数理解析研究所講究録 1260 巻 2002 年 24-32

24

Geometric approach to certain systems ofconservation laws*

Mikio TSUJI ( 辻 幹雄 ) \dag er
Department of Mathematics, Kyoto Sangyo University(京都産業大学理学部数学教室)

Abstract
Our problem: Global theory for nonlinear hyperbolic equations and systems.
Difficulty: Appearance of singularities in finite time.
Our aim: Extension of solutions beyond the singularities.
Our method: Resolution of singularities, i.e., analysis in cotangent space.
Conclusion: Some questions on the mathematical theory to fluid mechanics.

1Our program.
In this talk we will consider the Cauchy problem for certain systems of hyperbolic type.
It is well known that the Cauchy problem with smooth data has asmooth solution
in aneighbourhood of the initial curve, and that singularities appear generaly infinite time. But, even if singularities may appear in solutions, physical phenomena canexist with the singularities. Moreover it seems to us that the singularities might cause
various kinds of interesting phenomena. We are interested in the global theory for theabove Cauchy problem. Therefore we would like to extend the solutions beyond their
singularities. Then what happens ?This is the subject of this talk.
Let us consider the following equation:$F(x,$y, z,p, q,r,s,$t)=Ar$ $+Bs+C$ $t+D(rt -s^{2})-E=0$

(1.1)

.

where

$p=\partial z/\partial x,q=\partial z/\partial y,r$$=\partial^{2}z/\partial x^{2},s$$=\partial^{2}z/\partial x\partial y$

, and

$t$

$=\partial^{2}z/\partial y^{2}$. Here weassume that , , , and are real smooth functions of $(x,y, z,p, q)$ .
Before stating our program, we will give some historical comments on the methodof integration of (1.1). It had been first investigated by the french school, especiallyby G. Monge, G. Darboux [2] and E. Goursat $[4, 5]$ . One of their principal methods
$A$

$B$

$C$

$D$

$E$

is how to reduce the solvability of (1.1) to the integration of first order partial differential equations. For this approach they assumed strong conditions on equations.
The first paper which succeeded to remove their assumptions is H. Lewy [10]. Alittle’2000 Mathematics Subject Classification: Primary; Secondary
supported by Grant-in-Aid for Scientific Research (C), No. 13640226$35\mathrm{L}70$$\uparrow \mathrm{P}\mathrm{a}\mathrm{r}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{u}\mathrm{y}$$53\mathrm{C}21,58\mathrm{C}27,58\mathrm{G}17$.

25
later, J. Hadamard [6] gave another proof. But their assumptions on the coefficients$\{A, B, C, D, E\}$ were alittle strong. The best result on the existence and uniquenesstheorem is given by P. Hartman and A. Wintner [7].
Now we review our method. Equation (1.1) is regarded as asmooth surface defined$\{(\#, y, z,p, q, r, s, t)\}$ . As
and are first order derivain eight dimensional space
tives of $z=z(x, y)$ , we put the relation $dz=pdx+qdy$ . Moreover, as , and aresecond order derivatives of $z=z(x, y)$ , we introduce the relations $dp=rdx+sdy$ and$dq=sdx+tdy$ . Let us call $\{dz=pdx+qdy, dp=rdx+sdy, dq=sdx+tdy\}$ the “contact
structure of second order”. Asolution of (1.1) is regarded as amaximal integral submanifold of the contact structure of second order in the surface $\{(x, y, z, p,q, r, s, t)\in$$\mathrm{R}^{8};F(x, y, z, p, q, r, s, t)=0\}$ . For lifting the solution into the cotangent space, we define equation (1.1) in the cotangent space. To do so, we decompose (1.1) as aproductbe the solutions of $\lambda^{2}+B\lambda+(AC+DE)=0$ . Then, in
andof one forms. Let
the case where $D\neq 0$ , equation (1.1) is written as follows:$\mathrm{R}^{8}=$

$p$

$q$

$r$

$s$

$t$

$\lambda_{2}$$\lambda_{1}$$\omega_{1}\wedge\omega_{2}=\varpi_{1}\wedge\varpi_{2}=D\{Ar+Bs+Ct+D(rt-s^{2})-E\}dx\Lambda dy$(1.2)

where $\omega_{1}=Ddp+Cdx+\lambda_{1}dy,\omega_{2}=Ddq+\lambda_{2}dx+Ady$ , $\varpi_{1}=Ddp+Cdx+\lambda_{2}dy$ , and$\ovalbox{\tt\small REJECT} 2=Ddq+\lambda_{1}dx+Ady$.

Definition 1.1 A regular geometric solution of (1.1) is a submanifold of dimension 2.
y, z,p,$q)\}$ on which it holds dz $=pdx+qdy$ anddefined in$\omega_{1}\wedge\omega_{2}=0$$\mathrm{R}^{5}=\{(x,$Therefore we must solve aPfaff problem for getting ageometric solution. The Pfaffian
problem was also well studied by many people, especially by E. Cartan in analytic
space. Adifference between their fundamental study and ours is that we consider it in-space.Therefore we need some condition which is corresponding to “hyperbolicity”.
Next we project the geometric solution to the base space and construct areasonableweak solution which satisfies some physical conditions. This is our program for the
global theory to nonlinear hyperbolic equations. In fact, we could establish the global
theory for single first order partial differential equations ([15]) from this point of viewand fill, by the classical characteristic method, the gap between the theory of viscosity
solutions and the singularities of solutions.
This note is the brief outline of our recent researches. The detailed paper will be
published elsewhere.
$C^{\infty}$

2System of conservation laws.
Putting $u(x, y)={}^{t}(p, q)$ ,following Cauchy problem$\mathrm{H}(\mathrm{u})={}^{t}(f(q),p)$in$\frac{\partial}{\partial x}u-\frac{\partial}{\partial y}H(u)=0$$u(0, y)=u_{0}(y)$

and

on

$\mathrm{u}\mathrm{o}(\mathrm{y})={}^{t}(p_{0}(y), q_{0}(y))${x$>0, y\in \mathrm{R}^{1}\}$$\{x=0, y\in \mathrm{R}^{1}\}$,, we consider the
(2.1)

(2.2)

26
Suppose that $f(q)$ is in. As we consider the hyperbolic case, we assume that$f’(q)>0$ . This is asystem of conservation laws called as “-system”. It is well known
that, even if the initial data are sufficiently smooth, singularities generally appear in
the solution of (2.1)-(2.2). Therefore we will construct a“solution with singularities”
caUed “weak solution”. Let us recall here the definition of weak solutions of (2.1)-(2.2)
introduced by P. D. Lax [8].
$C^{2}(\mathrm{R}^{1})$Definition 2.1 A bounded and measurable 2-vector function
solution of (2.1)-(2.2) if it satisfies (2.1)-(2.2) in the weak sense, $i.e.$ ,$u=\mathrm{z}(\mathrm{x},y)$is a weak$\int_{\mathrm{R}_{+}^{2}}\{u(x, y)\frac{\partial\Phi}{\partial x}(x,y)-H(u)\frac{\partial\Phi}{\partial y}(x,y)\}dxdy+\int_{\mathrm{R}^{1}}u_{0}(y)\Phi(0,y)dy=0$

for any 2-vector function$\Phi(x, y)\in C_{0}^{\infty}(\mathrm{R}^{2})$

(2.3)

.

If there exists aweak solution $u=u(x,y)$ in the sense of the above definition,
the second equation of (2.1) means that there exists acontinuous function $z=z(x, y)$satisfying $dz=pdx+qdy$ (see [14]). We substituteandinto thefirst equation of (2.1), we get$p=\mathrm{d}\mathrm{z}/\mathrm{d}\mathrm{x}$$q=\mathrm{d}\mathrm{z}/\mathrm{d}\mathrm{y}$$F(q,r,t)= \frac{\partial^{2}}{\partial x^{2}}z-\frac{\partial}{\partial y}f(\frac{\partial z}{\partial y})=r-f’(q)t=0$

.

(2.4)

This is asecond order hyperbolic equation. Here we construct ageometric solution of(2.4) by using the result stated in \S 1. See [14] for the detailed proof.

Remark. We can directly introduce the notion of a“geometric solution” to (2.1)-(2.2)
without going back to (1.1). See [14].
We can prove that the Cauchy problem (2.1)-(2.2) admits aregular geometricsolution globally. Here we have used the terminology “globally” in the sense that the
projection of the geometric solution to the base space $R^{2}=\{(x, y)\}$ coincides with thewhole space . In this case the geometric solution is written down by$R^{2}$

and$p+\Lambda(q)=\psi_{1}(\beta)$

here

,A

$\mathrm{f}(\mathrm{q})=\sqrt{f’(q)}$$\Lambda(q_{0}(\alpha))$problem:$(\mathrm{q})\equiv \mathrm{f}(\mathrm{q})$. Here $x=x(\alpha,\beta)$ andp $-\Lambda(q)=\psi_{2}(\alpha)$$\psi_{1}(\beta)=\mathrm{P}\mathrm{o}(\mathrm{a})+\Lambda(ffi(\beta))$$y=y(\alpha,\beta)$$\lambda(q)\frac{\partial x}{\partial\alpha}+\frac{\partial y}{\partial\alpha}=0$,$x(\alpha, \alpha)=0$, and

are the solutions of the following Cauchy$- \lambda(q)\frac{\partial x}{\partial\beta}+\frac{\partial y}{\partial\beta}=0$,$\psi_{2}(\alpha)=p_{0}(\alpha)-$$y(\alpha, \alpha)=\alpha$

.

,

(2.5)
(2.6)

As system of equations (2.5) is essentially asystem of linear wave equations concerning$x=x(\alpha,\beta)$ and $y=y(\alpha,\beta)$ , theCauchy problem (2.5)-(2.6) has globally aclassicalsolution. We have believed that aweak solution of (2.1)-(2.2) would be constructedby the projection of the geometric solution to the base space. In fact, we could do27
so for first order partial differential equations. See [15]. We think that R. Thorn [11]developed the theory of catastrophe to understand the natural phenomena from thispoint of view. But we have proved the following

Theorem 2.2 ([14]) Assume that $f’(q)>0$ and $f’(q)\neq 0$ . Then we can not generally
construct a single-valued weak solution of the Cauchy problem (2.1)-(2.2) by cutting offsome part of the above geometric solution.
After getting this result, we have come back to the starting point and have reconsidered the fundamental notions used in the “mathematical theory on fluid mechanics”.
In the following section we will present several questions on this subject.

3Rarefaction waves.
Many people have considered the Cauchy problem (2.1)-(2.2) in various kinds of func, etc, etc. The most well-known method would,
tional spaces, for example
they have reduced (2.1)-(2.2) to Riemethod,be the differnce method. For using thismann problem. For solving it, they have introduced “rarefaction wav\"e.
$L^{1}(\Omega)$$L_{loc}^{1}(\Omega)$

Example. Let $u=u(t, x)$ be ascalar function and consider the Cauchy problem asfollows:(3.1),$\frac{\partial u}{\partial x}+u\frac{\partial u}{\partial y}=0$$u_{+}$$(y>0)$(3.2)$u(0,y)=\{$$u_{-}$

whereby$u_{+}$

and$u_{-}$$(y<0)$are constants. In the case where $u_{+}<u_{-}$ , the weak solution is given$u_{+}$

in $\{(x, y);y>\gamma x\}$
(3.3)

$u(x, y)=\{$$u_{-}$

in $\{(x, y);\gamma x>y\}$where $\gamma=(u_{+}+u_{-})/2$ . The jump discontinuity along the line $y=\gamma x$ is called a“shock wav\"e, or simply “shock”. Concerning this point, we do not have any question.
If $u_{+}>u_{-}$ , then the solution accepted in the mathematical theory is given by$u_{+}$$u(x, y)=\{$$y/x$$u_{-}$

in $\{(x,y);y>u_{+}x\}$in $\{(x, y);u_{+}x>y>u_{-}x\}$(3.4)

in $\{(x,y);u_{-}x>y\}$This is called as “centred rarefaction wave” or simply “rarefaction wav\"e. Thissolution has been well accepted in the mathemathecal theory for first order partial
equations. One of the reasons comes from the viscosity theory. Let us add the viscosity
term to (3.1) as follows:

28
(3.5)

$\frac{\partial u}{\partial x}+u\frac{\partial u}{\partial y}=\mu\frac{\partial^{2}u}{\partial x^{2}}$where is apositive constant. Write the solution of (3.5)-(3.2) by. Whentends to +0: Up(x, ) converges to the function given by (3.4). This means that,though the initial data has ajump discontinuity, it disappears immediately. It seems
to us that this phenomenanon would be strange in the mechanics of continuum. In
this sense, equation (3.5) also might be inappropriate for the description of continuum
mechanics. Then, how should we understand this ?
In the mathematical model of continuum mechanics, we think that the jump discontinuity would change continuously, that is to say that it would gradually become
small and finally disappear. It is possible to construct such asolution. Assume $u_{+}>u_{-}$ .
Let be any positive constant. We define afunctionby$u_{\mu}(x, y)$

$\mu$

$y$

$\mu$

$T$

$\varphi\tau(y)$$u_{+}$,$y>u_{+}T$,$y<u_{-}T$(3.6)$\varphi_{T}(y)=\{$$u_{-}$

where (
is in
and $\varphi_{T}’(y)>0$ for $y\in(3-5)u_{+}T$ ). We put the initialcondition at atime $x=T$ by $u(T,y)=\varphi\tau(y)$ . We consider the Cauchy problem for(3.1) under this initial condition. Then, though the solution of this Cauchy problem has
the jump discontinuity in aneighbourhood of $x=0$ , the singularity disappears in finite
time and the solution becomes smooth for all $x>x_{0}$ . But this solution has several
inappropriate properties. The first one is that it does not satisfy the entropy condition
introduced by Lax [8]. Therefore the uniqueness of weak solution does not follow. If we
may accept the above solution, we must introduce anew “entropy condition”.
The principal reason why we insist the above thing is that, if the equation might
describe the mechanics of continuum, the jump discontinuity should change continuously.
$\mathrm{p}\mathrm{r}(\mathrm{y})$$C^{2}(R^{1})$$x_{0}$

Remark. In the case where $u_{+}>u_{-}$ , we define afunction$u_{+}$$\varphi\tau(y)=\{$

,

$y/T$

$u_{-}$,$\varphi\tau(y)$by$y>u_{+}T$,$u_{+}T>y>u_{-}T$(3.7)$y<u_{-}T$

Then the solution of the Cauchy problem for (3.1) with $u(T,y)=\varphi\tau(y)$ is equal to therarefaction wave (3.3). Therefore the rarefaction wave is obtained if we combine, in theinitial data
, two points $(x,y,u)=(T,u_{+}T,u_{+})$ and $(x,y,u)=$ ( $T,u_{-}T$,u-) by astraight line. If we may combine these two point smoothly and monotonely, we get aweak solution whose jump discontinuity changes continuously.
$\varphi_{T}(y)$294Piecewise smooth solutions.
We suppose that the initial data are sufficiently smooth. What kinds of solutions should
we look for ?Many people believe that the weak solution would become piecewisesmooth. Recently we have found apaper [9] on this subject. First we write his result. We
consider the Cauchy problem (2.1)-(2.2). For the diagonalization of (2.1), we introduce
the Riemann invariants, that is to say, we do the change of dependent variables asfollows:(4.1)$w=(p-\Lambda(q))/2$ , $v=(p+\Lambda(q))/2$ .
Then

$w$

and

$v$

satisfy$\frac{\partial w}{\partial x}+h(v-w)\frac{\partial w}{\partial y}=0$,$\frac{\partial v}{\partial x}-h(v-w)\frac{\partial v}{\partial y}=0$

.

(4.2)

where $h(v-w)=\lambda(q)$ . Assume that the initial data satisfy$w(0, y)=c$ ,

where

$c$

(4.3)

$v(0, y)=v_{0}(y)$is aconstant. Then M.-P. Lebaud [9] proved the following

Theorem 4.1 ([2]) Une solution faible entropique de (4.2)-(4.3) presente un choc au
le long de la courbe $y=\gamma(x)$ . La courbe de choc $y=\gamma(x)$ est de classe
point
et les fonctions et sont continues \‘a droite et \‘a gauche de celle-ci.
$(x_{0}, y_{0})$$C^{1}$$z$$w$

is astarting point of shock which is determined by the initial
In Theorem 4.1,
data. The proof of this theorem is very long and complicate. In fact, the author spentalmost all the pages of [9] to prove Theorem 4.1. But, if we might consider this problem
from our point of view, we can understand the result easily. As $w=w(x, y)$ is one of.
Riemann invariants, $w(x, y)=c$ . Therefore $v=v(x, y)$ satisfies the single consevatibn
law as follows:(4.4).
$(x_{0}, y_{0})$$\frac{\partial v}{\partial x}-h(v-c)\frac{\partial v}{\partial y}=0$

For (4.4), we can easily construct aweak solution which satisfies the entropy condition
introduced by Lax. See [15]. Hence we get “une solution faible entropique de (4.2)(4.3)”.
After the statement of this theorem, the author added the explanations as follows:
La solution de (2.1) construite verifie les conditions de Lax [1]. This is wrong. This says,
as it is well known, that the notion of “weak solution” is not conserved by the changeof variables. We will briefly explain the meaning of this problem. As the preparation
for this, we will give several lemmas concerning the Cauchy problem (2.1)-(2.2).
andLemma 4.2 Assume that $f’(q)>0$ and $f’(q)\neq 0$ , and that the initial datasolution. Then the Cauchy problem (2.1)-(2.2) has uniquely aare inin a neighbourhood of $x=0$ .
$p_{0}(y)$$q_{0}(y)$$C^{1}$$C^{1}(R^{1})$Proof. First we get p
p and q. Next we get x$=p(\alpha,\beta)$

and q
and y

$=\mathrm{x}(\mathrm{a},/3)$

by solving (2.5) with respect to$=y(\alpha,\beta)$ by solving the Cauchy problr$=q(\alpha,\beta)$

30(2.5)-(2.6). Then the Jacobian $D(x, y)/D(\alpha,\beta)$ does not vanish in aneighbourhood of, i.e., in aneighbourhood of $x=0$ . In fact it holds from (2.6)$\{(\alpha, \alpha);\alpha\in R^{1}\}$,$\frac{\partial x}{\partial\alpha}(\alpha, \alpha)+\frac{\partial x}{\partial\beta}(\alpha,\alpha)=0$$\frac{\partial y}{\partial\alpha}(\alpha,\alpha)+\frac{\partial y}{\partial\beta}(\alpha,\alpha)=1$

.

(4.5)

From (2.5) and (4.5) it follows$\frac{\partial x}{\partial\alpha}(\alpha,\alpha)=-\frac{1}{2\lambda(q)}$,$\frac{\partial x}{\partial\beta}(\alpha,\alpha)=\frac{1}{2\lambda(q)}$,$\frac{\partial y}{\partial\alpha}(\alpha,\alpha)=\frac{\partial y}{\partial\beta}(\alpha,\alpha)=\frac{1}{2}$.

Hence we get $(D(x, y)/D(\alpha,\beta))(\alpha,\alpha)=-\{2\lambda(q)\}^{-1}\neq 0$ .
Therefore we can solve the system of equations $\{x=x(\alpha,\beta), y=y(\alpha,\beta)\}$ withrespect toand . Substituting $\alpha=\alpha(x,y)$ and $\beta=\beta(x,y)$ into $p=p(\alpha,\beta)$ and$q=q(\alpha,\beta)$ , we obtain the solutions of(2.1)-(2.2). For the proof of the uniqueness ofsolution, we write (2.1) in adiagonal form as (4.2) by using (4.1). Concerning the system
written in the form as (4.2), T. Wazewski [16] already proved the local uniqueness ofsolution.
$\alpha$$\beta$$C^{1}$

Lemma 4.3 Equation (2.1) has the property that the propagation speed is finite.
Proof. p
and q
is obtained by (2.5). As (2.5) are linear wave
equations, they have the property that the propagation speed is finite.
$=p(\alpha,\beta)$$=q(\alpha,\beta)$

Remark. Similar properties as Lemma 4.2 and Lemma 4.3 can be proved for general
quasi-linear systems. See P. Hartman and A. Wintner [7]. Though it seems to the authorthat this paper has been forgotten almost completely, it contains the fundamental
results concerning nonlinear hyperbolic equations and systems of two variables.

Theorem 4.4 Assume that $f’(q)>0$ and $f’(q)\neq 0$ , and that the initial data Po(y)andare in. Let $u=u(x, y)$ be a piecewise smooth function. If the number
$u(x,
y)$
has a jump discontinuity is only one, $u=u(x, y)$ can notof curve on which
generally become a weak solution of (2.1)-(2.2).
$q_{0}(y)$$C^{1}(R^{1})$Idea of the proof. As we have informed in \S 2, the Cauchy problem (2.1)-(2.2) has aregular geometric solution globally. We project the geometric solution to the base space.
In aneighbourhood of apoint where the projection is regular, the projected solution
becomes aclassical solution of (2.1). Moreover Lemma 4.2 assures the uniqueness ofthe classical solution. Therefore, the solution which satisfies the condition stated in
the theorem must be obtained by the projection of the geometric solution. But wehave already proved in [14] that aweak solution of (2.1) can not be generally obtainedby cutting off some part of the geometric solution. In fact we can construct such an
example. Hence we get the conclusion of this theorem.
What we would like to propose: The most typical mathematical method for solving
the problem of singularity would be the “resolution of singularity”. We think that, evenin physical problems, the phenomena of singularities would be understood from thepoint of view of “resolution of singularity”.

31
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[5] E. Goursat, Cours d’analyse mathematique, tome 3(Gauthier-Villars, Paris, 1927).
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[15] T. D. Van, M. Tsuji and N. D. Thai Son, The characteristic method and its generalizations for first-Order non-linear partial differential equations (Chapman &, USA, 1999)$\mathrm{H}\mathrm{a}11/\mathrm{C}\mathrm{R}\mathrm{C}$32[16] T. Wazewski, Sur Vunicite et la limitation de Vintegrales de certains systemes
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[17] H. Whitney, On singularities of mappings of Euclidean spaces I, Ann. Math., 62,(1955).
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Mikio TSUJI
Department of Mathematics
Kyoto Sangyo University
Kyoto 603-8555, Japan.
-mail:mtsuji@cc.kyotO-su. ac.jp$\mathrm{E}$