Pointwise Green s Function Estimates Toward Stability for Degenerate Viscous Shock Waves

Transcription

1 Communications in Partial Differential Equations, 3: 73, 006 Copyright Taylor & Francis, LLC ISSN print/ online DOI: 0.080/ Pointwise Green s Function Estimates Toward Stability for Degenerate Viscous Shock Waves PETER HOWARD Department of Mathematics, Teas A& M University, College Station, Teas, USA We consider degenerate viscous shock waves arising in systems of two conservation laws, where degeneracy describes viscous shock waves for which the asymptotic entates are sonic to the hyperbolic system the shock speed is equal to one of the characteristic spee. In particular, we develop detailed pointwise estimates on the Green s function associated with the linearized perturbation equation, sufficient for establishing that spectral stability implies nonlinear stability. The analysis of degenerate viscous shock waves involves several new features, such as algebraic nonintegrable convection coefficients, loss of analyticity of the Evans function at the leading eigenvalue, and asymptotic time decay of perturbations intermediate between that of the La case and that of the undercompressive case. eywor Conservation laws; Degenerate shock waves; Stability. Mathematics Subject Classification 35L65; Introduction We consider degenerate viscous shock waves arising in the system: u t + fu u uf u0 u 0. that is, solutions of the form ū st ū st, ū st tr, ū± u ± u± tr, that satisfy the Rankine Hugoniot condition s f ku + u+ f ku u u + k u k k Received April, 004; Accepted August 7, 005 Address correspondence to Peter Howard, Department of Mathematics, Teas A & M University, College Station, TX 77843, USA; phoward@math.tamu.edu 73

2 74 Howard and for which s Spectrumdfu ± a ± k k. Throughout the analysis, we will make the following assumptions on the structure of. and the wave ū: H0 f C ; H La degeneracy Either a <s<a and a+ <s a+ right side degenerate or a s<a and a+ <s<a+ left side degenerate; H First order degeneracy For either case a ± k s right or left side degeneracy, we assume there hol l ± k d fu ± r ± k r± k 0 where l ± k and r± k denote the left and right eigenvectors of dfu ± respectively, and d fu ± denotes the operator d fu ± v v u u f u ± u± v + u u f u ± u± v v + u u f u ± u± v u u f u ± u± v + u u f u ± u± v v + u u f u ± u± v Under assumption H, both ū st and ū st decay to the degenerate side entate with rate st see Howard and Zumbrun, 004. Though H H are stated for clarity in terms of an arbitrary wave speed s, the generality of f allows us to take s 0 without loss of generality by shifting to a moving coordinate frame st and replacing f with f su, which continues to satisfy our only general requirement H0. Throughout the article, then, we will consider a standing wave ū. We note that assumptions H and H describe the most generic degenerate case. In particular, H asserts that there is only degeneracy on one side, and not associated with both characteristics, while H is analogous to the condition for single equations f u ± 0 see Howard, 00a,b. Finally, we mention that our restriction to the case of identity viscosity and two equations follows from technical restrictions in the Evans function analysis of Howard and Zumbrun 004, and that we regard the cases of general viscosity including partial regularization and an arbitrary number of equations as interesting directions for future work. Under the assumption of conditions H0 H, we develop detailed estimates on the Green s function of the linearized perturbation equation sufficient for establishing that spectral stability implies nonlinear stability for degenerate viscous shock waves arising in.. The full development of nonlinear stability is carried out in a companion article, Howard, 005. We remark that the only systems analysis to date in the case of degenerate viscous shock profiles regar the partially regularized p-system, v t u 0 u t pv u which can be reduced through coupling to a scalar analysis and analyzed by energy metho see Nishihara, 995. Our interest in the degenerate case is motivated both by the physical application to detonation waves see Howard, 00a and the references therein and by

3 Pointwise Green s Function Estimates 75 its position as a boundary case between La and undercompressive waves. The critical new feature in the analysis of degenerate waves, necessarily absent in the case of nondegenerate waves for which s Spectrumdfu ±, is the algebraic nonintegrable decay to entate of the degenerate wave see hypothosis H. As a consequence of this slow decay, the convection coefficients for the perturbation equation found by linearizing. about ū are also nonintegrable, and the consequent asymptotic analysis is considerably more complicated than that of the nondegenerate case. More important, the Evans function associated with the linearized operator has terms of the form ln, and hence analyticity is lost at the critical point 0, around which long time behavior is determined. This loss of analyticity has two critical consequences: the study of the point spectrum of the linear operator near 0 is more delicate; and the time decay of the linear semigroup operator e Lt is reduced. While the former of these has been considered in Howard and Zumbrun 004 in the case of conservation laws and in Santede and Scheel 004 in the case of reaction diffusion systems the latter has not to our knowledge been addressed. This, then, is the critical issue of the current analysis. It is well known that solutions ut of., initialized by u0 near a standing wave solution ū, will not generally approach ū, but rather will approach a translate of ū determined by the amount of mass measured by u0ūd carried into the shock as well as the amount carried out to the far field along outgoing characteristics. In our framework, a local tracking function t will serve to approimate the shift of this translate at each time t. Following Howard and Zumbrun 000, we build this shift into our model by defining our perturbation vt as vt ut + t ū. We will say that ū is orbitally stable with respect to some measure if for v0sufficiently small in that measure we have vt 0ast. Substituting vt ut + t ū into., we obtain the perturbation equation v t Lv + Qv + tū + v. where Lv v Av, A dfū, and Qv Ov is a smooth function of v. Restricting the discussion without loss of generality to the case of right-side degeneracy see H, we observe that according to hypotheses H0 H, we have, for A ± lim ± A, that A C and k A A + O k k A A Oe k 0 degenerate side k 0 nondegenerate side for some >0. Integrating., we have after integration by parts on the second integral and observing that e Lt ū ū, vt + Gt yv 0 ydy + tū t + 0 G y t s yqvs y + svs ydy

4 76 Howard where Gt y represents a Green s function for the linear part of.: G t + AG G G0y y I.3 The goal of the current analysis is to develop pointwise estimates on Gt y sufficient for establishing that an iteration on vt will close. The estimates on Gt y will be divided into those terms for which the dependence is eactly ū referred to as the ecited terms, and denoted Et y ū et y and those for which the dependence is not eactly ū. Typically, the ecited terms do not decay in t, and represent mass that accumlates in the shock layer, shifting the shock. Our approach will be to choose our shift t to annihilate this mass, so that we track the shock in time. Following Howard and Zumbrun 000, we write for which we have vt + t + 0 Gt y Gt y +ū et y Gt yv 0 ydy +ū + et yv 0 ydy + tū G y t s yqvs y + svs ydy Choosing t to eliminate the linear ecited terms, we have and consequently vt + t Gt yv 0 ydy + t + 0 et yv 0 ydy.4 G y t s yqvs y + svs ydy We mention that in the iteration on vt, we only require an estimate on t, so in practice we consider the time derivative of.4. Typically, we analyze Gt y through its Laplace transform, G y, which satisfies the ODE t G AG G y I and can be estimated by standard metho. Letting + and + represent the necessarily two linearly independent asymptotically decaying solutions at + of the eigenvalue ODE L.5 L as in. and and similarly the two linearly independent asymptotically decaying solutions at, we write G y as a linear combination, { + G y N y + + N y > y N + y + N + y < y

5 Pointwise Green s Function Estimates 77 where we observe the notation ± ± k N k k N k ± k ± N k N k k ± N k k k ± N k k ± N k Insisting on the continuity of G y across y and a step in G y leading to the Dirac delta, we have + N + + N N + N N + + N N + N + I.6 Equations.6 represent eight equations and eight unknowns, which decouple into two sets of four equations and four unknowns. Solving by Cramer s formula, we have, for eample, N y + det det + + Clearly, then, G y will be well behaved so long as + + W det We define the Evans function as D W0, which in our setting is equivalent to the wedge-product formulations of Aleander et al. 990, Evans 97/975, Gardner and Zumbrun 998, Jones 984, and apitula and Santede 998. More precisely, in the wedge-product formulation, we denote ± ± k k k ± and set Y ± ± ± where denotes a general wedge product on the space of vectors in 4 i.e., denotes an associative, bilinear operation on vectors v 4 for which v v 0. In this case, we obtain through straightforward calculation that D e 0 trydy Y Y +.7 which is the form considered in Aleander et al. 990, Evans 97/975, Gardner and Zumbrun 998, Jones 984, and apitula and Santede 998. As described below, the matri is simply the matri that arises when.5

6 78 Howard is written as a first order system. One advantage of.7 is that the two sicomponent vectors Y ± are the asymptotically decaying solutions of where is defined by Y ± Y ± Y ± ± ± ± ± + ± ± ± ± + ± ± ± ± ± In this way, the formulation.7 may remain valid in cases when our formulation is not. In the current setting, however, the vectors k ± can be distinctly defined, and our formulation is valid. In order to understand the behavior of the Evans function, consider an eigenvector, V, of the linear operator Lv v Av Since V must decay at both ±, it must be a linear combination of +, and +, and also of and. Consequently, these four solutions must be linearly dependent, and their Wronskian must be zero. In general, away from essential spectrum, zeros of the Evans function correspond with eigenvalues of the operator L, an observation that has been made precise in Aleander et al. 990 in the case pertaining to reaction diffusion equations of isolated eigenvalues, and in Zumbrun and Howard 998/00, Gardner and Zumbrun 998 in the case pertaining to conservation laws of nonstandard effective eigenvalues embedded in essential spectrum of L. The latter correspond with resonant poles of L, as eamined in the scalar contet in Pego and Weinstein, 99. In Howard and Zumbrun 004, the authors established that under assumptions H0 H, D can be written as an analytic function plus a small error, D D a + O 3/ ln as 0 where D a O is analytic in a neighborhood of 0. Following Howard and Zumbrun 004, we introduce the following stability condition. D has precisely one zero in Re 0 necessarily at 0 and D a0 0 While condition is generally quite difficult to verify analytically see, for eample, Jones, 984 in the contet of the FitzHugh Nagumo equations, and Dodd, 996; Freistuhler and Szmolyan, 00; Goodman, 986; Humpherys and Zumbrun, 00; awashima and Matsumura, 985; awashima et al., 986; Matsumura and Nishihara, 985, and Plaza and Zumbrun, 004 in the case of conservation laws, it can be checked numerically see Brin, 00; Oh and Zumbrun, 003. A condition that len itself more readily to eact study is the stability inde, typically defined as sgn D a0 sgn lim D

7 Pointwise Green s Function Estimates 79 For +, we have D a, so that in the event that, D must have a positive real root, which guarantees instability. In the case that +, the question of stability remains undecided. It is shown, for eample, in Howard and Zumbrun 004 that the degenerate wave arising in the contet of the fully regularized p-system v t u v u t v + v 3 u satisfies +. Finally, we observe that if stability condition hol, there eists a contour d defined through d k d 0 + id k d k.8 with d 0 and d both positive constants, so that aside from the eigenvalue 0, the point spectrum of L lies entirely to the left of d. We are now in a position to state the main result of the article. Theorem.. Suppose ū is a standing wave solution to. and suppose H0 H hold, as well as stability criterion. Then for some positive constants M,, P +, 0 and, depending only on dfū and the spectrum of L, the Green s function Gt y described through.3 satisfies the following estimates. i y 0: Gt y Ot / e yat a + Ot / e a ya t +ū e t y + O + a 3/ y a t + t / lne + t a G y t y Ot e yat a + Ot e a ya t + O + a 5/ y a t + t / lne + t where a I { a a y a t +ū y e t y I { a a y a t e t y Oe y+at + OI y a t y e t y Ot / e y+at t e t y Ot / e y+at + O +y + a t+t/ 3/ lne + t I y a t + O +y + a t+t/ 3/ lne + t I y a t ii y 0: Gt y Ot / e yat a + Ot / e a ya t + O + a 3/ y a t + t / lne + t a +ū e t y I { a a y a t

8 80 Howard G y t y Ot e yat a + Ot e a ya t +ū y e t y + O + a 5/ y a t + t / lne + t I { a iii 0 < y: a a y a t Gt y Ot / e a + O + a a s a t +ū e + t y + Ot / e a t I { a s a t 3/ + t/ + O + a t+t/ 3/ O +yi { G y t y [ Ot + t /4 Ot +y ] e a where iv y 0 : lne + t I { a y a s a s a t a t a s a t a s a t +ū y e + t y + O + a a s a t 5/ + t/ lne + t I { a y a s + O +a t+t/ 3/ I { a t a s y e + t y Oe a s + t + OI { y e + t y Ot / e + + t e + t y Ot / e + + a s + t a s + Oe y t + Ot / + t / +ye y a s + t 3/ + t/ a s +t lne + t I { + Ot / + t / e y a s + t 3/ + t/ lne + t I { a s a s t t a t Gt y Ot / + t /4 O + e yat I y a t +ū e t y + O +y + a t/ O + e Iy a t

9 Pointwise Green s Function Estimates 8 + Ot / + t /4 lne + to + e yat I y a t + O +y + a t/ lne + to + e Iy a t G y t y Ot + t /4 O + e yat I y a t +ū y e t y v 0 < y : Gt y + O +y + a t3/ O + e Iy a t + Ot + t /4 lne + to + e yat I y a t + O +y + a t3/ lne + to + e Iy a t Ot / O + O +ye y + P + ū Iy 0 t I y 0 t +ū e + t y + Ot / lne + t O + O +ye y + Ot / + t /4 O + e a s + + Ot / + t /4 lne + to + e G y t y Ot O + O +ye y vi 0 < y a s +t I { a s + +ū y e + t y + Ot + t / lne + to + e y + Ot + t / O + O +ye y + Ot lne + to + e y + Ot + t /4 O + e Gt y a s a s +t t I { a s + Ot 3/4 lne + toe y a s + a s +t I { a s t Ot / O + O +ye y + P + ū Iy 0 t I y 0 t +ū e + t y + Ot / lne + t O + e y a s +t + Ot / e + O + a s + t 3/ + t/ lne + t I { a s t t

10 8 Howard + Ot / + t /4 O + e a s + + Ot / + t /4 lne + to + e a s +t I { a s + y + Ot / + t /4 O + e a s +t I { + Ot / + t /4 lne + to + e + + O + + O G y t y a s + t / a s + t / lne + t Ot O + O +ye y +ū y e + t y + O + a s + t + Ot + t /4 O + e a s a s a s +t t a s +t I { O + e I { a s t I { a s t t O + e I { a s +t + Ot e 5/ + t/ lne + t a s + + Ot + t /4 lne + to + e a s +t I { a s + y + Ot + t /4 O + e a s +t I { + Ot + t /4 lne + to + e + + O + + O a s + t 3/ a s + t 3/ lne + t a s a s a s +t t a s +t I { O + e I { a s t I { a s t t O + e I { For 0 y<, the integrals can be replaced by zero, and similarly for. a s a s a s I a s a s t t a s t t t We observe that the etensive number of summan in the cases, y 0 arises from the various interactions between the degenerate and nondegenerate characteristic spee, and also from the two different decay rates + and + lne + t. In practice, most of these terms give a negligible contribution in the nonlinear iteration of Howard 005, but we record them here for completeness.

11 Epressions of the form Pointwise Green s Function Estimates 83 will be fully eplained in the analysis; we point out a s here only that they are clearly necessary inasmuch as the behavior of a s is only understood asymptotically, and the algebraic nature of the approach of a s to a + precludes the possibilty of working as in the nondegenerate case with asymptotic entates. We also mention that the behavior for small t that is, y t for some fied constant is of the form of heat kernels Ot / e y which can be subsumed into the estimates of Theorem.. Of particular interest are the estimates on Gt y with decay rate t /4. In the case 0 < y, the ODE Green s functions associated with degenerate decay in and nondegenerate decay in y takes the form S y O / O e 3s In the case y 0, we can proceed here as though analyzing the heat equation to immediately recover an estimate of the form Ot / O e Iy0 The difficulty arises in the limit as goes to zero, in which case he have an estimate of the form S y O / e 3s The eponent 3s is O and thus near 0 gives slower decay in t than e y. We observe, however, that this reduced t decay occurs only for y a t, and this allows the loss of decay in t to be compensated by decay in y. Regarding the ecited terms, we note the relation et y e t yi y 0 + e + t yi y>0 Also, we observe that in the case 0 < y, the term P + ū I y 0 t arises naturally in the analysis, and is bounded by the epression ū Oe y Lt In the cases y >0, the term that arises naturally in the analysis is and we replace this with P + ū I y 0 t P + ū I y 0 t P + ū I y 0 t + P + ū I y 0 t I y 0 t.9 where the first epression on the right-hand side of.9 is an epression of e + y t and the second appears in G.

12 84 Howard. ODE Estimates In this section we establish the critical estimates on k ±, as well as on a choice of asymptotically growing solutions of.5 and dual solutions to both. Our eigenvalue ODE.5 takes the form v a v a v v v a v a v v. where a kj uj f k ū ū. Writing V v, V v, V 3 v, V 4 v,we have the first order system V V. We will also be interested in the associated integrated equation w a w a w w w a w a w w.3 In particular, we will find it convenient when possible to compute growth and decay solutions of. by computing growth and decay solutions of.3 and computing their derivatives. We stress that from this point of view, no assumptions regarding integrability need be made. It is a question, rather, of scaling. Writing.3 as a first order system with W w, W w, W 3 w, W 4 w, we have W W where a a 0 a a which has four eigenvalues k satisfying a a a + a + 4 with associated eigenvectors a a a + a + 4 P V V V 3 V 4 r r r r r r r 3 r 4

13 Here, a a are the eigenvalues of namely Pointwise Green s Function Estimates 85 a a A a a a tr A tr A 4 det A a tr A + tr A 4 det A tr with associated eigenvectors r k a ka a a kr k a tr. a At +, we have + O, + O, 3 O, and + 4 O, prompting our designation of and 3 as nondegenerate modes, and and 4 as degenerate modes. According to our assumption H and without loss of generality, taking s 0 and right side degeneracy, we have det A + 0 tr A + <0 from which we observe that there eists a constant sufficiently large so that > tr A 4 det A>0 In the analysis that follows, we often proceed in the case, observing that for, all growth and decay solutions can be regarded as O in. We note, prior to the statements of our lemmas, that the asymptotic limits ±of the k represent the growth and decay rates of solutions to both our unintegrated equation. and our integrated equation.3. Setting we have, for sufficiently small, ± k lim k ± Re <0 Re Re 4 +>0 Re Re + <0 Re+ O 3 O 3 O Finally, we mention that throughout the analysis order, terms involving refer to behavior as, while order terms involving refer to behavior as 0. The following lemma is proven in Howard and Zumbrun 004. Lemma.. Under the assumptions of Theorem., there eists some constant r sufficiently small so that for r we have the following estimates on degenerate solutions of the integrated equation.3.

14 86 Howard For 0, k W + k e ū k u + k + E k W + k+ e ū k u + k + ūk ū k u + + E k+ + ūk k ū k u + E k k W + 4k e+ ū k u + k + E 4k W + 4k+ e+ ū k u + k + + ūk ū k u + k where represents min, and + E 4k+ + ūk ū k u + E 4k k E k E 4k O / ln O k E k E 4k O / O k 3 4 Additionally, we have for /, the error estimates E jk E jk 0+ O / O Proof. The only statement in Lemma. not established in Howard and Zumbrun 004 is the / estimate on E jk. Following the proof of Lemma 4. in Howard and Zumbrun 004, we write w from Equation.3 as w ū u + u, for which u e + E u e + E 3 We have, then, the ODE for E, E E E 3. Integrating, we find E e E 0+ 0 e E 3 d from which the estimate is clear. The remaining cases are similar. We remark that the ln behavior in Lemma. appears in the analysis of single equations as well, and appears to be sharp see Howard, 00a; Howard, 00b; Pego and Weinstein, 99; Santede and Scheel, 004. Differentiating the estimates of Lemma., we obtain estimates on growth and decay solutions of the unintegrated equation.. Lemma.. Under the assumptions of Theorem. and for large enough so that assures tr A 4 det A 0, there eists some constant r sufficiently small

15 Pointwise Green s Function Estimates 87 so that for r, we have the following estimates on solutions of the unintegrated equation.. i 0: k e k+ V k+ + Oe k e k V k + Oe where k, and k and V k are eigenvalue eigenvector pairs for the matri lim. ii nondegenerate solutions For : + + tr + e s r + + O + + tr tr + e s r O + + tr + e 3s r + + O + + tr tr + e 3s r O while for 0 each term is O. iii degenerate solutions For 0, k, and : + k e ū k u + k + ūk ū k u + k + E k+ + ūk ū k u + E k k + k e ū k + a k ū E + a k ū E + O / O + OO + k e+ ū k u + k + + ūk ū k u + k + k e+ ū k + a k ū E 4 + a k ū E 4 +O / O + OO where the E kj are as in Lemma.. + E 4k+ + ūk ū k u + E 4k k Proof. The estimates of Lemma. on degenerate solutions follow directly from Lemma. and Equation.3. For the nondegenerate solutions, standard theory suffices. See, for eample, Zumbrun and Howard, 998/00. Lemma.3. The Evans function associated with Equation., defined by D det

16 88 Howard satisfies + D det O 3/ ln a r 0 0 D a + O 3/ ln 0 where ± k ± k ± k tr Proof. Though Lemma.3 has been proven in Howard and Zumbrun 004, we restate a brief account of the argument for later reference. This development follows that of Gardner and Zumbrun 998. Fast decay ODE solutions k ± 0 Oe. The fast decay solutions in this analysis are and +. For the first, proceeding as in Gardner and Zumbrun 998, we integrate on to obtain A A where is analytic in. Similarly, for + + A+ s 4 we have + where + is analytic in. Slow decay ODE solutions + 0 O. The only slow decay solution in this analysis is. Integrating again on, we have A + A r 3 where 3 is analytic in, and r is the eigenvector of A associated with the eigenvalue a. Degenerate decay ODE solutions + 0 ū. ODE solution on + we have by construction, + A+ + tr W + W+ where by Lemma O/ ln. Now, we compute + + D det + + y0 Integrating our degenerate + + det A A A a r + 3 A + 4 y0

17 Pointwise Green s Function Estimates 89 [ ] I det A I + + a r + 3 [ ] + + det + + a r + 3 y0 From this final epression we see immediately that D0 0. The standard approach toward gaining higher order information on the Evans function at 0 involves differentiating this final epression with respect to see for eample, Gardner and Zumbrun, 998. Since in the case of a degenerate wave, the Evans function is not analytic at 0, we cannot follow this approach. As observed in Howard and Zumbrun 004, however, the claim of Lemma.3 can be established directly from our detailed ODE estimates of Lemma.. In addition to the estimates of Lemma., we require estimates on the solutions dual to ± k and ± k. The ODE dual to. takes the form y0 z + a z + a z z z + a z + a z z.4 whose solutions we will denote ± k, ± k. Denoting Z z, Z z, Z 3 z, and Z 4 z, we have the associated first order system Z Z.5 where a a 0 a a We will employ the following lemma from Zumbrun and Howard 998/00. Lemma.4. Suppose V satisfies.. Then Z is a solution of.5 if and only if ZSV constant, where S A I I 0 According to Lemma.4, we can describe dual solutions through the relations + j S+ k + j S+ k 0 j k + j S+ k + j S+ k 0 j k + S+ + S+ + S+ + S+ 3/.6

18 90 Howard and similarly, j S k j S k 0 j k j S k j S k jk j k.7 We remark that the distinguished relations + S+ + S+ 3/ are taken as a result of the coalescence of + and + at 0 see Lemma.iii. Accordingly, we have the following estimates on solutions of.4. Lemma.5. Under the assumptions of Theorem., there eists some constant r sufficiently small so that for r, we have the following estimates on solutions of.4. The order epression O a refers to a term meromorphic in s. i 0: k e k+ Ṽ k+ + Oe k e k Ṽ k + Oe where k, the k are as in Lemma., and Ṽj are the eigenvectors of lim, scaled so that Ṽ j ± V k ± j k Moreover, 3, 4, 3 and 4 are all O a. ii nondegenerate solutions For, k : + k e s O a + O / ln + k + k+ e s O a + O / ln + k e 3s O a + O / ln + k + k+ e 3s O a + O 3/ ln while for each epression is O, ecept + k, which is O. iii degenerate solutions 0: + k e O a + O / O + O / ln where k. + k + k+ e OO + k e O a + O / O + O / ln + k + k+ e OO Proof. The case 0 is nondegenerate, and since the solutions do not coalesce at 0, straightforward see, for eample, Zumbrun and Howard, 998/00. The only new assertion regar the O behavior of y and y. Such

19 Pointwise Green s Function Estimates 9 behavior is most directly observed by choosing the fast growth solution as the derivative of a solution to the integrated equation.3. In this way, we insure by construction that Then, according to.7, we must have A Oe M tr tr where, omitting independent variables for brevity, a ± a ± + ± 3 a ± a ± + ± 4 ± ± a M ± ± a ± + ± 3 a ± a ± + ± 4 ± ± a ± a ± + ± 3 a ± a ± + ± 4 ± ± a ± a ± + ± 3 a ± a ± + ± 4 ± ± Applying Cramer s rule and transposing the matrices, we find 3 det a a + 3 a a + 3 a a + 3 a a + 4 a a + 4 a a + 4 det det a a + 3 O O a a + 4 O O det Oe The calculations for 4, 3, and 4 are similar. For 0, the analysis is considerably more delicate, and we carry out a full analysis for the cases + and + only. For +, from.6 we have the relation Applying Cramer s rule, we find M + tr tr det a + a a + a a + a a + a a + a a + a det

20 9 Howard We observe now that by our construction from the integrated equation.3, we have the reduced form det W + W + 3 W + 4 W + + W + 3 W det Computing directly, we observe that the numerator has the form W W W W + 4 W + W + W W W + + W + 4 For the terms in parentheses, we have cancellation, eemplified by the calculation, + W W + ū u + [ + ū ū u + ū + ū u + + E 4 + ū ū u + E + E 4 ] + E 44 + ū ū u + E 4 + E ū u + [ + E 4 E 44 E 4 + E + E 4 E 4 E E 44 ] O O / + O O ln with similar estimates on the remaining terms. We conclude that the numerator has the form e 3s [ O O 3/ + O O ln ] Computing directly i.e. epanding the determinants, according to the estimates of Lemmas. and., we find det e s+ 3 s ord 3/ ord + ordord 3 Ẽ where by ord we mean strict order, bounded above and below hence a term we can divide by, and by Ẽ we mean Ẽ E 4 + E E 4 E O / ln O Combining these representations, we find that + 3 e s O a + O / ln Integrating + 3 on +, we obtain a similar estimate on +. For, we have + M + tr 0 3/ 0 0 tr

21 Applying Cramer s rule, we find Pointwise Green s Function Estimates 93 + det / a + a a + a a + a det We observe now that by our construction from the integrated equation.3, we have the reduced form det W + + W + 3 W + 4 3/ det Epanding as in the case +, we have det W + W + 3 W + 4 e + s+ 3 O + O / O + O / ln from which our estimate on similar. + is immediate. The remaining cases are Lemma.6. Under the assumptions of Lemma.5, we observe the following critical cancellation estimates, for which, as in Lemma.5, the epression O a refers to terms meromorphic in a neighborhood of 0. i For k, 3/ E 4k 0 E k 0 + m O a + O / ln ii d + + 3/ + e P+ ū + O / ln O + O / O O iii d / + iv d + e O a + O / ln O + O / O O O / e O ln + O / ln O + O / O O

22 94 Howard where m and d + are defined through the relation d + det + 3/ + det 3/ + m for which m O ln. Proof. The cancellation in Lemma.6 arises from the coalescing of the decay and growth degenerate solutions. In the case of nondegenerate waves, independence of these solutions plays a critical role in the determination of the behavior of the scattering coefficients see, for eample, Zumbrun and Howard, 998/00. Here, rather, the scattering coefficients must be understood first so that the etent of the dependence can be observed. We begin by observing that the estimate.8 on the scattering coefficient d + and on m is a direct result of the relationship O/ ln which is immediate from the estimates of Lemma.. In particular, since d + does not depend on, we can evaluate the determinant quotient at any value of, and we evaluate it at 0. Taking k for definiteness in i, we first compute d + + 3/ + 3/ + m + 3/ + 3/ + me ū u + + ū ū u + + E 3 + ū ū u + E 3/ e ū u + In the case, we have + ū ū u + + E 43 + ū ū u + E 4 3/ + m + O ū u + + ū ū u + + E 3 + ū ū u + E 3/ + O ū u + ū + ū u + + E 43 + ū ū u + E 4 O a O +ū E E 4 3/ + m + O / ln O + O / O O.9 According to Lemma.3, spectral stability implies d / + 0 O a + O / ln

23 Pointwise Green s Function Estimates 95 Combining these last two equalities, we conclude i. Recalling from Lemma. the relations E jk E jk 0+ O / O, we have ū E E 4 3/ + m ū E 0 E 4 0 3/ + m + O / O 3/ ū O a + O / ln + O a O from which we conclude ii. In the case, we observe that from which ii follows immediately from.9. For estimate iii, we observe as in the proof of Lemma.5 that det W + + W + 3 W + 4 3/ det / det [ W O] with also det / W + W + W + 3 det / det [ W O] We have, then, d / + 3/ + a + a + det [ + d + 3/ + + d + 3/ + ] + O Combining ii with the estimate det e s+ 3 s ord 3/ ord + ordord 3 Ẽ and repeating the analysis for + and +, we obtain iii.

24 96 Howard Finally, for estimate iv we note that det W + W + 3 W + 4 W + + W + 3 W + 4 3/ det / det [ + W W + W W + + W + 4 W O ] with also det W + W + W / Combining, we have W + W + W + 3 det / det [ + W + + W + W W + + W + W O ] d / + 5/ W + 3 det [ d W + + W + + 3/ + W W + + O] 5/ W + 3 det [ d W + + W + + 3/ + W + + W O] 5/ W + 3 det [ W d + 3/ + + d W + 3/ W O] 5/ W + 3 det [ W d + 3/ + + d W + 3/ W O] from which we have iv. 3. Estimates on G y In this section we develop estimates on G y, first for the critical case of near zero which will correspond with large t behavior.

25 Pointwise Green s Function Estimates 97 Lemma 3. Small G y estimates. Under the assumptions of Theorem., there eists r sufficiently small so that for r, and for some constant sufficiently large, we have the following estimates, for which terms containing O a are meromorphic to the right of d, while the remaining terms are analytic to the right of d and away from the negative real ais. i y 0: G y O a e y + O a e 3 y + O / ln e 3 y +ū O a + O / ln e y y G y O a e y + O a e 3 y + O a e y ii y 0: + O 3/ ln e 3 y +ū O a + O / ln e y G y O a e 3 y + O a e 3 y + O / ln e 3 y +ū O a + O / ln e y y G y O a e 3 y + O a e 3 y + O a e 4 y iii 0 < y: + O 3/ ln e 3 y +ū O a + O / ln e y G y O a e 3 3s + O / ln e 3 3s + Oe 3 y +ū O a + O / ln +ū O a + O / ln e y y G y O a e 3 3s + O 3/ ln e 3 3s iv y 0 < : e 3s + OOye 3 y +ū O a + O / ln e 3s +ū Oye y G y O / O e y + O / ln O e y +ū O a e y y G y O / O e y + O / ln e y v 0 < y : G y +ū O a e y e y [ O / O Oy +ū O a + O / ln O Oy ]

26 98 Howard + e [ y 3s O / O + O O / ln ] + O a e y s +ū O a e 3s y G y e [ y O Oy + O / ln O + O / O Oy + O ln O ] + O / O e 3s +ū O a e 3s + O a e y s + O / ln e y s vi 0 < y: G y e [ y O / O Oy +ū O a O + O / ln O ] + e [ y 3s O a + O / ln ] + e [ y 3s O / O + O O / ln ] +ū O a e 3s y G y e y O Oy + O a e 3s + O / O e 3s +ū O a e 3s + O 3/ ln e 3s For 0 y, the epressions e 3s can be replaced by O and similarly for. Remarks on Lemma 3.. The fundamentally new aspect of the estimates of Lemma 3. with respect to analogous results for nondegenerate waves is the loss of analyticity in a neighborhood of 0. On the degenerate side >0, our linear equation behaves like a heat equation, and we epect such loss of analyticity, at least with terms of the form. Even on the nondegenerate side, however, the Green s function is constructed from solutions that decay at both ± and our epansion coefficients carry and ln behavior. Proof of Lemma 3. Following Zumbrun and Howard 998/00, we establish estimates on G y through the useful representation { G G y + M + y > y G G y M + y < y where ± ± ± ± ± denote solutions to the eigenvalue ODE., while ± ± ± ± ±

27 Pointwise Green s Function Estimates 99 denote solutions to the dual ODE.4. According to Zumbrun and Howard 998/00 and the metho described there, we have the additional representations, { G G y 0 + y y Sy y 3. G G y y y Sy y and 0 S G G y + + y G G y y S + 0 y y 3. In order to see the validity of these relationships, we briefly recall the development of Zumbrun and Howard 998/00. Observing the jump condition, [ ] G G y 0 I S I A we have G G y + y M + + y y M + y Sy or in matri form M + + y 0 y y 0 M + y Sy from which we conclude the critical relation M y y Sy y 0 M + y In order to establish 3., we rewrite this last relationship as M + + y y Sy 0 y 0 M + y and for >y multiply on both sides by + 0 and for <y multiply on both sides by 0 y. The case of 3. is similar. Case i y 0. In the case y 0, we compare representations G G y k d kj j y + k e kj j y G G y jk jk

28 00 Howard and G G y G G y S + y 0 j S E + j j y to obtain the relation k d kj + k e kj S E + j 3.3 k k Multiplying 3.3 on the left by l S, and recalling relations.7, we find e lj l E + j for which elj is the jth component of the vector J satisfying J + l Proceeding by Cramer s rule, we have for j e l det l tr tr + { tr + tr det l tr tr + tr + tr 0 l and similarly for j, so that elj j l, where j l represents the ronecker delta. Alternatively, multiplying 3.3 on the left by l S we find d lj l E + j Proceeding again by Cramer s rule and observing the estimates of Lemma.5 O and O, we find the estimates d O a + O 3/ ln, d O a + O / ln, d O a + O / ln, and d O a + O / ln. Collecting these observations, we have y Oa e y y O a e y y Oa e y y O a e y

29 Pointwise Green s Function Estimates 0 d y Oa + O 3/ ln e 3 y d y O a + O 3/ ln e 3 y d y Oa + O / ln e 3 y d y O a + O 3/ ln e 3 y d y Oa + O / ln e 4 y d y O a + O / ln e 4 y d y Oa + O / ln e 4 y d y O a + O / ln e 4 y Finally, observing that 0 does not decay as, we conclude that 0 Pū for some projection constant P, from which the estimates of Case i are immediate. Case ii y 0. The analysis of Case ii is almost identical to the analysis of Case i. The only new terms are and, for which we find y O ae 3 y y O a e 3 y y Oa e 4 y y O a e 4 y Case iii 0 < y. In the case 0 < y, we compare the representations G G y k d kj + j y 3.4 G G y jk and G G y 0 + y y Sy G G y k k Etr +k + y y Sy to obtain the relation d kj + j j y Etr +k + y y Sy We observe that the scattering coefficients d kj here are not necessarily the same as the coefficients dkj from Cases i and ii. Multiplying this last equation on

30 0 Howard the right by Sy + l y, wefind d kl + l y Sy+ l y Etr +k + y y + l y Proceeding as in Case i and the proof of Lemma.3, we find d det+ + + det + + O a + O / ln d det+ + + det + + O a + O / ln On the other hand, for d and d we have, due to our scaling.6, d 3/ det+ + + det + + d 3/ det+ + + det + + In this case, we observe that the fast growth solution + can without loss of generality be constructed by determining the fast growth solution to the integrated equation.3 and computing its derivative. Constructed in this way, + clearly satisfies + A according to which + det det Proceeding as in Lemma.3 and observing cancellation between the degenerate solutions + and + as in Lemma.6, we conclude and similarly, d O a + O / ln d O a + O / ln Gathering these observations, we have the estimates, d + y d + y d + y Oe 3 y d + y OOye 3 y Oa + O / ln e 3 3s O a + O 3/ ln e 3 3s

31 Pointwise Green s Function Estimates 03 d + y Oa + O / ln e 4 d + y O a + O / ln e 4 3s d + y Oa + O / ln e 4 y d + y Oye 4 y 3s Case iv y 0 <. In the case y 0 <, we compare representations G G y + k d kj j y G G y jk and G G y y y Sy G G y k + k Etr k + y y Sy to obtain the relation d kj j j y Etr k + y y Sy 3.5 Multiplying Equation 3.5 by Sy l y on the right, we find d kl E tr k + y y l y Proceeding as in Case iii, we determine d O a + O / ln, d O a + O / ln, and d O a + O / ln. Moreover, for the critical term d, we have d det + det + + According to our scaling, + 0 and 0 are both proportional to ū, and the numerator vanishes at 0, giving d O / ln + O ln Collecting these observations, we find + d y Oa + O / ln e s y + d y O a + O / ln e s y + d y O / ln + O ln e + d y O a + O / ln e s y s y

32 04 Howard + d + d + d y O / O + O e y y O / O + O e y y + d y [ Oa O + O / O + O / ln O ] e y [ O / O + O ] e y Case v 0 < y. In the case 0 < y, we compare representations G G y + k d+ kj + j y + + k e+ kj + j y 3.6 G G y jk jk and G G y y y Sy G G y k + k Etr k + y y Sy 3.7 to obtain the relation d + kj + j y + e + kj + j j j y Etr k + y y Sy 3.8 Multiplying Equation 3.8 by Sy + l y on the right and employing relations.6, we obtain e + kl + l y Sy+ l In the case k l, we have y Etr k + y y + l y 3/ e y y + y Observing that J + y y + y is a matri equation, we apply Cramer s rule to determine e + 3/ J det + 3/ + det 3/ + + Similarly, e + e+ 0, and e+. We analyze the d+ kj similarly, arriving at d + O/ ln, d + O/ ln, d + O a + O / ln, and d + 3/ + m, where m O ln. This last decomposition is as in the case of Lemma.6.

33 Pointwise Green s Function Estimates 05 Combining these observations and epanding G y in detail, we have for 0 y G G y + d+ + y + + d+ + y G G y + + d+ + y + + d+ + y y + 3/ + + y Computing directly from Lemma. and from Lemma.6, we have + d+ + y O / ln e s 3s + d+ + y O/ ln e s 3s + d+ + y O / ln + OyO ln e s y + d+ + y O/ ln Oye s y + + y O ae y s + O / ln e y s + + y O ae y s + O / ln e y s + d+ + y O a O + O / O + O O / ln e 3s + d+ + y O + O / ln + O / O e 3s and finally, according to Lemma.6, + d + + y + 3/ + y [ e y O / O Oy + P+ ū ] + O / ln O Oy + d y + 3/ y e [ y O Oy + O / ln O + O / O Oy + O ln O ] Case vi 0 < y. The case 0 < y is similar to Case v, with only two new terms, namely, + + y Oe 3s + O / ln e 3s + + y O ae 3s + O / ln e 3s

34 06 Howard and, according to Lemma.6, + d 3/ + + y e [ y O / O Oy + O a O + O / ln O ] + d 3/ + + y e y O Oy Lemma 3. Large G y estimates. For bounded to the right of d and for M, some constant M sufficiently large, there eist constants C, >0 so that G y C / e / / y y G y Ce / / y Proof. The large behavior of G y depen on viscosity rather than convection and can be developed eactly as in the nondegenerate case. See in particular Zumbrun and Howard 998/00, p Lemma 3.3 Medium G y estimates. For bounded to the right of d and for r R, any fied constants R>r>0, there eists C>0 so that G y C y G y C Proof. The medium estimates follow from decay properties rather than decay rates and can be developed eactly as in the nondegenerate case. See in particular Zumbrun and Howard 998/00, p Estimates on Gt y We now employ the estimates of Lemmas 3., 3., and 3.3 to derive estimates on the Green s function Gt y through the inverse Laplace transform representation Gt y e t G i yd where the contour of integration must encircle the poles of G which occur at point spectrum for the operator L. Before beginning the detailed proof of Theorem., we give a brief overview of the approach taken and set some notation. The analysis is divided into two cases, y t and y t, where is a constant chosen sufficiently large in the analysis. Generally speaking, in the case y t, we use the estimates of Lemma 3., while in the case y t, we use the estimates of Lemma 3., though in all cases contours must eventually involve large estimates. In the cases for which we use Lemma 3., the estimates on G y are divided into a number of terms that can be integrated separately against e t. For every such term, the contour of integration,, will be chosen to depend on t and y. In the

35 Pointwise Green s Function Estimates 07 Figure. Contours in the case of analyticity at 0. event that y t, it will typically be advantageous to take a parabolic contour that crosses the real ais far to the right of the imaginary ais, while for y t, it will typically be advantageous to take a contour to the left of the imaginary ais see Figure. In either case, we only follow our contour of choice until it strikes the contour d, defined in.8, which aside from the point 0 lies to the right of the point spectrum of L. Throughout the analysis, then, for chosen contour, we will use the notation to indicate the truncated portion of the contour that stops at d. While there are a great many terms in G y to analyze, the analysis of several is similar. Two particularly important terms are the scattering and ecited terms from the case 0 y y t, S y O / O Oye y E y ū O a e y and Though several estimates will be more technical than these, the fundamental ideas are all contained here. For the first we take the heat-equation-like contour defined through k y Lt + ik for which k y L t y + ik k y d i k dk Lt Lt

36 08 Howard Figure. Contours in the case of a branch along Re <0. where L is chosen large enough so that for k sufficiently small, this contour remains in a small neighborhood of the origin. We follow this contour until it strikes d, and then follow d out to the point at see Figure. Observe in particular that though our contour of choice in this case always crosses the real ais to the right of the imaginary ais, it moves rapidly into essential spectrum. Letting ±k represent the values of k for which we strike d, we have e t S yd O Oy +k k +k O Oy e y k e kt Ok / e ky d L t tk t y Lt Ot / O Oye y i y Lt y Lt k + ik dk Finally, along d, we observe that Re 0 and Ok / O, so that our integral decays at eponential rate in time, e t, >0. Since y t, we have e t y t e t which lea to our final estimate on this term e y t Ot / O Oye y

37 Pointwise Green s Function Estimates 09 For E y, we again begin along the contour defined through k y + ik, along which we have Lt ū O a e t y +k d ū e y i y k k t Lt k y + ik y k L t t dk O t O Ot / e y y In the event that y 0 t, we observe that t / y 0, so that we have an estimate by recalling that we are considering the case 0 y Ot / O O ye y Ot / O Oye y In the event that y 0 t, we proceed as in Howard 00a,b and divide the integrand into an analytic term and an error, as P + ū e i t y d P + ū e i t d + i P + ū e t e y d 4. Here, we have observed that for O a meromorphic in in a neighborhood of the origin, we have whence O a P + + O a E y ū O a e y P +ū e y + O e y the second of which can be subsumed into S y. For the first integral on the righthand side of Equation 4., we employ analyticity of numerator and denominator to proceed similarly as in the nondegenerate analysis of Zumbrun and Howard 998/00 and shift our contour to the left of the imaginary ais, using Cauchy s integral formula to compute the residue. Our estimate on this term becomes P + ū I y 0 t For the second integral, we first consider the strip of over which y, for which we have ū e t e y d O e t O y d O Oy O / e t d O OyOt / e y Lt

38 0 Howard where we have made use of the observation that for y 0 t, Ce y Lt. On the other hand, for y >, we have / O y and consequently the estimate ū e t e y d O e t O / O yd as above. Our final estimate becomes Ot / O Oye y + P + ū I y 0 t Note in particular that in this analysis we have determined critical tracking information regarding how rapidly mass in the far field contributes to a shift of the shock wave. According to our choice of shift t in Equation.4, we will have a linear term of the form 0 t t P + v 0 ydy 0 for which we observe that the window of mass that has accumulated in the shock layer at time t is 0 0 t. Proof of Theorem. In the small time case y t, some sufficiently large, we can proceed via the large estimates of Lemma 3., which are eactly the same estimates as in Zumbrun and Howard 998/00. In this case, we take the contour defined through y k + ik t from which an analysis precisely as that of the heat kernel yiel estimates of the form Gt y Ct / e y 4t which can be subsumed into the estimates of Theorem.. For the case y t, we proceed in a number of subcases. Case i y 0. According to Lemma 3., we have five integrals to evaluate in the case y 0, beginning with the integran O a e y, O a e 3 y, and ū O a e y Each of these arises in the case of nondegenerate waves and can be analyzed as in Zumbrun and Howard 998/00. Summarizing, we have O a e t+ y d Ot / e yat a O a e t+ 3 y d Ot / e a ya t ū O a e t y d ū Oe y+a t + OI y a t 4.

39 Pointwise Green s Function Estimates For the term O / ln e 3 y, we must alter the analysis employed above so as to avoid, where we lose analyticity. Epanding and 3, we have a a a + a a a + O 3 Recall a 3 > 0 + a a + O 3 Recall a 3 < 0 Our principal contour will be chosen as in Zumbrun and Howard 998/00 by the relation + y k + a a a y k 3 a 3 + y a a R + a y R 3 a + i + y k 3 a a where R is the critical point where the contour crosses the real ais. Epanding k R + k + k +, we immediately find i + O R and y a 3 a 3 a + y a + O R for which <0, some fied. We choose R optimally by minimizing the eponent function g R R t + a + y a R + a 3 y a 3 R for R p a a y a t t p a t a y 0 a 3 t Note in particular that our contour of choice is entirely determined by our choice of R. Following Zumbrun and Howard 998/00 in the case 0 that is, for p a y a a t, we take + M p R p t / t / p 0 p t / to obtain an estimate by a Ot 3/4 ln te a ya t

40 Howard which can be subsumed into estimates 4.. The difficulty arises in the case < 0 p that is, a y a a t, when the nondegenerate analysis would select R < 0, according to R p t / p t / p 0 p t / which is precluded by our branch along. At this point, we arrive at the primary new feature of the degenerate-wave contour analysis. The idea in the nondegenerate case was, for t y, to take contours that remain entirely in the negative real half-plane and thus take advantage of eponential time decay, which dominates the eponential growth in and y. In the degenerate case, we cannot avoid passing our contours through the positive real ais, and our new approach will be to move our contours quickly into the negative real half-plane. We first observe that we can integrate along essential spectrum to obtain an estimate ess Either Re 0orRe 3 0 Ot 3/4 ln ti { a a y a t Alternatively, we can move more rapidly into essential spectrum by taking the heat-equation-like contour k t + ik which we denote D, until it strikes the non-degenerate contour described above. Along D we have k t + ikt k d it kdk and Re k t 4 + k 4 6k t so that O / ln e t+ 3 y d C +k it kt + iklnt + ik D k e k t a + y a +k 4 a y 3 a 3 dk We observe here that for k sufficiently large, we cannot get a good estimate along this contour. The inde k, then, where we cross the nondegenerate contour, is

41 Pointwise Green s Function Estimates 3 critical. Indeing our nondegenerate contour by l, we have l R + il + O R + y a 3 a 3 a for which the intersection with D occurs for + y a l + O R + Ol 3 k t R + Ot for which we recall that in this case R < 0. In the first case, t / 0, we p choose R t /, for which for t we have k Ct /4. In this case, since a y a a t, the eponent k4 y a 3 a is bounded, and we can integrate 3 over k Ct /4 to obtain an estimate by In the case t /, we choose p O a 3/ y a a t ln t I { a a y a t R p t a + y a y a 3 a 3 for which our degenerate contour intersects with our nondegenerate contour for For this range of k, we have k t a k t a + y + k 4 a a 3 + y a y a 3 a 3 y a 3 + Ot k y a 3 a 3 k t a for which we can integrate over k as above. In the final case, that Re k t a + y + k 4 a a 3 y a 3 t a + y a y k a 3 a 3 + y a k y a 3 a 3 + Ot, we observe p k p for which we have decay as in the previous cases for k. We can choose d 0 for the contour d defined in.8 sufficiently small so that we strike d for k. The nondegenerate analysis of Zumbrun and Howard 998/00 applies along d, and we obtain an estimate that can be subsumed into those above.

42 4 Howard The remaining term in this case, O / ln e y, can be analyzed as in the case O / ln e 3 y with set to zero. The estimates on G y t y follow similarly. Case ii y 0. According to Lemma 3., we have five integran to evaluate in the case y 0, though the only integrand not eamined in the analysis of Case i is O a e 3 y, which arises in the nondegenerate case and has been analyzed in Zumbrun and Howard 998/00, with an estimate by Ot / e yat Estimates on G y t y in this case follow similarly. Case iii 0 < y. According to Lemma 3., we have seven integrals to evaluate in the case 0 < y, beginning with the integrand O a e 3 3s. Since 3 is analytic in in a neighborhood of 0, this integral can be analyzed similarly as in the nondegenerate case. We must, however, keep track of the y-dependence in 3 through the Taylor epansion 3 s a s + a s 3 + O3 for which we take y. In the event y, we can subsume y behavior into O a and proceed with an estimate for the case y 0. Consequently, we choose our contour k through a for which + a + a s k + a s R + a 3 a 3 a s 3 a s 3 k R + ik a k R + ik + O R + yk + O R + Ok 3 + a s where y a 3 a + a s 3 a s 0 < 0 Similarly as in the analysis of Case i, we choose our principal value of R minimize g R R t + + a a s R + a 3 a s 3 R to

43 Pointwise Green s Function Estimates 5 so that R p a a s a t t p a y a t a s 3 We can now proceed as in Case i with ± ± ±M p R p t/ p ±t / 0 ±t/ p The final estimate becomes Ot / e a a s a t For the integrand ū O a e 3s, we can proceed again as in the case of nondegenerate waves, with 3 treated as above. We find y O a ū e t 3s d ū Oe a s +a t + OI a s t For the integran O / ln e 3 3s and ū O / ln e 3s, we proceed as in the analysis of O / ln e 3 y in Case i. We obtain estimates that can be subsumed into those above, as well as the additional estimate [ ] Ot 3/4 ln t O a a s a t3/ ln t I { a a s a t which we observe for t bounded away from zero can be alternatively recorded in the useful form, O + a a s a t 3/ + t/ lne + t I { a The fundamentally new terms in this case are Oe 3 y and ū O a + O / ln e y a s a t In both cases, we first observe that for 0, we may take advantage of the p observation that along the contour chosen for the integrand O a e 3 3s, and for sufficiently small, we have Re y Re 3s, so that the estimates obtained can be subsumed into those above. For < 0, we begin with the p integrand Oe 3 y, for which we proceed similarly as in the Case i analysis of O / ln e 3 y, by taking the degenerate contour defined through