Decay estimates for the solutions of the system of crystal acoustics for ...
数理解析研究所講究録 1412 巻 2005 年 1-12
Decay estimates for the solutions of the system of crystal acoustics for cubic crystals. Otto Liess University of Bologna Department of Mathematics Italy 1. The main aim of this report is to study decay estimates for global solutions of the system of linear crystal elasticity in three space dimensions for cubic crystals in the nearly isotropic case. In order to state the problem, we recall that the system of linear elasticity is of form
.
$\frac{\partial^{2}u_{i}}{\partial t^{2}}=.\sum_{\wedge---}^{3}$
$c_{ijpq^{\frac{\partial^{2}u_{p}}{\partial x_{j}\partial x_{q}}}}$
(1)
, $i=1,$ 2, 3
will be real constants (called “stiffness” constants) and depend For crystals, the on the crystal under consideration. At a first glance the system thus apparently have to satisfy the , but since the depends on the 81 $(=3^{4})$ constants conditions , , , , $c_{ijpq}=c_{jipq}=c_{ijqp}=c_{pqij}$ , $c_{ejpq}$
$c_{ijpq}$
$c_{ijpq}$
$\forall i$
$\forall j$
$\forall p$
$\forall q$
the number of “essential” constants is at most 21. Additional restrictions will come from the fact that the system has to be hyperbolic (the condition is that the form $\sum 3$
$\sigma=(\sigma_{ij})arrow$
$c_{ijpq}\sigma_{ij}\sigma_{pq}$
,
: ,$p,q=1$
$i,$
, $li$ , ) and must be positive definite on symmetric tensors, i.e., when the number of essential constants will decrease further when the crystal under consideration has additional symmetries. Thus, when the crystal is cubic, as we will suppose in this report, then the number of essential constants is 3. (Cf. [13].) $\sigma_{ij}=\sigma_{j:}$
$\forall j$
With the system (1) we now associate the initial conditions: $u_{i}(0, x)=f_{i}(x)$
,
$\frac{\partial u_{i}(0,x)}{\partial t}=g_{i}(x)$
,
$x\in R^{3}$
, $i=1,$ 2, 3.
(2)
Here we shall assume that the functions $fiy$ , $i=1,2,3$ , are functions on and have compact support. Since this problem is hyperbolic (and with constant . It is moreover standard to coefficients) it clearly admits global solutions on observe that the functions $xarrow u_{i}(t, x)$ are compactly supported in for any fixed , so it will in particular make sense to consider partial Fourier transforms in , with a parameter. We also recall that the characteristic variety of the system in is (1) is by definition $\{(\tau, \xi);\tau\in R, \xi\in R^{3}, P(\tau,\xi)=\det A(\tau, \xi)=0\}$, where the matrix $R^{3}$
$C^{\infty}-$
$g_{i}$
$R^{4}$
$x$
$t$
$x$
$t$
$A$
$A( \tau, \xi)=(\tau^{2}-\sum p,qc_{1p1q}\xi_{p}\xi_{q}-\sum_{p,q}-\sum_{p,q}c_{3p1q}c_{2p1q}\xi_{p}\xi_{q}\xi_{p}\xi_{q}\tau^{2}-\sum_{-}p,qc_{2p2q}\xi_{p}\xi_{q}-\sum_{p,q}\mathrm{c}_{3p2q}\xi_{p}\xi_{q}\sum_{p,q}c_{1p2q}\xi_{p}\xi_{q}$
$\tau^{2}-\sum p,qc_{3p3q}\xi_{p}\xi_{q}--\sum c_{1p3q}\xi_{p}\xi_{q}\sum_{p,q}^{p,q}c_{2p3q}\xi_{p}\xi_{q})$
(“ $\det A$ ” is the determinant of and thus $P$ is the characteristic polynomial of the system.) The polynomial $P$ is immediately seen to be homogeneous and of degree $=0$ six. “Hyperbolicity” then implies that for every the equation has 6 real roots if multiplicities are counted. We also note right away that while in the general theory of partial differential equations tradition has it to work directly with the characteristic surface, in elasticity theory it seems more natural to work in terms of the s0-called “slowness surface , which, by definition, is the surface $\mathit{5}=\{\xi\in R^{3}; P(1, \xi)=0\}$ . It is useful to write the slowness surface in what is called “Kelvin’s” form. We do so for the particular case of cubic crystals, when Kelvin’s form is (cf [4]): $A$
$\xi$
$\in R^{3}$
$P(\tau, \xi)$
$\mathrm{r}$
(3)
$\frac{b\xi_{1}^{2}}{1-c|\xi|^{2}+(b-a)\xi_{1}^{2}}+\frac{b\xi_{2}^{2}}{1-c|\xi|^{2}+(b-a)\xi_{2}^{2}}+\frac{b\xi_{3}^{2}}{1-c|\xi|^{2}+(b-a)\xi_{3}^{2}}=1,$
which we shall also write as
$G(\xi)=0,$
where .
$G( \xi)=\sum_{i=1}^{3}b\xi_{i}^{2}(1-c|\xi|^{2}+(b-a)\xi_{i+1}^{2})(1-c|\xi|^{2}+(b-a)\xi_{i+2}^{2})-\prod_{j=1}^{3}(1-c|\xi|^{2}+(b-a)\xi_{j}^{2})$
(4) Here the , , are real constants which can be calculated in terms of the 3 “essential” stiffness constants of a cubic crystal. (Cf. [4] and [11].) The fact that (4) is the slowness surface of a cubic crystal gives some restrictions on the , , . We mention here the following ones calculated in [11]: $c>0,3c-b+a>0$ , $a\neq 0,$ a-l- $c>0.$ As in [4] we shall also assume that $b>0.$ The quantity $d=b-a$ is a measure of the “anisotropy” of the crystal and in a number of arguments we shall have to assume that is small. We shall say then that we are in the “nearly isotropic” case. When vanishes, the equation $G(\xi)=0$ reduces to $a$
$b$
$c$
$a$
$b$
$c$
$d$
$d$
$(1-c|\xi|^{2})^{2}(1-(c+b)|\xi|^{2})=0.$
(5)
The slowness surface is thus the union of two spheres, one of which being double. This is a manifestation of the fact that in the isotropic case every solution of the
system (1) is a sum of a “transversal wave” and a “longitudinal wave” and that the components of these waves satisfy the classical scalar wave equation (with wave speed which depends on the type of the wave). In the case of effectively anisotropic cubic crystals, the structure of the slowness surface is more complicated and the surface will always have singular points. We shall review the known results about the geometry of the slowness surface for cubic crystals later on. Our interest in the slowness surface comes from the fact that solutions of the system of crystal elasticity can be expressed in terms of Fourier-integrals living on the slowness surface.
Our main estimate for solutions of the Cauchy problem (1), (2) is Theorem 1. Assume that (1) is the system of crystal elasticity for some given cubic crystal Also assume that we are in the nearly isotropic case. Then there is a constant $c’>0$ and an integer such that $k$
$|$
?j
,
$(t, x)| \leq c’(1+|t|)^{-1/2}\sum\sum[|D_{x}^{\alpha}7_{j}|_{|L^{1}(R^{3})}+3|D:g_{j}|_{|L^{1}(R^{3})}]$ $\forall(t, x)\in R^{4}$
,
(6)
$j=1|a|\leq k$
for any solution of the
Cauchy problem (1), (2),
for which the
$f_{j}$
and
$gj$
are
$C^{\infty}(R^{3})$
and have compact supports. (If
$\varphi$
:
$R^{3}arrow C$
is given, we denote by ? $|$
$|_{1}L^{1}(R^{3})$
its
$L^{1}$
-nor, i.e.,
$|\varphi||\mathrm{Z}^{1}$
$(R^{3})=$
$/73|\varphi(\xi)$ $|d\xi.)$
Remark 2. After this authors talk at the conference, I was told by T. Sonobe that aparently the estimate in theorem 1 can be improved, at least in the case $b=$ 2a. .) However, while it still seemed (at the time when the discussion took (Cf. place) difficult to predict what the “optimal” estimate could be, it seems clear that with respect to the case of the wave equation there will be “loss” of decay. $f\mathit{2}\mathit{0}]$
$a$
2. The first equation for which decay properties as in theorem 1 have been studied in a systematic way has been the wave equation: cf. e.g. Segal [15], Strauss [22], von Wahl [24], Klainerman [5], Racke [14], Sideris [16]. In the 3-dimensional case, , when the results obtained in these papers give then a decay of type is typical for the wave equation in two dimensions. whereas a decay of order Similar results have been obtained for a number of related hyperbolic equations, such as the Klein-Gordon equation, or, sometimes, for more general classes of constant coefficient hyperbolic operators or systems, cf. von Wahl [24], Costa [3], Sideris [17], Sideris-Tu [18], Sugimoto [23]. All papers mentioned so far have in common that they refer to the case of operators with characteristics of constant multiplicity. These results also imply that if the crystal under consideration is $c|t|^{-1}$
$ct^{-1/2}$
$tarrow\infty$
. isotropic, then the conclusion in (6) can essentially be improved to $|u(t, x)$ With respect to the isotropic case, in theorem 1 we therefore have a “loss” of decay of one dimension. It is interesting to note then that the same loss of decay does also appear for the system of crystal optics (cf. Liess [10]). Actually, there are a number of analogies which relate crystal elasticity to crystal optics, to the point that in both cases the loss of decay is related to the fact that the corresponding slowness surfaces have singular points and imbedded curves along which the total curvature vanishes. However, the structure of the singular points in the case of crystal elasticity is more complicated and much less seems to be known on the geometry of the slowness surface for crystal elasticity when compared with the situation in crystal optics. $|\leq c|t|^{-1}$
Let us mention here also briefly that decay estimates for solutions of the system of crystal elasticity in a somewhat different setting have been considered before in a number of papers: cf. e.g. Buchwald [1] (and papers cited therein) and Stodt [21]. (The results of Stodt are described also in Racke [14].) None of these papers however addresses the difficulties related to singular points on the slowness surface. 3. The overall strategy to prove a result like theorem 1 is well-understood nowadays and is in particular similar with the one used in the related case of crystal optics in [10]. Starting point is that the solutions of the Cauchy problem (1), (2) admit rather explicit representations in terms of Fourier integrals involving the Fourier transforms of the Cauchy data , , $j=1,2,3$ , of and : cf. Duff, [4]. It is then also no surprize that rather than arriving at an estimate of the form (6), we shall obtain at first the estimate $f_{j}$
$|u(t, x)$
$|\mathrm{S}$
$\hat{g}_{j}$
$f_{j}$
$g_{j}$
$c’(1+|t|)-1/2 \sum_{k=1}^{3}\{\sup_{\xi\in R^{8}},\sum_{|\sigma|\leq d,|\beta|\leq d’}|\xi\sigma| [|’ \mathrm{j}\hat{f}_{k}(\xi)|+|’\xi\hat{g}_{k}(\beta\xi)|]\}$
for some constants
,
$\forall(t, x)\in R^{4}$
,
$\mathrm{e}\mathrm{s}.\mathrm{t}\mathrm{i}-(7)$
, $d’$ . The fact that this estimate implies the stronger mate (6) can be proved as in the corresponding passage on page 65 in [10]. Also cf. Klainerman [8]. An important point is that due to homogeneities in the representation formulas, we can reduce the estimates (7) to estimates of Fourier transforms of densities (with parameters) which live on the slowness surface. It is then possible to localize these estimates on the slowness surface and the contribution of a suitably localized portion near some point $P$ on the slowness surface will depend on the geometric properties of the surface near $P$ . $c’$
,
$d’$
The main steps in the argument are as follows: a) at first we have to study curvature properties and the behaviour near singular points of the slowness surface for cubic crystals, with special emphasis on the nearly isotropic case;
b) the next step is to study the representation formulas for the solutions of the Cauchy problem (1), (2) ;
finally, one can reduce the estimate (7) to an estimate of some parametric integrals living on the slowness surface. We should also mention that once this reduction is done, the continuation of the argument depends on wether we are in the regular part of the slowness surface or wether we are near a singular point. Furthermore, the situation will depend on wether the singular point is “conical” or “uniplanar”. To treat the uniplanar case, one can at a crucial step apply a result on estimates of Fourier integrals defined on surfaces near uniplanar singularities which has been discussed in [12]. The case of conical singularities is to some extend parallel to the uniplanar case, so in the end one is left with a (rather elementary) discussion of the contribution of the regular part of the slowness surface. c)
4. We now review some definitions from classical differential geometry and recall the main result in [12]. To fix the terminology, we consider at first a surface . (In our case we defined in some open neighborhood of some fixed point $\xi^{0}=0.$ We assume shall have $n=3.$ ) To simplify notations, we shall assume that $\varphi(\xi)=0,$ $W$ function for some is given by the equation further that ? defined is a “nod\"e, in the sense that on . We shall assume that . We also assume that coordinates 4 can be found for but and apply the Malgrange preparation theorem to write such that , where A locally near 0 in the form $\psi(\xi)\neq 0$ and denote near 0. For convenience we assume that and Finally, we denote by $=a^{2}(\xi’)-4b(\xi’)$ the local discriminant of ?. by . This is a geometric invariant when $J_{k’}\varphi(0)=0$ for $k’0$ , $\gamma+1>0$ , $\alpha-\beta>0,$ . (These conditions have been introduced in [12]; they are precisely uniplanar singularity with two the ones needed if we are to have a sheets which touch near the singular point only in the singular point itself.) The following result is obtained in [12]: $\alpha$
$2\alpha^{2}>$
$\beta$
$\gamma$
$\mathrm{d}^{2}(\gamma+1)$
$\mathrm{n}\mathrm{o}\mathrm{n}- \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\dot{\mathrm{r}}\mathrm{a}\mathrm{t}\mathrm{e}$
Theorem 5. Assume that
curve
$\beta$
$>0\gamma+1>0,$
$\alpha-\beta>0;2\alpha^{2}>\beta^{\mathit{2}}(\gamma+1)$
$\Gamma^{+}=\{(\xi_{1}, \xi_{2})\in R^{2};\alpha(\xi_{1}^{2}+ 422)+\beta\sqrt{\xi_{1}^{4}+2\gamma\xi_{1}^{2}\xi_{2}^{2}+\xi_{2}^{4}}=1\}$
point Furthermore,
will have no
. Then the
has no
inflection
$\Gamma^{-}=\{(\xi_{1}, \xi_{2})\in R^{2};\alpha(\xi_{1}^{2}+\xi_{2}^{2})-\beta\sqrt{\xi_{1}^{4}+2\gamma\xi_{1}^{2}\xi_{2}^{2}+\xi_{2}^{4}}=1\}$
inflection point, if and only if
$(\alpha-\beta\gamma)(-\alpha\sqrt{2(1+\gamma)}+\beta(3-\gamma))0,$ although this is not strictly speaking necessary in arguments which refer directly to surfaces defined as in (3), and is not assumed in [4]. $\sqrt{c}$
$” d$
$d$
shall denote th slowness surface henceforth by and shall denote by , , , the “outer”, “middle” and “inner” sheet of . More precisely, when we denote by generic points on the unit sphere, then we will have 3 values $\rho>0$ , and belongs to . We denote these values by su that the point respectively, where the numbering is made in such a way that P2{ . In view of the assumption that $b>0,$ we will have for small is smooth and for all . It is then immediate to see that that it is also standard to observe that it must be convex. (Cf. [2], [4], and also [11].) and all three surfaces are spheres. (Also see the coincides with When $d=0$ , discussion above.) As in the case of crystal optics for biaxial crystals, both surfaces 5 and become effectively singular at a finite number of points as soon as $d60$ . can be singular only when they touch and it is and Indeed, it is clear that classical (cf. e.g., [4], [11]), that this happens precisely on the six points they have , where $|=|4$: $|=|43|$ . on the coordinate axes and on the eight points on For later use we mention that $S^{2}$
$S\mathrm{y}1$
$S$
$5.\mathrm{W}\mathrm{e}$
$\mathrm{e}$
$S$
$S^{3}$
$\omega$
$\mathrm{c}\mathrm{h}$
$\rho_{1}(\omega)$
$S$
$\mathrm{k}$
$\mathrm{h}(\omega)$
$0
Decay estimates for the solutions of the system of crystal acoustics for cubic crystals. Otto Liess University of Bologna Department of Mathematics Italy 1. The main aim of this report is to study decay estimates for global solutions of the system of linear crystal elasticity in three space dimensions for cubic crystals in the nearly isotropic case. In order to state the problem, we recall that the system of linear elasticity is of form
.
$\frac{\partial^{2}u_{i}}{\partial t^{2}}=.\sum_{\wedge---}^{3}$
$c_{ijpq^{\frac{\partial^{2}u_{p}}{\partial x_{j}\partial x_{q}}}}$
(1)
, $i=1,$ 2, 3
will be real constants (called “stiffness” constants) and depend For crystals, the on the crystal under consideration. At a first glance the system thus apparently have to satisfy the , but since the depends on the 81 $(=3^{4})$ constants conditions , , , , $c_{ijpq}=c_{jipq}=c_{ijqp}=c_{pqij}$ , $c_{ejpq}$
$c_{ijpq}$
$c_{ijpq}$
$\forall i$
$\forall j$
$\forall p$
$\forall q$
the number of “essential” constants is at most 21. Additional restrictions will come from the fact that the system has to be hyperbolic (the condition is that the form $\sum 3$
$\sigma=(\sigma_{ij})arrow$
$c_{ijpq}\sigma_{ij}\sigma_{pq}$
,
: ,$p,q=1$
$i,$
, $li$ , ) and must be positive definite on symmetric tensors, i.e., when the number of essential constants will decrease further when the crystal under consideration has additional symmetries. Thus, when the crystal is cubic, as we will suppose in this report, then the number of essential constants is 3. (Cf. [13].) $\sigma_{ij}=\sigma_{j:}$
$\forall j$
With the system (1) we now associate the initial conditions: $u_{i}(0, x)=f_{i}(x)$
,
$\frac{\partial u_{i}(0,x)}{\partial t}=g_{i}(x)$
,
$x\in R^{3}$
, $i=1,$ 2, 3.
(2)
Here we shall assume that the functions $fiy$ , $i=1,2,3$ , are functions on and have compact support. Since this problem is hyperbolic (and with constant . It is moreover standard to coefficients) it clearly admits global solutions on observe that the functions $xarrow u_{i}(t, x)$ are compactly supported in for any fixed , so it will in particular make sense to consider partial Fourier transforms in , with a parameter. We also recall that the characteristic variety of the system in is (1) is by definition $\{(\tau, \xi);\tau\in R, \xi\in R^{3}, P(\tau,\xi)=\det A(\tau, \xi)=0\}$, where the matrix $R^{3}$
$C^{\infty}-$
$g_{i}$
$R^{4}$
$x$
$t$
$x$
$t$
$A$
$A( \tau, \xi)=(\tau^{2}-\sum p,qc_{1p1q}\xi_{p}\xi_{q}-\sum_{p,q}-\sum_{p,q}c_{3p1q}c_{2p1q}\xi_{p}\xi_{q}\xi_{p}\xi_{q}\tau^{2}-\sum_{-}p,qc_{2p2q}\xi_{p}\xi_{q}-\sum_{p,q}\mathrm{c}_{3p2q}\xi_{p}\xi_{q}\sum_{p,q}c_{1p2q}\xi_{p}\xi_{q}$
$\tau^{2}-\sum p,qc_{3p3q}\xi_{p}\xi_{q}--\sum c_{1p3q}\xi_{p}\xi_{q}\sum_{p,q}^{p,q}c_{2p3q}\xi_{p}\xi_{q})$
(“ $\det A$ ” is the determinant of and thus $P$ is the characteristic polynomial of the system.) The polynomial $P$ is immediately seen to be homogeneous and of degree $=0$ six. “Hyperbolicity” then implies that for every the equation has 6 real roots if multiplicities are counted. We also note right away that while in the general theory of partial differential equations tradition has it to work directly with the characteristic surface, in elasticity theory it seems more natural to work in terms of the s0-called “slowness surface , which, by definition, is the surface $\mathit{5}=\{\xi\in R^{3}; P(1, \xi)=0\}$ . It is useful to write the slowness surface in what is called “Kelvin’s” form. We do so for the particular case of cubic crystals, when Kelvin’s form is (cf [4]): $A$
$\xi$
$\in R^{3}$
$P(\tau, \xi)$
$\mathrm{r}$
(3)
$\frac{b\xi_{1}^{2}}{1-c|\xi|^{2}+(b-a)\xi_{1}^{2}}+\frac{b\xi_{2}^{2}}{1-c|\xi|^{2}+(b-a)\xi_{2}^{2}}+\frac{b\xi_{3}^{2}}{1-c|\xi|^{2}+(b-a)\xi_{3}^{2}}=1,$
which we shall also write as
$G(\xi)=0,$
where .
$G( \xi)=\sum_{i=1}^{3}b\xi_{i}^{2}(1-c|\xi|^{2}+(b-a)\xi_{i+1}^{2})(1-c|\xi|^{2}+(b-a)\xi_{i+2}^{2})-\prod_{j=1}^{3}(1-c|\xi|^{2}+(b-a)\xi_{j}^{2})$
(4) Here the , , are real constants which can be calculated in terms of the 3 “essential” stiffness constants of a cubic crystal. (Cf. [4] and [11].) The fact that (4) is the slowness surface of a cubic crystal gives some restrictions on the , , . We mention here the following ones calculated in [11]: $c>0,3c-b+a>0$ , $a\neq 0,$ a-l- $c>0.$ As in [4] we shall also assume that $b>0.$ The quantity $d=b-a$ is a measure of the “anisotropy” of the crystal and in a number of arguments we shall have to assume that is small. We shall say then that we are in the “nearly isotropic” case. When vanishes, the equation $G(\xi)=0$ reduces to $a$
$b$
$c$
$a$
$b$
$c$
$d$
$d$
$(1-c|\xi|^{2})^{2}(1-(c+b)|\xi|^{2})=0.$
(5)
The slowness surface is thus the union of two spheres, one of which being double. This is a manifestation of the fact that in the isotropic case every solution of the
system (1) is a sum of a “transversal wave” and a “longitudinal wave” and that the components of these waves satisfy the classical scalar wave equation (with wave speed which depends on the type of the wave). In the case of effectively anisotropic cubic crystals, the structure of the slowness surface is more complicated and the surface will always have singular points. We shall review the known results about the geometry of the slowness surface for cubic crystals later on. Our interest in the slowness surface comes from the fact that solutions of the system of crystal elasticity can be expressed in terms of Fourier-integrals living on the slowness surface.
Our main estimate for solutions of the Cauchy problem (1), (2) is Theorem 1. Assume that (1) is the system of crystal elasticity for some given cubic crystal Also assume that we are in the nearly isotropic case. Then there is a constant $c’>0$ and an integer such that $k$
$|$
?j
,
$(t, x)| \leq c’(1+|t|)^{-1/2}\sum\sum[|D_{x}^{\alpha}7_{j}|_{|L^{1}(R^{3})}+3|D:g_{j}|_{|L^{1}(R^{3})}]$ $\forall(t, x)\in R^{4}$
,
(6)
$j=1|a|\leq k$
for any solution of the
Cauchy problem (1), (2),
for which the
$f_{j}$
and
$gj$
are
$C^{\infty}(R^{3})$
and have compact supports. (If
$\varphi$
:
$R^{3}arrow C$
is given, we denote by ? $|$
$|_{1}L^{1}(R^{3})$
its
$L^{1}$
-nor, i.e.,
$|\varphi||\mathrm{Z}^{1}$
$(R^{3})=$
$/73|\varphi(\xi)$ $|d\xi.)$
Remark 2. After this authors talk at the conference, I was told by T. Sonobe that aparently the estimate in theorem 1 can be improved, at least in the case $b=$ 2a. .) However, while it still seemed (at the time when the discussion took (Cf. place) difficult to predict what the “optimal” estimate could be, it seems clear that with respect to the case of the wave equation there will be “loss” of decay. $f\mathit{2}\mathit{0}]$
$a$
2. The first equation for which decay properties as in theorem 1 have been studied in a systematic way has been the wave equation: cf. e.g. Segal [15], Strauss [22], von Wahl [24], Klainerman [5], Racke [14], Sideris [16]. In the 3-dimensional case, , when the results obtained in these papers give then a decay of type is typical for the wave equation in two dimensions. whereas a decay of order Similar results have been obtained for a number of related hyperbolic equations, such as the Klein-Gordon equation, or, sometimes, for more general classes of constant coefficient hyperbolic operators or systems, cf. von Wahl [24], Costa [3], Sideris [17], Sideris-Tu [18], Sugimoto [23]. All papers mentioned so far have in common that they refer to the case of operators with characteristics of constant multiplicity. These results also imply that if the crystal under consideration is $c|t|^{-1}$
$ct^{-1/2}$
$tarrow\infty$
. isotropic, then the conclusion in (6) can essentially be improved to $|u(t, x)$ With respect to the isotropic case, in theorem 1 we therefore have a “loss” of decay of one dimension. It is interesting to note then that the same loss of decay does also appear for the system of crystal optics (cf. Liess [10]). Actually, there are a number of analogies which relate crystal elasticity to crystal optics, to the point that in both cases the loss of decay is related to the fact that the corresponding slowness surfaces have singular points and imbedded curves along which the total curvature vanishes. However, the structure of the singular points in the case of crystal elasticity is more complicated and much less seems to be known on the geometry of the slowness surface for crystal elasticity when compared with the situation in crystal optics. $|\leq c|t|^{-1}$
Let us mention here also briefly that decay estimates for solutions of the system of crystal elasticity in a somewhat different setting have been considered before in a number of papers: cf. e.g. Buchwald [1] (and papers cited therein) and Stodt [21]. (The results of Stodt are described also in Racke [14].) None of these papers however addresses the difficulties related to singular points on the slowness surface. 3. The overall strategy to prove a result like theorem 1 is well-understood nowadays and is in particular similar with the one used in the related case of crystal optics in [10]. Starting point is that the solutions of the Cauchy problem (1), (2) admit rather explicit representations in terms of Fourier integrals involving the Fourier transforms of the Cauchy data , , $j=1,2,3$ , of and : cf. Duff, [4]. It is then also no surprize that rather than arriving at an estimate of the form (6), we shall obtain at first the estimate $f_{j}$
$|u(t, x)$
$|\mathrm{S}$
$\hat{g}_{j}$
$f_{j}$
$g_{j}$
$c’(1+|t|)-1/2 \sum_{k=1}^{3}\{\sup_{\xi\in R^{8}},\sum_{|\sigma|\leq d,|\beta|\leq d’}|\xi\sigma| [|’ \mathrm{j}\hat{f}_{k}(\xi)|+|’\xi\hat{g}_{k}(\beta\xi)|]\}$
for some constants
,
$\forall(t, x)\in R^{4}$
,
$\mathrm{e}\mathrm{s}.\mathrm{t}\mathrm{i}-(7)$
, $d’$ . The fact that this estimate implies the stronger mate (6) can be proved as in the corresponding passage on page 65 in [10]. Also cf. Klainerman [8]. An important point is that due to homogeneities in the representation formulas, we can reduce the estimates (7) to estimates of Fourier transforms of densities (with parameters) which live on the slowness surface. It is then possible to localize these estimates on the slowness surface and the contribution of a suitably localized portion near some point $P$ on the slowness surface will depend on the geometric properties of the surface near $P$ . $c’$
,
$d’$
The main steps in the argument are as follows: a) at first we have to study curvature properties and the behaviour near singular points of the slowness surface for cubic crystals, with special emphasis on the nearly isotropic case;
b) the next step is to study the representation formulas for the solutions of the Cauchy problem (1), (2) ;
finally, one can reduce the estimate (7) to an estimate of some parametric integrals living on the slowness surface. We should also mention that once this reduction is done, the continuation of the argument depends on wether we are in the regular part of the slowness surface or wether we are near a singular point. Furthermore, the situation will depend on wether the singular point is “conical” or “uniplanar”. To treat the uniplanar case, one can at a crucial step apply a result on estimates of Fourier integrals defined on surfaces near uniplanar singularities which has been discussed in [12]. The case of conical singularities is to some extend parallel to the uniplanar case, so in the end one is left with a (rather elementary) discussion of the contribution of the regular part of the slowness surface. c)
4. We now review some definitions from classical differential geometry and recall the main result in [12]. To fix the terminology, we consider at first a surface . (In our case we defined in some open neighborhood of some fixed point $\xi^{0}=0.$ We assume shall have $n=3.$ ) To simplify notations, we shall assume that $\varphi(\xi)=0,$ $W$ function for some is given by the equation further that ? defined is a “nod\"e, in the sense that on . We shall assume that . We also assume that coordinates 4 can be found for but and apply the Malgrange preparation theorem to write such that , where A locally near 0 in the form $\psi(\xi)\neq 0$ and denote near 0. For convenience we assume that and Finally, we denote by $=a^{2}(\xi’)-4b(\xi’)$ the local discriminant of ?. by . This is a geometric invariant when $J_{k’}\varphi(0)=0$ for $k’0$ , $\gamma+1>0$ , $\alpha-\beta>0,$ . (These conditions have been introduced in [12]; they are precisely uniplanar singularity with two the ones needed if we are to have a sheets which touch near the singular point only in the singular point itself.) The following result is obtained in [12]: $\alpha$
$2\alpha^{2}>$
$\beta$
$\gamma$
$\mathrm{d}^{2}(\gamma+1)$
$\mathrm{n}\mathrm{o}\mathrm{n}- \mathrm{d}\mathrm{e}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{e}\dot{\mathrm{r}}\mathrm{a}\mathrm{t}\mathrm{e}$
Theorem 5. Assume that
curve
$\beta$
$>0\gamma+1>0,$
$\alpha-\beta>0;2\alpha^{2}>\beta^{\mathit{2}}(\gamma+1)$
$\Gamma^{+}=\{(\xi_{1}, \xi_{2})\in R^{2};\alpha(\xi_{1}^{2}+ 422)+\beta\sqrt{\xi_{1}^{4}+2\gamma\xi_{1}^{2}\xi_{2}^{2}+\xi_{2}^{4}}=1\}$
point Furthermore,
will have no
. Then the
has no
inflection
$\Gamma^{-}=\{(\xi_{1}, \xi_{2})\in R^{2};\alpha(\xi_{1}^{2}+\xi_{2}^{2})-\beta\sqrt{\xi_{1}^{4}+2\gamma\xi_{1}^{2}\xi_{2}^{2}+\xi_{2}^{4}}=1\}$
inflection point, if and only if
$(\alpha-\beta\gamma)(-\alpha\sqrt{2(1+\gamma)}+\beta(3-\gamma))0,$ although this is not strictly speaking necessary in arguments which refer directly to surfaces defined as in (3), and is not assumed in [4]. $\sqrt{c}$
$” d$
$d$
shall denote th slowness surface henceforth by and shall denote by , , , the “outer”, “middle” and “inner” sheet of . More precisely, when we denote by generic points on the unit sphere, then we will have 3 values $\rho>0$ , and belongs to . We denote these values by su that the point respectively, where the numbering is made in such a way that P2{ . In view of the assumption that $b>0,$ we will have for small is smooth and for all . It is then immediate to see that that it is also standard to observe that it must be convex. (Cf. [2], [4], and also [11].) and all three surfaces are spheres. (Also see the coincides with When $d=0$ , discussion above.) As in the case of crystal optics for biaxial crystals, both surfaces 5 and become effectively singular at a finite number of points as soon as $d60$ . can be singular only when they touch and it is and Indeed, it is clear that classical (cf. e.g., [4], [11]), that this happens precisely on the six points they have , where $|=|4$: $|=|43|$ . on the coordinate axes and on the eight points on For later use we mention that $S^{2}$
$S\mathrm{y}1$
$S$
$5.\mathrm{W}\mathrm{e}$
$\mathrm{e}$
$S$
$S^{3}$
$\omega$
$\mathrm{c}\mathrm{h}$
$\rho_{1}(\omega)$
$S$
$\mathrm{k}$
$\mathrm{h}(\omega)$
$0
