on L p , stationary viscous shock waves present a critical case for which standard semigroup methods do not su ce to determine stability. More precisely, there is no spectral gap between stationary modes and essential spectrum of the linearized operator about the wave, a fact which precludes the usual analysis by decomposition into invariant subspaces. For this reason, there have been until recently no results on shock stability from the semigroup perspective
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... ept in the scalar or totally compressive case Sat , K.2 , resp., each of which can be reduced to the standard semigroup setting by Sattinger's method of weighted norms. We o v ercome this di culty in the general case by the introduction of new, pointwise semigroup techniques, generalizing earlier work of Howard H.1 , Kapitula K.1-2 , and Zeng Ze,LZe . These techniques allow us to do hard" analysis in PDE within the dynamical systems semigroup framework: in particular, to obtain sharp, global pointwise bounds on the Green's function of the linearized operator around the wave, su cient for the analysis of linear and nonlinear stability. The method is general, and should nd applications also in other situations of sensitive stability. Central to our analysis is a notion of e ective" point spectrum which can be extended to regions of essential spectrum. This turns out to be intimately related to the Evans function, a well-known tool for the spectral analysis of traveling waves. Indeed, crucial to our whole analysis is the Gap Lemma" of GZ,KS , a technical result developed originally in the context of Evans function theory. Using these new tools, we can treat general over-and undercompressive, and even strong shock w a v es for systems within the same framework used for standard weak i.e. slowly varying Lax waves. In all cases, we show that stability is determined by the simple and numerically computable condition that the number of zeroes of the Evans function in the right complex half-plane be equal to the dimension of the stationary manifold of nearby traveling wave solutions. Interpreting this criterion in the conservation law setting, we quickly recover all known analytic stability results, obtaining several new results as well. 8 STABILITY OF VISCOUS SHOCK WAVES C1 Re B 0. C2 A real, distinct, nonzero. C3 ess L f Re , I m 2 g f Re , Img,for some 0. Note: unless otherwise speci ed, L refers to L 2 spectrum. All of our linear results will be proved at the level of generality o f C0 C3, the assumptions H0 H4 entering only at the nonlinear level. Conditions C0 C2 follow in straightforward fashion. Condition C3 follows from the fact that, given C0, ess L lies on and to the left of the rightmost boundary of L + L , , where L v := ,A v x + B v xx denote the limiting, constant coe cient operators approached by L as x ! 1 , and by Fourier Transform calculation L ess L = f 2 , ikA , k 2 B : k realg: This standard result may be found in, e.g., He,CH , or see the calculations later in this paper. The nal step is to observe that Re ,ikA ,k 2 B is less than , 1 k 2 by H3, while I m , ikA ,k 2 B is at most of order k for k bounded, k 2 for k large. Note that the tangent manifold, Spanf@ u =@ j g, of f u g at u consists of stationary solutions of 1.6, or equivalently 1:7 Spanf@ u =@ j g K e r L : Just as for the nonlinear equations, this precludes asymptotic stability. Instead, we study linearized orbital stability, de ned analogously to De nition 1.3 as approach to the tangent manifold. De nition 1.5. Fix a norm, kk, and a set A of admissible perturbations. Then u is linearly orbitally stable with respect to A if v; t ! Spanf@ u =@ j g as t ! 1 whenever v; 0 2 A , where v is the solution of 1.6. 1.1.4. The pointwise Green's function method. A fundamental difculty in the study of stability of viscous shock waves is the accumulation at the imaginary axis, C3, of the essential spectrum of the linearized operator L, that is, the lack of a spectral gap between stationary and time-decaying modes of 1.6. From the dynamical systems viewpoint, considering 1.2 as an ODE on an appropriate Banach space, this means that f u g is a nonhyperbolic rest manifold for which standard semigroup methods do not yield stability. Indeed, as is familiar from nite-dimensional ODE, this is a critical case in which one can conclude neither stability nor instability without further investigation. Moreover, stability if it holds is at algebraic rather than exponential rate. Of course the type of nonhyperbolicity considered here, being entirely connected

doi:10.1512/iumj.1998.47.1604
fatcat:ny5dehjxbjb3vdtnzlcvsdxvl4