数理解析研究所講究録
913 巻 1995 年 31-53

31

SINGULARITIES FOR VISCOSITY SOLUTIONSOF HAMILTON-JACOBI EQUATIONSSHYUICHI IZUMIYA1 AND GEORGIOS T. KOSSIORIS2
1. INTRODUCTION

In this note we study the generation and propagation of singularities (shock waves) of thesolution of the Cauchy problem for Hamilton-Jacobi equations$\frac{\partial y}{\partial t}+H(t, x_{1,\ldots,n}x, \frac{\partial y}{\partial x_{1}}, \ldots, \frac{\partial y}{\partial x_{n}})=0$

(P)

$\{$

$y(\mathrm{o}, x_{1}, \ldots, x_{n})=\emptyset(_{X}1, \ldots, x_{n})$,-functions.
where $H$ and areplay an important role in various fields e.g., calculus of variaequations
Hamilton-Jacobi
tions (see e.g., [21]), optimal control theory (see e.g., [9]) and differential games (see e.g., [8]and references cited therein).
For small time the solution of $(P)$ is classically determined using the characteristic
method. The geometric solution of (P) has been defined in ([13], [14]) in the framework of
one-parameter Legendrian unfoldings and it is constructed by the method of characteristics.
Although is initially smooth there is in general a critical time beyond which characteristics
cross. The geometric solution past the critical time is multi-valued, that is singularities
appear. The classification of singularities of has been studied in [13] (see also [15]) In
Section 2 we give a survey on the geometric framework $([13],[151,[16])$ .
The theory of viscosity solutions (see [5]) has provided the right weak setting for the studyof (P). Existence and uniqueness of the solution of (P) in the viscosity sense have been
established in [6]. The single-valued viscosity solution is continuous and coincides with thesmooth geometric solution until the first critical time. After the characteristics cross, theviscosity solution develops shock waves i.e., curves across which the gradient of the viscosity
solution is discontinuous. The shock surfaces are referred to as singular surfaces in theliterature of optimal control and differential games (see e.g., [3], [12]).
The method of constructing the weak solution by selecting the proper single-valued branch
for Hamilton-Jacobi equations. Nakane in [20] has conwas introduced by Tsuji
past the first critical time in case that $H$ is convex
solution
structed the weak semi-concave
past the. The case of scalar conservation laws inwith respect tofirst critical time has been studied by Nakane in [19]. In [4] Bogaevskii has shown that the$\phi$$C^{\infty}$

$t$

$y$

$y$

$y$

$([_{-}^{;}2], [23])$$\nabla y=$$\mathbb{R}^{n}$$( \frac{\partial_{\mathrm{V}}}{\partial x_{1}}, \cdots , \frac{\partial y}{\partial x_{n}})$1Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo 060, Japan
2Department of Mathematics, University of Crete, 71409 Iraklion, Crete, Greece

32

potential solution of the Burgers system with vanishing viscosity is given by the minimum
function of a certain family of smooth functions and given a classification for $n=1,2,3$ . It
corresponds to the viscosity solution of the Hamilton-Jacobi equation when the Hamiltonian.
is given by
Hamiltonian in a neighborhood of the first critical
The viscosity solution of (P) fortime has been constructed in [18] (see also [17], [20]) by selecting a continuous single-valued
branch of the graph of the geometric solution. In which the shock curves of the weak solution
corresponds to the intersection of the branches of the graph of the multi-valued geometricsolution. In order to study the evolution of the shock curves we follow the evolution of theintersections of the branches defining the shock. After that we solve local Riemann problems
for each stage.
Here, we give proofs for some of the results. Further discussions will appear in elsewhere.
unless stated otherwise.
All maps considered hereclass
$H(p_{1}, \ldots p_{n})=\frac{1}{2}p_{1}^{2}+\cdots+\frac{1}{9,\sim}p_{n}^{2}$$\circ\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{l}$a.re$C^{\infty}$

2.

$\mathrm{G}\mathrm{E}\mathrm{O}_{1}\backslash 1\mathrm{E}\mathrm{T}\mathrm{R}\mathrm{I}\mathrm{c}$SOLUTIONSIn this section we give a survey on the geometric framework and present the necessary
notations which was described in $([13],[15],[16])$ .
Let
be the 1-jet bundle of functions of -variables which may be consideredis a, whereaswith a natural coordinate systemgiven. We also have a natural projection :coordinate system ofby $\pi(x, y,p)=(x, y)$ .
is said to be a Legendrian immersion germ
An immersion germ :, where $\theta=dy-\sum_{i=}^{n}1pidXi$ .
(i.e., Legendrian submanifold germ) if $\dim L=n$ and
of
and
it
is denoted by $W(i)$ . We also
The image ofis called the wave front set
and the canonical 1-form $0$ on that space. Let
consider the 1-jet bundle
and $(i, x_{1}, \ldots, x_{n}, y, S,p_{1}, \ldots,p_{n})$be a canonical coordinate system on. Then, the canonical 1-form is given
the corresponding coordinate system on. $dx-s\cdot dt=\theta-s\cdot di$ .
by $\coprod(t, x, y, s,p)=$We define the natural projection II :$(t, x, y)$ . We call the above 1-jet bundle an unfolded 1-jet bundle.
A Hamilton-Jacobi equation is defined to be a hypersurface$n$$J^{1}(\mathbb{R}^{n}, \mathbb{R})$$(x_{1}, \ldots , x_{n}, y,p_{1}, \ldots,p_{n})$$\mathbb{R}^{2n+1}$$\pi$$\mathbb{R}^{n}$$i$$(x_{1}, \ldots , x_{n})$$J^{1}(\mathbb{R}^{n}, \mathbb{R})arrow \mathbb{R}^{n}\cross \mathbb{R}$$(L_{0}, u_{0})arrow J^{1}(\mathbb{R}^{n}, \mathbb{R})$$i^{*}\theta=0$$i$$\pi \mathrm{o}i$$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$(t, x_{1}, \ldots , x_{n})$$\mathbb{R}\cross \mathbb{R}^{n}$$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$\mathrm{b}\mathrm{y}\ominus=dy-\sum_{i1}^{n}=p_{ii}$$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})arrow(\mathbb{R}\cross \mathbb{R}^{n})\cross \mathbb{R}$(G-H-J)$E(H)=\{(t, x, y, s,p)\in J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})|s+H(t, x,p)=0\}$

A geometric (multi-valued) solution of $E(H)$ is a Legendrian submanifoldinlying in $E(H)$ . In this case the wave front set $W(i)$ is “the graph” of thegeometric solution which is generally a hypersurface with singularities.
, we define
In order to study (P) we need the following framework: For any

in$L$$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R}).$$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$c\in(\mathbb{R}, 0)$$E(H)_{\mathrm{C}}=\{(_{C}, X, y, -H(C, x,p),p)|(x, y,p)\in J1(\mathbb{R}n, \mathbb{R})\}$Then,$E(H)_{\mathrm{c}}$.

is a $(2n+1)$ -dimensional submanifold of
a contact structure on $E(H)_{c}$ . We define a mapping$dz- \sum^{n}i=1pidXi\circ\sigma \mathrm{i}\mathrm{v}\mathrm{e}\mathrm{S}$$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$\mathrm{a}\mathrm{n}\mathrm{d}\ominus_{\mathrm{c}}=|E(H)c=$$\iota_{c}$:$J1(\mathbb{R}^{n}, \mathbb{R})arrow$33

by $\iota_{\mathrm{c}}(x, y,p)=(c, x, y, -H(C, X,p),p)$ . The mapping
and the following diagram is commutative:$E(H)_{c}$$i_{c}$

is a contact diffeomorphism$J^{1}(\mathbb{R}^{n},\mathbb{R})\mathbb{R}^{n}\cross \mathbb{R}\pi\downarrow--\underline{\iota_{\mathrm{c}}}\mathbb{R}^{n}\cross \mathbb{R}E(H)\downarrow\pi_{\mathrm{C}}\mathrm{c}$.
We say that a geometric Cauchy problem (with initial condition $L’$ ) associated with thetime parameter $(GcPT)$ is given for an equation $E(H)$ if there is given an n-dimensional. Sincefor some
submanifold : $L’\subset E(H)$ with $i^{*}\ominus=0$ andis the characteristic vector field given by$X_{H}\not\in TE(H)_{c}$ , we have $X_{H}\not\in TL’$ , where$i(L’)\subset E(H)_{\mathrm{c}}$$i$$c\in(\mathbb{R}, 0)$$X_{H}$.

$X_{H}= \frac{\partial}{\partial t}+\sum_{=i1}n\frac{\partial H}{\partial p_{i}}\frac{\partial}{\partial x_{i}}+(\sum_{=i}n1p_{i}\frac{\partial H}{\partial p_{i}}-H)\frac{\partial}{\partial\tau/}-\frac{\partial H}{\partial t}\frac{\partial}{\partial s}-\sum_{1i=}\frac{\partial H}{\partial x_{i}}\frac{\partial}{\partial p_{i}}n$

By using the classical characteristic method, we can show that there exists a unique geometricsolutions around $L’$ .
We remark that Cauchy problem (P) is a GCPT. The initial submanifold is given by$L_{\phi,0}=\{(0,$ $x,$ $\phi(_{X),H}-(0, x, \frac{\partial\phi}{\partial x}), \frac{\partial\phi}{\partial x})|x\in \mathbb{R}^{n\}}\subset E(H)_{0}$.

The problem of studying the singularities of the graph of the geometric solution is formulated as follows:

Geometric Problem. Classify the gen eric bifurcati$ons$ of wave fronts of$\pi_{t}|$:$L\mathrm{n}E(H)tarrow \mathbb{R}^{n}\cross \mathbb{R}$with respect to the param eter (i.e., the generic bifurc ations of wave fronparameter).
solu tions along the
$t$

$ts$

of geom etric$t\mathrm{i}me$

Following [16], in order to study the singularities of the geometric solution we identify geometric solutions with one-parameter Legendrian unfoldings. Let be an $(n+1)$ -dimensionalbebe a submersion germ and :smooth manifold, :$p_{t}=\ell|\mu^{-1}(t)$is ais a Legendrian family if
a smooth map germ. We say that the pair
. Then we have the following simple but very
Legendrian immersion germ for any
important lemma.
$R$

$\mu$

$(R, u\mathrm{o})arrow J^{1}(\mathbb{R}^{n}, \mathbb{R})$$\ell$$(R, u_{0})arrow(\mathbb{R},t_{0})$$(\mu, l)$$t\in(\mathbb{R}, t_{0})$Lemma 2.1. Let
such that$C_{u_{0}}^{\infty}(R)$be a$(\mu,\ell)$$f^{*}\theta=h\cdot d\mu$Define a map germ$\mathcal{L}$:$L\mathrm{e}_{\mathrm{o}}^{\sigma}endr\mathrm{i}\mathrm{a}\mathrm{n}$, wherefamily. Then there $eAst$ a uniq $ue$ element $h\in$of $sm$ ooth function germs at .
is the$C_{u_{0}}^{\infty}(R)$$(R, u_{0})arrow J^{1}(\mathbb{R}\cross \mathbb{R}n, \mathbb{R})$$u_{0}$$\mathrm{r}\mathrm{i}n_{\mathrm{o}}\sigma$by$\mathcal{L}(u)=(\mu(u), x\mathrm{o}\ell(u),$ $y\mathrm{o}\ell(u),$ $h(u),p\mathrm{o}\cdot\ell(u))$.

34

We can easily show that is a Legendrian immersion germ. If we fix 1-forms and , theLegendrian immersion germ is uniquely determined by the Legendrian family. We
call a Legendrian unfolding associated with the Legendrian family.
In order to study the evolution of the shock waves of the viscosity solutions of (P), we haveto classify the generic types of the appearingi.e., how a singularity is generated,
how one type can change into another and how different types ofinteract. We
study how various branches of the multi-valued
intersecting
at a point bifurcate in time for an arbitrary Hamiltonian $H(t, x,p)$ in [15]. We classify thebybifurcations of the branches of thethe bifurcations of singularities ofmulti-Legendrian unfoldings which are expressed in terms of.
$z_{i})(i=1,\ldots,r)$
Let:
be Legendrian unfoldings with$\Pi(z_{i})=0$ whereare distinct. We calla multi-Legendrian unfolding. Let
be multi-Legendrian
and. We say that theseare-Legendrian equivalent if there exist contact diffeomorphism germs$\mathcal{L}$$\ominus$$\theta$$(\mu,\ell)$$\mathcal{L}$$\mathcal{L}$$(\mu, \ell)$$\sin\circ\cdot \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{S}$$\sin\circ \mathrm{o}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}$$\circ\circ\cdot \mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}\nu V_{t}=(\{t\}\cross \mathbb{R}^{n}\cross \mathbb{R})\cap W(i)$$0\sigma \mathrm{r}\mathrm{a}\mathrm{p}\mathrm{h}$$\mathrm{c}\mathrm{l}\mathrm{a}\mathrm{S}\mathrm{S}\mathrm{i}\mathrm{f}\mathrm{y}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma$$\mathrm{m}\mathrm{u}\mathrm{l}\mathrm{t}\mathrm{i}- 0\sigma \mathrm{e}\mathrm{m}\mathrm{s}$$(R, u_{0})arrow(J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R}),$$\mathcal{L}_{i}$$z_{1},$$\ldots,$$(\mathcal{L}_{1}, \ldots, \mathcal{L}_{f})$$z_{r}$$(\mathcal{L}_{1}, \ldots, \mathcal{L}_{r})$$(\mathcal{L}_{1}’, \ldots, \mathcal{L}_{r}’)$$\mathrm{u}\mathrm{n}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma \mathrm{s}$$P_{(r)}$$I\mathrm{i}_{i}’$:$(J1(\mathbb{R}\cross \mathbb{R}n, \mathbb{R}),$ $z_{i})arrow(J^{1}(\mathbb{R}\mathrm{X}\mathbb{R}^{n}, \mathbb{R}),$$Z_{i}’)$$(i=1, , .., r)$of the form $I1_{i}’(t, x, y, S,p)=(\phi_{1}(t), \phi 2(\iota, X, y), \phi 3(t, x, y), \phi^{i}4(t, X, y, s,p), \phi i\mathrm{s}(t, X, y, s,p))$ anda diffeomorphism germ
: $(R, u_{0})arrow(R, u_{0}’)$ such thatfor any $i=$
, . It is clear that if two multi-Legendrian unfoldings are-Legendrian equivalent,
then there exists a diffeomorphism:of the form$\Phi(t, x, y)=(\phi_{1}(t), \phi_{2}(t, x, y), \phi_{3}(i, x,y))$ such that
. Thus theabove equivalence describes how bifurcations of wavefronts (i.e. graphs of solutions) interact.
We can define the notion of stability with respect to the-Legendrian equivalence in$[1],[24]$stability(see
the same way as for the ordinary Legendrian). Motivated by Arnol’dZakalyukin’s theory $([1],[24])$ , we can construct multi-generating families of multi-Legendrian
unfoldings and give a classification of-Legendrian stable Legendrian unfoldings by using
the classification of multi-families of function germs in Zakalyukin [24]. We get a list ofclassifications for $n=1,2,3$ in [15]. However, we only present the list of classifications for$n=1$ . For the case $n=2,3$ , see [15].
$\Psi$$1,$$\ldots$$I_{1_{i}}’\mathrm{o}\mathcal{L}_{i}=\mathcal{L}_{i}’0\Psi$$r$$P_{(r)}$$0\circ\cdot \mathrm{e}\mathrm{r}\mathrm{m}\Phi$$(\mathbb{R}\cross(\mathbb{R}^{n}\cross \mathbb{R}), 0)arrow(\mathbb{R}\mathrm{x}(\mathbb{R}^{n}\cross \mathbb{R}), 0)$$\Phi(\bigcup_{i=1}^{\Gamma}W(\mathcal{L}_{i}))=\bigcup_{i=1}^{r}W(\mathcal{L}_{i})$$P_{(r)}$$P_{(r)}$

Theorem 2.2 [15]. Suppose that $n=1$ . Then a generic multi-Legendrian unfolding is-Legendrian uivalent to one of theunfoldings in the following list :$r=1$ ;
:
;
;
:
$(t,
4u^{3}+2ut,
3u^{4}+u^{-}’t, -u^{2}, u)$ .
:$P_{(r)}$$e\mathrm{q}$$0A_{1}$$(t, u, \mathrm{O}, \mathrm{O}, \mathrm{o})$$0A_{2}$$(t, 3u^{2},2u^{3}, \mathrm{o}, u)$$mult\mathrm{i}- Le\sigma e\mathrm{n}\mathrm{O}dri\mathrm{a}\mathrm{n}$$1A_{3}$$r=\underline{9}$;$0(0A_{1}0A_{1})$

1 $(0A_{1}0A_{1})$$1A_{2}0A_{1}$: $((t, u, -u, \mathrm{o}, -1), (t,u, u, \mathrm{o}, 1))$ ;: $((t, u, t\pm u^{2},1, \pm 2u), (t, u, 0,0,0))$ ;: $((t, 3u^{2}-t, 2u^{3}, u, u), (\mathrm{f}, u, -u, 0, -1))$ .

$r=3$ ;
$0A_{1}0A_{1}0A1$:$((t, u,t-u, 1, -1), (t, u, \mathrm{O}, \mathrm{O}, \mathrm{o}), (t, u, u, \mathrm{o}, 1))$.

When we consider the geometric solution, we can get rid of the germ 1 $(^{00}A_{11}A)$ from theabove list because the geometric solution is a-one immersions into the unfolded l-jet
space.
$\mathrm{o}\mathrm{n}\mathrm{e}-\mathrm{t}_{0}$35

On the other hand, we have to identif.v geometric solutions with one-parameter Legendrian
unfoldings in locally, so that we prove the following realization theorems.
Theorem 2.3 [13]. (1) The local soluti on of the GCPT for the Hamilton-Jacobi equati on$(G- H_{-}J)$ is a
.
:
drian
be a
drian
associated with
(2) Let:-fun ciionsuch that is a
. Then there exists a
parameter for theeralized uchy problem associ ated with thelocal olution of the$(Garrow
HJ)$
tial condi iion.
, where th
byHamilton-Jacobi equati onthat the class of Legendrian unfoldings supplies the correct
The above theorem
class to describe the geometric solutions of (GCPT) for Hamilton-Jacobi equations. Thus,
can be translated toof Legendrian
generic results for the
results in the class of all Hamiltonians and all initial conditions. However, we have to also
concern ourselves with what are the types of singularities that the geometric solution to agiven Hamilton-Jacobi equation might exhibit. For the purpose, we need a kind of nondegeneracy condition on the Hamiltonian function. We say that a Hamiltonian function$H(t, x,p)$ is non-degenerate at
if it
for some $1\leq i,j\leq n$ . Thiscondition is weaker than the condition that $H(t, x,p)$ is convex (or concave) with respect to. The foliowing theorem is a realization theorem
for-variables at
for genericfor a given Hamilton-Jacobi equation.
aterate Hamilt onian functi on
Theorem 2.4 $([15],[16])$ . Let $H(t, x,p)$ be astable
drian
be
a
and :
’
. Then there exists a
Legendrian unfoldi associ ated withwhich is a geometric soln tion of the Hamilton-Jaco equation $+H(t,.x,p)=0$ such that
.
and
are
equivalen .
We remark thatsingularity (even for general ) describes how the singularity appears
stable Legendrian unfoldings, so that thesefrom a smooth solution. These are$Le_{\mathrm{o}}\sigma ex1$$\mathcal{L}$$(\mathbb{R}\cross \mathbb{R}^{n}, \mathrm{o})arrow J^{1}(\mathbb{R}\mathrm{x}\mathbb{R}^{n}, \mathbb{R})$$\mathrm{u}\mathrm{n}fold\mathrm{i}\mathrm{n}_{\mathrm{o}}\sigma \mathcal{L}$$(\mathbb{R}\cross \mathbb{R}^{n}, 0)arrow J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$C^{\infty}$$(\pi_{1},\ell)$$Le_{\mathrm{o}}\circ\cdot e\mathrm{n}$$\mathrm{u}\mathrm{n}foldi\mathrm{n}_{\mathrm{o}}\sigma$$\circ\circ\cdot e\mathrm{r}mH(t, x_{1,\ldots,n}x,p_{1}, \ldots , p_{n})$$t\mathrm{i}me$$C\mathrm{a}$$s$$\mathcal{L}$$0\sigma e\mathrm{n}$$i\mathrm{n}\mathrm{i}$$is\circ\sigma \mathrm{j}\gamma en$$\mathrm{e}$$\ell(0, u)$$0\circ\cdot \mathrm{u}\mathrm{a}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{e}\mathrm{S}$$(t_{0}, x_{0,p_{0}})$$(t_{0}, x0,p_{0})$$(p_{1}, \ldots,p_{n})$$0\sigma \mathrm{e}\mathrm{n}\mathrm{e}\mathrm{r}\mathrm{i}\mathrm{c}$$\mathrm{u}\mathrm{n}\mathrm{f}\mathrm{o}\mathrm{l}\mathrm{d}\mathrm{i}\mathrm{n}\circ\cdot \mathrm{s}\circ$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}$$\frac{\partial^{2}H}{\partial p_{i}\partial_{Pi}}(t_{0}, x_{0},p\mathrm{o})\neq 0$$n\geq 9arrow$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{e}\mathrm{s}$$\mathrm{n}on-de\mathrm{o}\sigma en$$(t_{0}, x_{0},p0)$$\mathcal{L}$$(R, u\mathrm{o})arrow(J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R}),$$0\sigma e\mathrm{r}m$$(t_{0}, x_{0}, y_{0}, S_{0}, p_{0}))$$(\mu, \ell)$$n_{\mathrm{o}}\sigma$$Le_{\mathrm{o}}\sigma endrianunfold\mathrm{i}\mathrm{n}_{\circ}\sigma \mathcal{L}$$bi$$\mathcal{L}’$$P_{(1)^{-Le}\circ}\sigma e\mathrm{n}$$\mathcal{L}$

$s$

$t$

$P_{(1)}-Le_{\circ}\sigma \mathrm{e}ndr\mathrm{i}\mathrm{a}n$$n$$1A_{3}$$P_{(1)0}-\mathrm{L}\mathrm{e}\sigma \mathrm{e}\mathrm{n}\mathrm{d}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}$can be realized as geometric solutions at the non-degenerated point for a given Hamilton-singularity appears.
Jacobi equation. We can also specify the point at where theis nonappears at, thenTheorem 2.5 [16]. If an
at.
degenerate$1A_{3}$$1A_{3^{-}}s\mathrm{i}n\sigma \mathrm{O}u.la\Gamma ity$$(t_{0}, x_{0,p_{0}})$$H(t, x,\mathrm{p})$$(t_{0}, x_{0,p_{0}})$3.

$\mathrm{v}_{\mathrm{I}\mathrm{s}\mathrm{c}\mathrm{o}}\mathrm{s}\mathrm{I}\mathrm{T}\mathrm{Y}$SOLUTIONS

The viscosity solutions for nonlinear equations of first order have been introduced byCrandall and Lions [6]. Such solutions need not be differentiable everywhere, as the onlyis aregularity required in the definition is that of continuity. The function
viscosity solution of$y_{\mathfrak{v}}\in C(\mathcal{O})$(H-J)$\frac{\partial\tau/}{\partial t}+H(t, x, \frac{\partial y}{\partial x_{1}}, \ldots, \frac{\partial?/}{\partial x_{n}})=0$

in the open domain$\mathcal{O}\subset \mathbb{R}^{+}\cross \mathbb{R}^{n}$provided$\frac{\partial\psi}{\partial t}(t, x)+H(t, x, \frac{\partial\psi}{\partial x_{1}}(i, x), \ldots, \frac{\partial\psi}{\partial x_{n}}(t, X))\leq 0$, (resp.

$\geq 0$)36

attains a local maximum (resp. local minimum) at thefor whichfor any
is a viscosity solution of the Cauchy
point $(t, x)\in O$ . The function
andproblem $(P)$ if and only if it is a viscosity solution of (H-J) in the domain
satisfies the initial condition$\psi\in C^{1}(\mathcal{O})$$y_{\mathrm{U}}-\psi$$y_{\mathfrak{v}}\in C([0, \infty)\cross \mathbb{R}^{n})$$(0, \infty)\cross \mathbb{R}^{n}$$\lim_{tarrow 0+}y\mathfrak{o}(t, x)=\emptyset(_{X})$.

be referred as the viscosity criterion at the point $(t, x)$ . We next
The above inequality
state the viscosity criterion in a form which is more useful for the construction of the solution.
is open and that there is a smooth hypersurface
To this end, assume that
. Then weand, which divides into two open setsofhave the following theorem.
$\mathrm{w}\mathrm{i}\mathrm{U}$$\mathcal{O}\subset(0, \infty)\cross \mathbb{R}^{n}$$\mathbb{R}^{+}\cross \mathbb{R}^{\mathfrak{n}}$$\Gamma$$O^{+}$$\mathcal{O}$$\mathcal{O}^{-},$

inand
Theorem 3.1. Let$ty$of (H-J) inon
solutiis a viscosi. Thenconditions hold:an
are assical olutions of (H-J) inand\’o) If the vector, then$y_{\mathfrak{v}}=y_{\mathrm{t}\mathrm{l}}^{+}$$y_{\mathfrak{v}}\in C(O)$$C^{1}(O^{\pm}\cup\Gamma)$$a)y_{\mathrm{U}}^{+}$$\mathcal{O}^{+}\cup\Gamma,$$cl$$\mathcal{O}^{+}$

$s$

where
in
if and on$ly$ if the following$y_{\mathfrak{v}}=y_{\mathrm{U}}^{-}$$\mathcal{O}$$y_{\mathrm{U}}$$y_{\mathfrak{o}}^{-}$$\mathcal{O}=\Gamma\cup \mathcal{O}^{+}\cup \mathcal{O}^{-}$$O^{-}\cup\Gamma$$d\mathcal{O}^{-_{r}}es_{\mathrm{P}^{e}}Ct\mathrm{i}_{\mathrm{V}}ely$$\tilde{\eta}=(H(i, x, \frac{\partial y^{+}}{\partial x})-H(t, X, \frac{\partial y^{+}}{\partial x}), -(\frac{\partial y^{+}}{\partial x_{\mathrm{t}}}-\frac{\partial\tau_{l^{-}}}{\partial x_{\mathrm{t}}}, .

$y_{\mathrm{c}}^{\pm}\in$

,

in to

.., \frac{\partial_{J^{+}}}{\partial x_{n}},-\frac{\partial}{\partial}\iota_{-}^{-}xn))\mathrm{p}o\mathrm{i}\mathrm{n}ts$$\mathcal{O}^{+}$$H(t,$ $x,$

where$(1- \lambda)\frac{\partial y_{\mathfrak{d}}^{+}}{\partial x}+\lambda\frac{\partial y_{\mathfrak{v}}^{-}}{\partial x})-(1-\lambda)H(t, x, \frac{\partial y_{\mathrm{U}}^{+}}{\partial x})-\lambda H(t, x, \frac{\partial y_{\mathrm{c}}^{-}}{\partial x})\leq 0$$\lambda\in[0,1]$$j_{\mathrm{o}in}in_{\mathrm{o}}\sigma$. In pariicular,

the pointsofand$the\circ\sigma r\mathrm{a}\mathrm{p}h$$(H(t, X,)\partial x’\partial x)\underline{\partial}_{\mathrm{R}}y^{+}\underline{\partial}y\mathrm{R}^{+}$$Hl\mathrm{i}$es respecti vely \’oelow or ab.

$o\mathrm{v}e$(resp.

$\geq 0$),the line segment$(H(t, x, \frac{\partial?J^{-}}{\partial x}),\partial x)\underline{\partial}y\mathrm{B}^{-}$The proof of Theorem 3.1 is given in $([17],[18])$ as a direct application of Theorem 1.3 in[7]. The condition b) will be referred in the sequel as the viscosity criterion. The hypersurface
has the properties specified in the above theorem is thein the neighbourhood of whichshock surface. If the Hamiltonian is uniformly convex (or concave), we can automaticallyconstruct viscosity solutions from our normal forms, so that we can easily draw the picturesof shock surfaces for lower dimensional cases. In [4] Bogaevskii has shown that the potential
solution of the Burgers system with vanishing viscosity is given by the minimum function of acertain family of smooth functions. It corresponds to the viscosity solution of the Hamilton. He has
Jacobi equation when the Hamiltonian is given bydrawn the pictures of shocks for this case. Our pictures are same as his pictures, so we do
not present these in here (see [4]).
On the other hand, Bogaevskii used Florin-Hopf-Cole method $([10],[11])$ to detect thesystem. Here, we prove the analogous statement
solution and it works only for thewhen
the Hamiltonian $H(p_{1}, \ldots,p_{n})$ is convex and
the
case
in
as the Bogaevskii’s assertion
depends only on the momentum. In this case we apply Bardi-Evans’ result [2] to our situations
in stead of Florin-Hopf-Cole method. The geometric solution for (P) is given by$\Gamma$$y_{0}$$H(p_{1}, \ldots p_{n})=\frac{1}{2}p_{1}^{2}+\cdots+\frac{1}{2}p_{n}^{2}$$\mathrm{B}\mathrm{u}\mathrm{r}_{\mathrm{o}}^{\sigma}\mathrm{e}\mathrm{r}\mathrm{S}$(S)$L_{\phi,t}=\{(t, X(t, u), y(t, u), -H(p(t, u)),p(t, u))|u\in \mathbb{R}^{n}\}$

,

37

where
$x(t, u)=u+t \frac{\partial H}{\partial p}(\frac{\partial\phi}{\partial x}(u)))$

$\{$

,

$p(t, u)= \frac{\partial\phi}{\partial x}(u)$$y(t, u)=t \{-H(\frac{\partial\phi}{\partial x}(u\mathrm{I})+<\frac{\partial\phi}{\partial x}(u), \frac{\partial H}{\partial p}(\frac{\partial\phi}{\partial x}(u))>\}+\phi(u)$.

We consider a family of functions $F(t, x,p, q)=\phi(q)+<p,$ $(x-q)>-H(p)t$ , where
the canonical inner product on. We have
and$<,$$(t, x,p, q)\in \mathbb{R}\cross \mathbb{R}^{n}\cross(\mathbb{R}^{n}\cross \mathbb{R}^{n})$$\mathbb{R}^{n}$$>\mathrm{i}\mathrm{s}$$\Sigma(F)=\{(t, q+\frac{\partial H}{\partial p}(\frac{\partial\phi}{\partial q}(q))t, \frac{\partial\phi}{\partial q}(q), q)|(t, q)\in \mathbb{R}\cross \mathbb{R}n\}$where$\Sigma(F)$is the set defined to beby$J^{1}(\mathbb{R}\cross \mathbb{R}^{n}, \mathbb{R})$$\frac{\partial F}{\partial p:}=0$

and$\frac{\partial F}{\partial q_{i}}=0$,. We now define a map. It follows that$\Phi_{F}$:$\Sigma(F)arrow$$\Phi_{F}(t, x,p, q)=(t, x, F.(t, x,p, q), \frac{\partial F}{\partial t}, \frac{\partial F}{\partial x})$$\Phi_{F}(t, q+\frac{\partial H}{\partial p}(\frac{\partial\phi}{\partial q}(q))t,$$\frac{\partial\phi}{\partial q}(q),$$(t, q+ \frac{\partial H}{\partial p}(\frac{\partial\phi}{\partial q}(q))t,$$q)=$$-H( \frac{\partial\phi}{\partial q}(q))t+<\frac{\partial\phi}{\partial q}(q),$ $\frac{\partial H}{\partial p}(\frac{\partial\phi}{\partial q}(q))>+\emptyset(q),$ $-H( \frac{\partial\phi}{\partial q}(q)),$is equal toThis shows that the image of the map
family of.
We refer the following result of Bardi-Evans [2].
$\Phi_{F}$$L_{\phi)t}$$\frac{\partial\phi}{\partial q}(q))$.

, namely, $F$ is a global generating$L_{\phi,t}$

Theorem 3.2. Assume that the Hamilton $ianH(p_{1}, \ldots , p_{n})$ is convex, then$y(t, x) \equiv\inf \mathrm{s}\mathrm{u}\mathrm{p}qp\{\phi(q)+<p, (_{X}-q)>-H(p)t\}$is the un que visc osity soluti on of (P).
$\mathrm{i}$Then we have the following theorem as a corollary of the above theorem.
Theorem 3.3. Assume that $H$ is un iformly convex and has the minimum. Let. Thenth geometricof the$\phi$$\mathrm{e}$$Solut\mathrm{i}\sim on(\mathrm{S})$$L_{\phi,t}$be$\mathrm{c}au.Chy\mathrm{P}^{ro\mathrm{b}}l.em(\mathrm{P})$$y(t, x) \equiv\min_{y}\{y|(i, x, y)\in\Pi(L_{\phi,t})\}$

is the un ique viscosi $ty$ solution of (P).

Proof. Consider the family of functions $F(t, x,p, q)=\phi(q)+<p,$$H(p)$ is uniformly convex, we have

$(x$

.

$\sup_{p}\{\phi(q)+<p, (x-q)>-H(p)t\}=F|\Sigma_{p}(F)$ ,$-q)>-H(p)t$ . Since

38

where

$\Sigma_{p}(F)=\{(t, x,p, q)|\frac{\partial F}{\partial p}.\cdot=x_{i}-q_{i}-\frac{\partial H}{\partial p}.\cdot(p)t=0i=1, \ldots, n\}$. It follows that.

$\inf_{q}\sup_{p}\{\phi(q)+<p, (_{X-q})>-H(p)t\}=\inf_{q}\{F(t, q+\frac{\partial H}{\partial p}(p)t,p, q)\}$Since$\phi$

has the minimum , it is equal to.

$\min_{q}\{F(t, q+\frac{\partial H}{\partial p}(p)t,p, q)\}=\min_{q}\{F(t, X,p, q)|(t, x,p, q)\in\Sigma_{p}(F)\}$On the other hand, we define functions
Since $H(p)$ is uniformly convex, we have$f_{i}(t, x,p, q)= \frac{\partial F}{\partial p_{i}}=X_{i}-q_{i}-\frac{\partial H}{\partial \mathrm{p}:}(p)t(i=1, \ldots , n)$.

, at any
point $(t_{0}, x_{0},p0, q0)$ , so that there exist local smooth functions $g_{i}(t, x, q)(i=1, \ldots,n)$ near$(t_{0}, x0,p_{0}, q_{0})$ such that $\Sigma_{p}(F)=\{p_{i}=g_{i}(t, x, q)\}$ . Thus we have$\frac{\partial f_{1}}{\partial p_{i}}(t_{0,0,p0}x, q\mathrm{o})=-\frac{\partial^{2}H}{\partial p_{i}\partial pj}(p_{0})t_{0}\overline{\gamma}-\angle 0$$\frac{\partial F|\Sigma_{p}(F)}{\partial q_{i}}=\sum_{j}\frac{\partial F}{\partial p_{j}}(t, x, g(t, x, q), q)\frac{\partial g_{j}}{\partial q_{i}}(t, X, q)+\frac{\partial\phi}{\partial q_{i}}(q)-g_{i}(t, x, q)=\frac{\partial\phi}{\partial q_{i}}(q)-g_{i}(t, x, q)$so that$\Sigma(F)=\{(t, x,p, q)\in\Sigma_{p}(F)|\frac{\partial F|\Sigma_{\mathrm{n}}.(F)}{\partial q1}=0\}$,. It follows that$y(t, x) \equiv\min_{y}\{y|(t, x, y)\in\Pi(L_{\phi,\mathrm{t}})\}$$= \min\{F(t, x,p, q)|(t, x,p, q)\in\Sigma(F)\}$$(p,q)$$= \min_{q}\{F(t, X,p, q)|(t, x,p, q)\in\Sigma_{p}(F)\}$.

It is the unique viscosity solution for the Cauchy problem (P).
However, for general (non-convex) Hamiltonian, situations are quite different.

4. NoN CONVEX HAMILTONIANS IN ONE SPACE VARIABLE
In this section we stick to the Cauchy problen of Hamilton-Jacobi equation in one spacevariable as follows:$\frac{\partial y}{\partial t}+H(\frac{\partial?/}{\partial x})=0$

(P)

$\{$

$y(0, x)=\phi(_{X)}$,where $H$ and are-functions. Since $H(p)$ is not assumed to be uniformly convex (orconcave), we cannot use Theorem 3.3, so that the situations should be quite complicated even.
for the one space variables
In this case the geometric solution is given by$\phi$$C^{\infty}$$\mathrm{c}a\mathrm{s}\mathrm{e}$$L_{\phi,t}=\{(t, x(t, u),y(t,u), -H(p(t, u)),p(t, u))|u\in \mathbb{R}\}$

,

39

where

$x(t, u)=u+tH’(\phi’(u))$ ,$\{$$p(t, u)=\phi’(u)$$y(t, u)=t\{-H(\emptyset l(u))+\phi’(u)H’(\phi’(u))\}+\phi(u)$.

Before the first critical time that characteristics cross in the $(t, x)$ -plane,is the graphof the viscosity solution . After the characteristics cross,becomes singular. Theorem
Tlue first
2.2 describes the generic singularities ofappears in the form of.
See Figure la, where we show the shape of the appearing singularity.
$\nu V_{t}$$\nu V_{t}$$y_{\mathfrak{v}}$$\nu V_{t}.$$1A_{3}$$\sin_{\mathrm{o}}\sigma \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}$x(t.u)

Figure lb
Figure lc
Figure la
By Theorem 2.5, these appear at the convex or the concave points of the Hamiltonian, the viscosity solution is given byfunction. Away from the. In ([17], [18]) we
have constructed the unique viscosity solution past the first critical time by selecting a singleappears at the point.
valued branch of . Assume that the singularity of type
is three-valued on an interval
After the critical time , the wave front, wheresee Figure . Let
is defined on a$i=1,2,3$ be the three branches ofof. Thenneighborhood ofon aandintersect at one
point $\chi(t)\in(x_{1}(t), x_{2}(t))$ , for $t>t_{0}$ . We define the viscosity solution pastby selecting a
as follows:
continuous single-valued branch of$W_{t}$$\sin_{\mathrm{o}}\circ \mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{t}\mathrm{y}$$1A_{3}$$W_{t}$$(t_{0}, x_{0,p_{0}})$$(x_{1}(t),$ $X_{2());}t$$\mathrm{T}\prime V_{t}$$t_{0}$$1\mathrm{b}$$\nu V_{t}$$y_{i},$$x_{1}(t)$$x_{2}(t)$$\mathrm{n}\mathrm{e}\mathrm{i}_{\mathrm{o}}\sigma \mathrm{h}\mathrm{b}\mathrm{o}\mathrm{r}\mathrm{h}_{0}\mathrm{o}\mathrm{d}$$y_{2}$$y_{1}$$y_{1},$$y_{2}$$t_{0}$$\nu V_{t}$Theorem 4.1. There exists an$(x_{1}(t), x_{2}(t))$ , definby$\epsilon>0$such th at the fun ction$y_{\mathfrak{v}}(t, x),$$(t, x)\in(t_{0}, t_{0}\backslash +\epsilon)\cross$$\mathrm{e}d$(4.1)$y_{1}(t, x),$$x\leq x(t)$$y_{3}(t, x),$$x\geq x(t)$$y_{\mathfrak{v}}(t, x)=\{$

is the viscosity solution of $(P)$ in a$\mathrm{n}e\mathrm{i}\circ h\mathrm{o}\mathrm{b}\mathrm{o}\mathrm{r}ho\mathrm{o}d$of$x_{0}$,

past the time$t_{0}$.

In view of Theorem 2.5 the Viscosit.v criterion (see Section 3) is satisfied acrosswhile. Hence, by the uniqueness of thedefined by (4.1) is a classical solution away fromviscosity solution, (4.1) gives the viscosity solution of (P) past .
$\chi(t)$$y_{\mathfrak{v}}$$\chi(t)$$i_{0}$

40

By this construction, we have extended the viscosity solution beyond the first critical time
. According to Theorem 2.5 the shock is generated in a convex or concave domains of $H(p)$ ,so the viscosity criterion is automatically satisfied. The graph of the viscosity solution past
the first critical time is depicted by a full line in Figure , where we assume that $H$ is. The shock corresponds to theconvex in the neighborhood of the appearing singularity
shock is definedintersection of the two branches and it is called a genuine shock. The(orwaves)
speed
is given by theanditscharacteristics
as the intersection of two incoming
Rankine-Hugoniot condition$t_{0}$$1\mathrm{c}$$1A_{3}$$\circ\sigma \mathrm{e}\mathrm{n}\mathrm{u}\mathrm{i}\mathrm{n}\mathrm{e}$,$\chi’(t)=\frac{H(_{J_{\mathrm{U}},\mathcal{I}}\tau^{+},(t,\chi(t)))-H(?^{-}/_{\mathfrak{o}},x(t,\prime\chi(t)))}{?J_{1})x+(t,x(t))-y^{-}\mathfrak{d},x(t,\chi(t))}$. Therefore in order to follow the evolution of the shock
and
where
we have to study the following questions:intersecting at one point bifura) How different branches of the multi-valued graph ofcate in time.
b) If the two branches initially defining the shock continue to cross, whether the viscosity
criterion is satisfied across the intersection.
are given in
The normal forms of the generic bifurcations of different branches ofTheorem 2.2. We depict these bifurcations in Figure 2.
$y_{\mathfrak{v},x}^{\pm}=\underline{\partial}_{A,\partial x^{-}}y^{\pm}$$\chi’(t)=\frac{d_{\mathrm{Y}}}{dt}(t)$$\nu V_{t}$$\mathrm{V}V_{\ell}$

1 $(0A_{1}0A_{1})$$0(0A_{1}0A_{1})$$0A_{1}0A10A1$$1A_{2^{0}}A_{1}$FIGURE 2

41, we can choose the correct
If the viscosity criterion is satisfied at the time
branch of the graphs of the geometric solutions as viscosity solutions (see Figure 3).
$t_{\alpha}=t_{0}+\epsilon$$.-\Gamma^{\backslash }\backslash ^{-}\prime \mathrm{c}’\backslash$$arrow-\backslash \wedge^{---}\backslash \backslash \prime\prime$$-\cdot-arrow_{--}.\vee--$’’$’\backslash \mathrm{A}$$arrow–\vee\wedge^{--}\backslash \prime 4$$arrow-\cdot \text{ノ_{}-\backslash }\vee--$.

$\backslash$’$\iota$$\backslash$’.

$\backslash$$arrow$$\nwarrow^{---},\prime \text{ノ}$$arrow$$-\mathrm{Y}-\backslash \prime_{\backslash }’\backslash \backslash$.

FIGURE 3
viscosity
criterion can be violated across the intersection of
We will now investigate how theandtwo branches. Assume that a generated shock is defined by two intersecting branches$x<\chi(t)$forviscositythesolution
representing) the branch. We denote by(resp.
$y^{-}$$y^{+}$$\overline{y}$$y^{+}$

42(resp. $x>\chi(t)$ ). If the two branches remain intersected they evolve according to.
$H(p)$
In
thecase whenhas only one
the intersection of the two branches.
We denote byinflection point Kossioris [17] studied this problem and constructed the viscosity solutions.
We consider the general situation here. It is clear that for generic Hamiltonian function$H(p),$ $H$ has only Morse type critical points and no tritangent lines. So we assume that the
Hamiltonian has the above properties. By Theorem 2.2, we have the following theorem.
$0(_{-4_{1}}00A_{1})$$\chi(t)$Theorem 4.2. For a generic initi function , if the viscosity criterion is violated at ,then the only following 8 cases may occu :toof $H(p)$ at only one of(1) The normal form is $0(^{0}A1A01)$ andto theat other points between theseand the line is not
the points.
nottoof $H(p)$ at each(2) The normal form is $0(^{0}A1A01)$ andand there exists on one an other point between these points at where theoint.
above line is tangent to the$0(^{0}A1A01)$
to the graph of $H(p)$ at only one ofand(3) The normal form isand there exists only on another point between these points at where
the points
the above line is angent to the graph.
tangent to the graph of $H(p)$ at each point(4) The normal form is $0(^{0}A1A01)$ and.
of
not tangent to the graph of $H(p)$ at each(5) The normal form is $0(0A_{1}0A_{1})$ andpoints between these points at where theexactly
trvo
other
point
existsand there
above line is angent to the graph.
(6) The normal
is $1A_{2}0A_{1}and\overline{P^{+}P-}is$ tangent to the graph of $H(p)$ at only one of theand the line is not tangent to the graph at oiher points \’oetween these points.
oints
to the graph of $H(p)$ at only one(7) The norm
is $0A_{1}0A_{1}0A1$ andof the points
and is not tangent to the graph at other points between th $ese$ points.
notto the graph of $H(p)$ at each(8) The normal
is $0A_{1}0A_{1}0A_{1}$ andpoint
and there exists only one another point between these points at where theabove line is tangen to the graph.
and
Here,
in the $(p, H(p))$ -plan .
the line through$\mathrm{a}l$$\phi$$t_{\alpha}$$\mathrm{r}$$P^{\overline{+_{P^{-i}}}}\mathrm{S}ta\mathrm{n}^{\sigma}\circ ent$$P^{+},$$P^{-}$$tan_{\mathrm{o}}\sigma ent$

$the\circ\sigma r\mathrm{a}\mathrm{p}h$$\mathrm{o}\mathrm{P}^{h}\sigma \mathrm{r}a$$p_{\mathrm{o}\mathrm{i}n}tS$$\overline{P^{+}P^{-}}\mathrm{i}_{S}$$the\circ\sigma \mathrm{r}aph$$tan_{\mathrm{o}}\sigma ent$$p$$P^{+},$$P^{-}$$l.v$$gr\mathrm{a}pl\iota$$\overline{P^{+-_{\mathrm{i}\sigma}}P}stan\mathrm{e}nt\circ$$P^{+},$$P^{-}$$\mathrm{e}$$t$$\overline{P^{+}P^{-}}\mathrm{i}_{S}$$P^{+},$ $P^{-}$$\overline{P^{+}P^{-}}\mathrm{i}_{S}$$P^{+},$$P^{-}$$t$$f\mathrm{o}\mathrm{m}$$p$$P^{+},$$P^{-}$$alfo\mathrm{m}$$P^{+},$$\overline{P^{+}P^{-\mathrm{j}_{S}}}tan_{\mathrm{o}}\sigma ent$$P^{-}$$\overline{P^{+}P-}\mathrm{i}s$$fo\mathrm{m}$$P^{+},$$tan_{\mathrm{o}}\sigma \mathrm{e}nt$$P^{-}$$t$$P^{+}=(y_{x}^{+}(t_{\alpha}, \chi(t\alpha)),$ $H(y^{+}x(t\alpha’\chi(t_{\alpha}))),$$\overline{P^{+}P^{-}}d\mathrm{e}notes$$P^{-}=(y_{x}^{-(t_{\alpha}}, \chi(t\alpha)),$ $H(yx-(t\alpha’\chi(t_{\alpha})))$$P^{+},$ $P^{-}$$e$.
After that the singularities of the graph of the geometric solution bifurcate in the forms of$0(0A_{1}0A_{1}),$
neveror $0A_{1}0A_{1}0A1$ . Since the characteristics incross, we can get rid of the case $1(0A_{1}0A_{1})$ .
We already mentioned that the viscosity criterion is satisfied past the first critical time ,tangent to the graph of $H(p)$ . By theso that it is satisfied until the time whenat most a
assumptions on the Hamiltonian $H(p)$ , we may consider the case that
double tangent line for each normal form. We now distinguish each normal form. We denote
.
that
and
(A) $0(0A_{1}0A_{1})$ : In this case each branch of the graph of geometric solution is a nonsingular curve. We remark that $y^{\pm}(t,\chi(\prime t))=t\{-H(\phi’(u\pm))+\phi’(u\pm)H’(\emptyset(u\pm))\}+\phi(u\pm)$ .

Proof. By Theorem 2.2, we may assume that the first singularities appear in the form of$1(^{0}A_{1}0A_{1}),$$1A_{2}0A_{1}$$1A_{3}$$J^{1}(\mathbb{R}\cross \mathbb{R}, \mathbb{R})$$t_{0}$$\overline{P^{+_{P^{-}}}}\mathrm{i}_{\mathrm{S}}$$t_{\alpha}$$\overline{P^{+}P^{-}}\mathrm{i}\mathrm{s}$$p^{+}=y_{x}^{+}(t_{\alpha}, \chi(t\alpha))=\phi’(u_{+})$$p^{-}=y_{x}^{-}(t_{\alpha}, \chi(t\alpha))=\phi’(u-)$

43

Since the normal form $0(0A_{1}0A_{1})$ has trivial bifurcations along the time parameter, the condition $y^{+}(t, \chi(t))=\overline{y}(t, \chi(t))$ defines a codimension submanifold in the corresponding jet
space, so that we may ignore this condition. We now consider the following conditions whichcorrespond to all possible cases:, which defines a submanifold inof codi(a)$0$$2J^{1}(\mathbb{R}, \mathbb{R})$$\pm H’(\phi’(u_{+}))=,\frac{H(\emptyset\prime(u+))-H(\delta’(u-))}{\emptyset(u+)-\phi(u_{-)}},$,
mension 1. Of course, we have to consider the case that
however this case is essentially contained in the above, so that we may ignore such nonessentially different cases in the following arguments.
. This condition
such(b) There exists withof codimension 1.
defines a submanifold in
that
with
(c) There exists$\pm H’(\phi’(u_{-))}=,\frac{H(\phi’(u+))-H(\emptyset’(u-))}{\phi(u+)-\phi(u_{-})},$$u_{0}\neq u_{\pm}$$u_{0}$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t}\pm H’(\phi’(u_{0}))=\frac{H(\phi’(u+))-H(\phi’(u-))}{\varphi’(u+)-\phi J(u_{-)}}$$2J^{1}(\mathbb{R}, \mathbb{R})$$u_{0}\neq u\pm \mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}$$u_{0}$.

$\pm H’(\phi’(u+))=\pm H’(\phi’(u_{0}))=\frac{H(\phi’(u_{+}))-H(\delta’(u_{-}))}{\phi’(u_{+})-\emptyset\prime(u_{-})}=\frac{H(\phi’(u_{+}))-H(\emptyset\prime(u\mathrm{o}))}{\phi’(u+)-\phi\prime(u\mathrm{o})}$This condition defines a submanifold in(d)of codimension 2.
which are different from(e) There exist$3J^{1}(\mathbb{R}, \mathbb{R})$of codimension 3.
, which defines a submanifold in$\pm H’(\phi’(u_{+}))=\pm H’(\phi’(u_{-))}=,\frac{H(\delta’(u+))-H(\phi\prime(u-))}{\phi(u+)-\phi(u_{-)}},$$2J^{1}(\mathbb{R}, \mathbb{R})$$u_{0},$$u\pm \mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h}$$u_{1}$

that$\pm H’(\phi’(u_{0}))=\pm H’(\phi’(u_{1}))=\frac{H(\phi’(u+))-H(\phi\prime(u_{-}))}{\phi’(u_{+})-\emptyset\prime(u_{-})}$$= \frac{H(\phi’(u_{+}))-H(\emptyset\prime(u\mathrm{o}))}{\phi’(u_{+})-\phi’(u_{0})}=\frac{H(\phi’(u_{+}))-H(\emptyset\prime(u_{1}))}{\phi’(u_{+})-\phi^{l}(u_{1})}$.

is a multiof codimension 4. Here,
This condition defines a submanifoldincodimension
has
at
most
. Each submanifold
1-jet space of function germ, so that we can not avoid such conditions by the multi-jet transversality theorem.
bifurcates at the time , so that we: In this case the normal form(B)
for fixed . It defines a submanshould consider the condition
ifold inof codimension 1. By the same arguments as the above, we can avoid theconditions (c), (d) and (e). So we may consider the condition (a) or (b). We now show that
. On the $(t, x)$ -plane, we denote $(t, \chi(t))$
the condition (a) holds for the normal form
. Suppose that the point -corresponds to the cusp point at
the genuine shocks forthe time . Then there exists a smooth function $u(t)$ such that $\chi(t)=u(t)+tH’(\phi’(u(t)))$forand $u(t_{\alpha})=u_{-}$ , where we choose one of the branches of the graph of the geometric. It follows that we have
solution corresponding to$rJ^{1}(\mathbb{R}, \mathbb{R})$$\mathrm{o}\mathrm{f}_{4}J^{1}(\mathbb{R}, \mathbb{R})$$rJ^{1}(\mathbb{R}, \mathbb{R})$$\mathbb{R}arrow \mathbb{R}$$r$$1A_{2^{0}}A_{1}$$1A_{2^{0}}A_{1}$$t_{\alpha}$$y^{+}(t_{\alpha’\prime}\chi(t\alpha))=y^{-}(t_{\alpha}, \chi(t\alpha))$$t_{\alpha}$$2J^{1}(\mathbb{R}, \mathbb{R})$$1A_{2}044_{1}$$t\leq t_{\alpha}$$u$$t_{\alpha}$$t\leq t_{\alpha}$$u_{-}$$\chi’(t)=u’(t)(1+H’’(\phi’(u(t))\phi\prime\prime(t))+H’(\phi’(u(t))$.

Since the graph of the geometric solution has a singularity at , we have$1+t{}_{\alpha}H’’(\phi’(u_{-}))\phi’’(u-)=0$ . So we have $\chi’(t_{\alpha})=\lim_{t-t_{\alpha}}\chi’(t)=H’(\phi’(u-))$ .
$t_{\alpha}$$\frac{\partial x}{\partial u}(t_{\alpha},$$u_{-)}=$

44

On the other hand, by the Rankine-Hugoniot condition we have$\chi’(t)=’\frac{H(y_{\emptyset}^{+}x(i,\chi(t)))-H(y^{-}\mathfrak{v},x(t,\chi(t)))}{y_{\mathfrak{v},x}^{+}(t,x(t))-y_{0}x-(t,\chi(t))},$’

for$t\leq t_{\alpha}$. Since$\chi)(t)))=\phi’(u\pm)$ , we have$\lim_{tarrow t_{\alpha}}y_{0x}^{\pm}(t,$$\chi’(t_{\alpha})=\frac{H(\phi’(u+))-H(\phi’(u-))}{\varphi’(\mathrm{u}+)-\phi(u_{-})},$, sothat we have $H’(\phi’(u_{-}))=\wedge^{-}H\phi’(u))H(\phi’(u-))$ . This condition corresponds to the case (a)and we may get rid of the case (b).
(C) $0A_{1}0A_{11}0A$ : In this case the normal form also bifurcate at the point , so that we canget rid of the case (c), (d) and (e) by the similar reasons as those of the case (B). Since eachbranch of the normal form is non-singular, the remaining two cases may occur in generic.
This completes the proof.
$\phi’(\mathrm{u}+)-\phi’(u-)$$t_{\alpha}$

We can solve local Riemann problems and construct viscosity solutions for each case in
the above theorem. However, we only consider the cases (1) and (6) in this paper. We will
give the detailed considerations for all cases in elsewhere.
Case (1). We assume that the graph of the viscosity solution at the time
as in Figure .

$t\leq t_{\alpha}$$4\mathrm{a}$$\mathrm{H}(\mathrm{P})$.

$H’(y_{x}^{-}(t \alpha’\chi(t\alpha)))=’\frac{H(y_{x}^{+}(t\alpha\chi(t\alpha)))-H(y_{x}^{-}(t\alpha x(t_{\alpha})))}{y_{x}^{+}(t_{\alpha},\chi(t\alpha))-y_{x}-(t\alpha’\chi(t\alpha))},=x’(t_{\alpha})$We now distinguish two cases as follows:a) If$H’(y_{x}^{-}(t, x(t))) \geq\frac{H(y_{\mathcal{I}}(+t,x(t)))-H(yx-(t,x(t)))}{y_{x}^{+}(t,\chi(t))-y^{-(t,\chi(}xt))}$is depicted

45, then we can easily show that the viscosity
for sufficiently small
for. So we can choose single valued continuous branches of thecriterion is satisfied for
viscosity
solution.
geometric solution as theb) If$\epsilon>0$$t_{\alpha}\leq t<t_{\alpha}+\epsilon$$t<t_{\alpha}+\epsilon$$H’(y_{x}^{-((}t, xt)))< \frac{H(y_{x}^{+}(t,\chi(t)))-H(yx-(t,\chi(\iota)))}{y_{x}^{+}(t,\chi(t))-y\overline{x}(t,\chi(t))}$, then we can easily show that the viscosity
for sufficiently small
for, so that a new way to build the solution is required
criterion is violated for(cf., Figure 5).
$\epsilon>0$$t_{\alpha}\leq t<t_{\alpha}+\epsilon$$t_{\alpha}<t<t_{\alpha}+\epsilon$$\mathrm{H}(\mathrm{D})$$\mathrm{H}(\mathrm{P})$FIGURE 5
In this case we can use the techniques in [12] to construct the contact discontinuity shock
curve and then obtain new characteristics. Lets consider the relation $H’(q)= \frac{H(p)-H(q)}{p-q}$
and $H”(q_{0})\neq 0$ . By the implicit function
witharoundtheorem, there exists a smooth function around such that the above relation is equivalent
to $q=\psi(p)$ . We will first construct the contact discontinuity as the solution of the following
initial value problem.
$x_{\mathrm{c}}’(t)=H’(\psi(yx(+t, x_{c}(t))))$ ,$(q_{0},p\mathrm{o})$$q_{0}\neq p_{0},$$H’(q_{0})= \frac{H(p_{0})-H(q_{0})}{p0-q0}$$\psi$$p_{0}$$\{$$\chi_{\mathrm{C}}(t_{\alpha})=\chi(t\alpha)$

The characteristic which is started at a point.

$(\tau, \chi_{c}(\tau))$should be satisfied the following:$x’(t)=H’(p(t))$ ,$\{$$p’(t)=0$$y^{l}(t)=-H(p(t))+p(t)H’(p(t))$ ,with initial condition $x(\tau)=\chi_{c}(\tau),$ $y(\tau)=y^{+}(\tau, \chi_{c}(\tau))$ andsolution is exactly given as follows:,$p(\tau)=\psi(y_{x}^{+}(\tau, \chi \mathrm{c}(\mathcal{T})))$. So the$\tilde{x}(t)=\chi \mathrm{c}(\mathcal{T})+(t-\mathcal{T})H;(\psi(y^{+}x(\mathcal{T}, xc(\mathcal{T}))\rangle)$$\tilde{p}(t)=\psi(y^{+}\mathcal{I}(\tau, \chi \mathrm{c}(\mathcal{T})))$$\{$$\tilde{y}(t)=y^{+}(\tau, \chi_{\mathrm{C}}(\tau))$.

$+(t-\mathcal{T})\mathrm{t}-H(\psi(yx(+))))+\psi(yx+(\tau, \chi c(\mathcal{T})))H’(\psi(yx+\tau, \chi_{\mathrm{C}}(\mathcal{T}(\tau, xc(\tau))))\}$

46

By definition of the contact discontinuity, we have,$\chi_{\mathrm{C}}’’(t)=H’’(\psi(\phi(u_{+}(t))\frac{\partial\psi}{\partial p}(\phi’(u_{+}(t))\phi’’(u+(t))u_{+(}’t)$where $\chi_{c}(t)=u_{+}(t)+tH’(\phi(u_{+}(t)).$ Since$\frac{\partial\psi}{\partial p}=,,\frac{H’(P)-H’(q)}{H(q)(p-q)}$, we have.

$x_{c}’’(t)= \frac{H’(\phi\prime(u_{+}(t))-H\prime(\psi(\emptyset\prime(u+(t))))}{\emptyset^{l}(u_{+(t))-^{\psi(\phi’((t)))}}u+}\phi\prime l(u_{+}(t))u_{+}(\prime t)$

We also have$\chi’(t)=u_{+}’(t)\{1+tH’’(\phi’(u+(t))\phi’’(u_{+}(t))\}+H’(\phi’(u_{+}(t)))$.

It follows that$\chi_{\mathrm{c}}’’(t)=-,\frac{(H’(\emptyset\prime(u_{+}(t)))-H\prime(\psi(\phi’(u_{+}(t)))))2}{\emptyset(u_{+(t))-^{\psi(\phi((t)))}}u+},\frac{\phi’’(u_{+}(t))}{1+tH’\prime(\emptyset(u+(t))\phi’’(u_{+}(t))},\cdot$

Since
,

$\frac{\partial x}{\partial u}(t, u_{+}(t))=1+tH’’(\emptyset’(u_{+}(t)))\phi\prime\prime(u+(t))$

is convex if and only if
we may assume that $1+tH”(\emptyset’(u_{+}(t)))\phi’’(u_{+}(t))>0$ . So$\phi’’(u_{+}(t))>0$ . We suppose that $\phi’’(u_{+}(t))\leq 0$ and denote $\chi_{c}(t)=u_{+}(t)+tH’(\phi(u+(t))=$$u_{-}(t)+tH’(\phi(u_{-}(t)))$ , where) is the point corresponding to the charac(resp.
. We distinguish two cases as follows:teristic from the right (resp. left) side of$\phi’’(u_{-}(t))>0$
moves to the leftb-l) If
is monotone. Since $u_{-}’(t)<0,$, thendirection, so that the viscosity criterion is satisfied across .
, then 1+
and the viscosity criterion is violated across fornear . Differentiate the equality $\chi_{c}(t)=u_{-}(t)+tH’(\phi(u_{-}(t))$
with respect to , then we have$\chi_{\mathrm{c}}(t)$$u_{-}(t)$$u_{+}(t)$$(t, \chi_{\mathrm{c}}(t))$$\phi’(u_{-}(t))$$\phi’$$\chi$$\mathrm{b}-2)\mathrm{I}\mathrm{f}\phi’’(u_{-}(t))<0$$t>t_{\alpha}$$\chi$$tH”(\emptyset’(u_{-}(t)))\emptyset\prime\prime(u_{-}(t))>0$$t_{\alpha}$$t$$x’(t)-H’(\phi’(u-(t)))=\{1+tH’’(\emptyset’(u-(t)))\phi’’(u-(t))\}u_{-(}’t)$

.

Since
$x’(t)=, \frac{H(\phi’(u_{+}(t)))-H(\phi’(u_{-}(t)))}{\phi(u_{+}(t))-\phi\prime(u_{-}(t))}>H’(\phi’(u_{-}(t)))$,is increase, which is a contradiction.
we have $u_{-}’(t)>0$ , so that, the contact discontinuity curve
Hence, if the viscosity criterion is violated forconvex and the viscosity solution can be constructed.
$u_{-}(t)$$t>t_{\alpha}$$\chi$is47

We draw the oicture which is$\mathrm{i}\mathrm{l}\mathrm{l}\mathrm{u}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{n}\not\subset$

the situations as follows:

Case (6). The bifurcations of the graphs of the geometric solution at the time
as follows:

FIGURE 8$t_{\alpha}$

is depicted48

We use the same notation as the case (1). Sincecorresponds to the cusp point,. Let
we havebe the locus of the cusps, where
we denote $\sigma(t)=\sigma_{(}t)+tH’(\phi’(\sigma-(t)))$ as the family of characteristics come from the left
side, so that we have $1+tH”(\phi’(\sigma-(t)))\phi;’(\sigma_{-}(t))=0$ and
. It
that
$H”(\phi’(\sigma-(t)))\phi’’(\sigma-(t))<0$ and $\sigma’(t)=H’(\phi’(\sigma’-(t))$ . Differentiating the equation again,. Therefore
we getis strictly convex. We
denote $\chi(t)=u_{-}(t)+tH’(\phi’(u_{-}(t)))=u_{+}(i)+tH’(\phi’(u_{+}(t)))$ for, then we have
, so thatby the proof of
discontinuity
thecontact
Theorem 4.2. We also construct
exactly the same as thatof in the case (1). We need examine the following two subcases.
. Since botha) Assume thatand are convex near , we have
for
that
.
On the other hand, we have$u_{-}(t_{\alpha})$$1+t_{\alpha}H’’(\phi(u_{-(t}\alpha)))\phi\prime\prime(u_{-}(t)\alpha)=0$$(t, \sigma(t))$$\sigma_{-}^{J}(t)<0$$\mathrm{f}\mathrm{o}\mathrm{U}\mathrm{o}\mathrm{w}\mathrm{s}$$(t, \sigma(t))$$\sigma’’(t)=H’’(\emptyset^{l}(\sigma_{-}(t))\phi\prime\prime(\sigma-(t))\sigma_{-}^{;}(t)>0$$t\leq t_{\alpha}$$\sigma’(t_{\alpha})=H’(\phi^{;}(\sigma_{-}(t\alpha)))=H’(\phi(u-(t_{\alpha})))=\chi’(t_{\alpha})$$u_{-}(t_{\alpha})=\sigma_{-}(t_{\alpha})$$(t,xc(\prime t))$$\sigma(t)\geq\chi_{c}(t)$$t\geq t_{\alpha}$$\chi_{\mathrm{c}}$$t_{\alpha}$$\sigma$$\sigma’’(t_{\alpha})>\chi’’(t_{\alpha})$,$y(t, \sigma_{\pm}(t))=t\{-H(\phi’(\sigma\pm(t)))+\phi’(\sigma\pm(t))H’(\phi’(\sigma_{\pm}(t)))\}+\phi(\sigma\pm(t))$$\frac{d\mathrm{c}/}{dt}(t, \sigma+(t))=-H(\phi’(\sigma+(t)))+\phi’(\sigma_{+}(\iota))H’(\emptyset’(\sigma_{+}(t)))$$+\phi’(\sigma_{+}(t))\sigma’(+t)\{1+tH^{l}’(\phi’(\sigma_{+}(t)))\phi\prime\prime(\sigma+(t))\}$

and$\frac{dy}{dt}(i, \sigma_{-}(t))=-H(\phi’(\sigma_{-}(t)))+\phi’(\sigma_{-}(t))H’(\phi’(\sigma_{-(}t)))$.

.
Let $A(t)=y(t, \sigma+(t))-y(t, \sigma_{-(t)})$ for$\sigma_{-}(t)+tH’(\phi’(\sigma_{-(}t)))=\sigma_{+}(t)+tH’(\phi’(\sigma+(t)))$Differentiating the equality
with respect
to , we get$t\geq t_{\alpha}$$t$$\sigma_{+}’(t)\{1+tH’’(\phi’(\sigma_{+}(t)))\emptyset l’(\sigma+(t))\}=H’(\phi’(\sigma_{-}(t)))-H’(\phi’(\sigma+(t)))$.

It follows that$A’(t)=H’(\emptyset(\sigma_{-}(t)))\{\phi’(\sigma+(t))-\phi l(\sigma-(t))\}-(H(\phi’(\sigma_{+}(t)))-H(\phi^{;}(\sigma_{-}(t))))$.

Furthermore, we have$A”(t)=(H’(\phi’(\sigma_{-}(t))-H’(\phi’(\sigma+(t)))\phi^{;\prime}(\sigma_{+}(t))\sigma_{+(t)}’$$+H”(\phi;(\sigma_{-}(t)))\phi’’(\sigma-(t))\mathrm{t}\phi’(\sigma+(t))-\phi;(\sigma_{-}(t))\}\sigma_{-()}\prime t$

Since

,

$\sigma_{+}’(t)=,,,\frac{H’(\phi’(\sigma-(t))-H’(\phi’(\sigma+(t))}{1+tH(\phi(\sigma+(t))\varphi(\sigma+(t))},$.

and $\sigma’’(t)=H’’(\phi’(\sigma-(t)))\phi’’(\sigma-(t))\sigma_{-}’(t)$ , we have$A”(t)=, \frac{(H’(\phi’(\sigma-(t))-H\prime(\phi’(\sigma_{+}(t)))}{1+tH’(\emptyset(\sigma+(t))\emptyset(\sigma_{+}(t))},,,\underline’\phi\prime\prime(\sigma+(t))+\sigma(\prime\prime t)\{\phi’(\sigma+(t))-\phi’(\sigma-(t))\}$On the other hand, as we already calculated in the case (1) that.

$\chi_{\mathrm{c}}’’(t)=-,\frac{(H’(\phi’(u_{+}(t))-H\prime(\emptyset(\phi’(u+(t)))))2}{\phi(u_{+}(t))-\psi(\phi(u+(t)))},\frac{\phi’’(u+(t))}{1+tH’’(\phi(u_{+(t))}\phi^{\prime i}(u_{+(}t))},$.

49

At the point$t=t_{\alpha}$, we have$u\pm(t_{\alpha})=\sigma\pm(t_{\alpha})$and$\psi(\phi’(\sigma_{+}(t_{\alpha})))=\phi’(\sigma_{-}(t\alpha))$, so that$\chi_{c}’’(t_{\alpha})=-,\frac{(H’(\emptyset\prime(u_{+}(t\alpha))-H’(\psi(\phi\prime(u+(t_{\alpha})))))^{2}}{\phi(u_{+}(t_{\alpha}))-\psi(\phi(u_{+}(t_{\alpha})))},,\cdot$

Thus, we have.

$A”(t_{\alpha})=(\sigma’’(t\alpha)-x_{\mathrm{c}}’’(t_{a}lpha))(\phi’(\sigma+(t\alpha)-\phi’(\sigma_{-}(t\alpha))$Since$\sigma’’(t_{\alpha})>\chi_{c}’’(t)\alpha$and$\phi’(\sigma_{+}(t)\alpha<\phi’(\sigma_{-}(t_{\alpha})$, we have $A”(t_{\alpha})<0$ . This means that, so $y(t, \sigma_{+}(t))<y(t, \sigma_{-}(t))$ .
We also consider$A’(t)<0$ near$t_{\alpha}$$y_{+}(t)=t\{-H(\phi’(u+(i)))+\phi’(u_{+}(t))H’(\phi’(u_{+}(t)))\}+\phi(u_{+}(t))$

and,$y_{\alpha}(t)=t\{-H(\phi’(u_{\alpha}))+\phi’(u\alpha)H’(\emptyset^{l}(u\alpha))\}+\phi(u_{\alpha})$and $x(t, u_{\alpha})=u_{\alpha}+\iota H’(\emptyset’(u_{\alpha}))=x(t, u_{+}(t))=u_{+}(t)+tH’(\phi’(u_{+}(t)))$ .
where
Differentiating the last equality, we get$u_{\alpha}=\sigma_{-}(t_{\alpha})$$H’(\emptyset’(u_{\alpha}))=u_{+}’(t)+H’(\phi’(u_{+}(t)))+tH’’(\phi^{l}(u_{+}(t)))\phi’’(u+(t))u_{+}(\prime t)$.

Then$y_{+}’(t)=-H(\phi’(u_{+(t)})+\phi’(u_{+}(t))H’(\emptyset’(u_{+}(t)))$$+tH”(\phi’(u_{+}(t)))\phi’(u_{+}(t))\phi’’(u+(t))u_{+(t)}^{l}+\phi’(u_{+}(t))u_{+}’(t)$$=-H(\phi’(u_{+}(t)))+\phi’(u_{+}(t))H’(\emptyset’(u_{+}(t)))$.

So we obtain$\frac{d}{dt}(y_{\alpha}(t)-y_{+}(t))=H(\phi’(u_{+}(t)))-H(\phi’(u_{\alpha}))-H’(\phi’(u_{\alpha}))(\phi’(u_{+}(t))-\phi’(u_{\alpha}))$Since$\phi’(u_{+}(t))<\phi’(u_{\alpha})$and$\phi’(u_{\alpha})$.

is in the convex region of $H(p)$ , we have,$H’( \phi’(u_{\alpha}))<,\frac{H(\phi’(u_{+}(t)))-H(\emptyset\prime(u\alpha))}{\phi(u_{+}(t))-\phi\prime(u_{\alpha})}$. This means that $y_{\alpha}(t)<y_{+}(t)$ for.
so that we have
These situation describes that the two branches of the multi-valued graph have intersection
for $t>t-\alpha$ . This contradicts to the assumption that the singularity is 1
. In this subcase two shocks bifurcates fromforb) Here we assume thatthe point.
$\frac{d}{dt}(y_{\alpha}(t)-y_{+}(t))<0$$\sigma(t)<\chi_{c}(t)$$(t_{\alpha}, x_{\alpha})$$t>t_{\alpha}$$t>t_{\alpha}$

50

See Figure 9.

FIGURE 9
the intersection of the original characteristic fromThe left one is a new shock given
the left side and the new characteristic from the contact discontinuity (i.e., the rarefaction
waves).
By definition, we have$\mathrm{b}.\mathrm{v}$$\tilde{y}_{\alpha}(t)=(t-t\alpha)\{-H(\psi(yx+(t\alpha’\chi_{\mathrm{C}}(t_{\alpha}))))+\psi(yx(+tx\alpha’ \mathrm{c}(tlpaha)))H\prime\prime(\psi(y_{x}^{+}(t\chi_{\mathrm{c}}(\alpha’ t_{\alpha}))))\}$$+y^{+}(t_{\alpha},x(t\alpha))$and$y-(t)=t\{-H(\phi’(u_{-}(t)))+\emptyset^{l}(u_{-(t))H^{;}(\emptyset(u}\mathrm{t}t)))\}+\phi(u_{(}t))$,where $u_{\alpha}+tH’(\emptyset’(ualpha)))=x(t,u_{-}(t))=u_{-}(t)+iH’(\phi’(t))$.
Differentiating the last equality, we get$u’-(t)+H’(\emptyset’(u_{-}(t)))+tH’’(\emptyset’(u_{-}(t)))\phi\prime\prime(u-(t))u\mathrm{L}(t)=H’(\phi^{l}(u\alpha))$.

Then$y_{-}’(t)=-H(\phi’(u_{-}(t)))+\phi’(u_{-}(t))H’(\phi(u(t)))$$+tH”(\emptyset l(u_{-}(t)))\phi r(u_{(}t))\phi’’(u_{-}(t))u\mathrm{L}(t)+\phi’(u_{-}(t))u_{-}^{;}(t)$$=-H(\phi’(u_{-}(t)))+\phi’(u-(t))H’(\phi(u\alpha)$

So we obtain$\frac{d}{dt}(\tilde{y}_{\alpha}(t)-y_{-}(t))=H(\phi’(u_{-}(t)))-H(\phi’(u_{\alpha}))-H’(\emptyset’(u_{a}lpha))(\phi’(u_{-}(t))-\phi’(u_{\alpha}))$.

51

Since $u_{-}(t)<u_{\alpha}$ and both ofand$\phi’(u_{(}t)),$$\phi’(u_{\alpha})$are in the convex region of $H(p)$ , we have$\phi’(u_{(}t))>\phi’(u_{\alpha}))$$H’( \phi’(u_{\alpha}))>\frac{H(\phi’(u-(t))-H(\phi’(u_{\alpha}))}{\emptyset^{J}(u_{-}(t))-\phi\prime(u_{\alpha})}$so that we obtainforthat
We also consider$\frac{d}{dt}(\tilde{y}_{\alpha}(t)-y_{-}(t))<0$$\tilde{y}_{\alpha}(t)<y_{-}(t)$$t>t_{\alpha}$.

. Since$\tilde{y}_{\alpha}(t_{\alpha})=y_{-}(t_{\alpha})$,, the last inequ&ity means$\tilde{y}_{\alpha}(t, \sigma(i))=(t-\mathcal{T}(t))\{-H(\psi(y_{x}(+\mathcal{T}(t), x_{c}(\tau(t))))$$+\psi(y_{x}^{+}(T(t), \chi c(\mathcal{T}(t))))H’’(\psi(y^{+}x(\mathcal{T}(t), \chi_{\mathrm{c}}(\mathcal{T}(t)))))\}$$+y_{x}^{+}(\mathcal{T}(t), \chi c(\mathcal{T}(t)))$and$y-(t, \sigma(t))=t\mathrm{t}-H(\phi’(\sigma_{-}(t)))+\phi’(\sigma_{-}(t))H’(\phi(\sigma_{-}(t))\}+\emptyset(\sigma-(t))$

,

where
$y^{+}(\tau(t), \chi \mathrm{C}(\mathcal{T}(t)))=T(t)\mathrm{t}-H(\emptyset’(\tau(t)))+\emptyset’(\mathcal{T}(t))H’(\phi(\tau(t))\}+\phi(\mathcal{T}(t))$,$y_{x}^{+}(\tau(t), \chi_{\mathrm{c}}(\tau(t)))=\emptyset^{l}(u_{+(\tau}(t)))$and$\sigma(t)=\sigma_{-}(t))+tH’(\phi’(\sigma_{-}(t))=\chi_{C}(\mathcal{T}(t))+(t-\mathcal{T}(t))H’(\psi(\phi’(u_{+}(t)))$Since
above equality, we get$\chi_{\mathrm{c}}(\tau)=H’(\psi(\phi’(u_{+}(\tau)))$.

and $1+tH”(\emptyset’(\sigma_{-}(t)))\phi’’(\sigma_{-}(t))=0$ , differentiating the$H’(\phi’(\sigma_{-}(t)))=H’(\psi(\phi’(u+(\mathcal{T}(t))))$.

$+(t-T(t))H”( \psi(\phi(u+(\mathcal{T}(t))))\frac{d\psi}{dp}\emptyset’(u+(\mathcal{T}(t)))u_{+(\mathcal{T}(t))}’\tau’(t)$Then$\tilde{y}’(t, \sigma(t))=\tau l(t)\{-H(\emptyset’(u_{+}(_{\mathcal{T}(}t))))+\phi’(u+(\mathcal{T}(t)))H’(\phi’(u_{+}(\tau(t))))\}$$+\tau(t)\phi’(u_{+^{\tau}}(t))H\prime\prime(\phi’(u_{+}(\tau(t))))\phi’’(u+(\tau(t)))u_{+(}’\tau(t))\mathcal{T}’(t)$$+\phi’(u_{+}(\mathcal{T}(t)))u(’+\mathcal{T}(t))\mathcal{T}’(t)$$+(1-\mathcal{T}’(t))\{-H(\psi(\phi’(u+(\mathcal{T}(t)))))+^{\psi(\emptyset(}\prime u+(\mathcal{T}(t)))H’(\psi(\emptyset’(u+(\tau(t)))))\}$$+(t- \mathcal{T}(t))\psi(\emptyset’(u_{+}(\tau(t))))H\prime\prime(\psi(\phi’(u+(\mathcal{T}(t)))))\frac{d\psi}{dp}\phi’’(u_{+}(\tau(t)))\mathcal{T}’(t)$$=\mathcal{T}’(t)\{H(\psi(\emptyset;(u_{+}(\tau(t)))))-H(\phi’(u_{+}(\tau(t))))$$+(\phi’(u_{+}(\tau(t)))-\psi(\phi’(u_{+}(\tau(t))))H’(\psi(\emptyset l(u_{+}(\tau(t)))))\}$$+\psi(\phi^{l}(u_{+(\tau(}\iota))))H’(\phi’(\sigma_{-}(t))-H(\psi(\phi’(u_{+}(\tau(t)))))$.

52

By definition, we have$H’( \psi(\phi’(u+(\mathcal{T}(t)))=\frac{H(\phi\prime(u_{+}(\tau(t))))-H(\psi(\phi\prime(u_{+}(\tau(t)))))}{\phi’(u_{+}(\mathcal{T}(t)))-\psi(\phi’(u_{+}(\mathcal{T}(t))))}$,

so that$\tilde{y}’(t, \sigma(t))=\psi(\phi’(u_{+}(\tau(t))))H’(\phi’(\sigma(t)))-H(\psi(\phi l(u_{+}(\tau(t))))$.

Thus we have$\frac{d}{dt}(\tilde{y}(t, \sigma(t))-y-(t, \sigma(t)))=H’(\phi’(\sigma_{(t)})-H(\psi(\phi’(u+(\tau(t))))$$-H’(\phi’(\sigma(t)))(\phi;(\sigma_{-}(t))-\psi(\emptyset’(u+(\mathcal{T}(t)))))$

Since both$\phi’(\sigma_{-}(t))$$\psi(\phi’(u+(\tau(t))))$

for

and

$t>i_{\alpha}$$\psi(\phi’(u_{+((t}\mathcal{T}))))$. Therefore

belong to the concave region of $H(p),$.

$\emptyset’(\sigma-(t))<$,$\frac{H’(\emptyset\prime(\sigma(t))-H(\psi(\phi;(u_{+}(\tau(t))))}{\phi’(\sigma_{-}(t))-\psi(\phi’(u_{+}(\tau(t))))}<H’(\phi’(\sigma(t)))$. Since, the above
hence we have
. It follows that, there exists a uniqueforinequality means thatwith $x(t,u_{\alpha})<\chi_{r}(t)<\sigma(t)$ , such that $\tilde{y}(t, \chi_{r}(t))=y-(t, x_{r}(t))$ .
Then we can draw the picture of the graph of the viscosity solution forand the10)
bifurcation of the shock curves around . (cf., Figure$\frac{d}{dt}(\tilde{y}(t, \sigma(t))-y-(t, \sigma(t)))>0$$\tilde{y}(t, \sigma(t))>y_{-}(t, \sigma(t)))$$\tilde{y}(t_{\alpha}, \sigma(t\alpha))=y-(t_{\alpha}, \sigma(t\alpha))$$t>t_{\alpha}$$(t, \chi_{r}(t))$$t>t_{\alpha}$$t_{\alpha}$

FIGURE 10

For other cases, the detailed discussions will appear in elsewhere.
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$51\overline{/}-532$$geome\ell r\mathfrak{i}c$$\mathrm{I}<\mathrm{o}\mathrm{s}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{s}$$747-7\tau \mathrm{o}$$\mathrm{I}<\mathrm{o}\mathrm{s}\mathrm{S}\mathrm{i}\mathrm{o}\mathrm{r}\mathrm{i}\mathrm{s}$$So\iota ut\mathfrak{i}on$$singu\iota a\dot{\cap}teS$$singu\iota_{a}$