http://www.ictp.trieste.it/~pub_offIC/97/15

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

ANALYSIS OF SINGULARITIES IN THE NONLINEAR WAVESFOR QUASILINEAR HYPERBOLIC SYSTEMS

Kong De-xingInternational Centre for Theoretical Physics, Trieste, Italy.

ABSTRACT

It is well-known that classical solutions of quasilinear hyperbolic system in general havesingularities even if their initial data are smooth. In this paper we apply the singularitytheory of smooth mapping to show that the singularities of these solutions are in generalfold points, and at the points where the shock waves occur initially, the singular points arecusp points. At the same time, we prove that the blow-up points must be on the envelopecurve of characteristics in the same family. Finally, we give a clear structure of singularitiesof the solution for a system with the form of conservation laws.

MIRAMARE - TRIESTE

May 1997

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§1. INTRODUCTION

We consider a strictly quasilinear hyperbolic system of partial differential equations

ut + a(u)ux = 0, (1)

where u = u(t,x) is a vector with n components u\, • • • ,un and a(u) is an n X n squarematrix. We will discuss the singularities of the solution u of (1) which satisfies the initialdata

u(0,x) = <p(x). (2)

Throughout this paper, we will assume that both a(u) and <p(x) are smooth. The system(1) is called strictly hyperbolic if the matrix a(u) has n real and distinct eigenvalues Ai <• • • < \ n in the region under consideration. Associated to each eigenvalue A there are righteigenvector r{ and left eigenvector U respectively. We say that the system (1) is genuinelynonlinear if, in the u-space, the derivative of each A in the r{ direction does not vanish; (1)is linearly degenerate if, in the u-space, the derivative of each A in the r{ direction vanish(see [1]).

It is well known that, in general, the first derivative of the solution u will become un-bounded in finite time. For the case n = 1 and n = 2, see Lax's papers [2],[3] and Klainerman& Majda's [4]. For general n, see John [5], Liu [6], Li et al [7]-[9].

In this paper, we will study singularities of the solution of Cauchy problem (l)-(2) by thegeometric means. In §2 we will introduce the geometric theory on mapping. In §3 and §4we will deal with the single equation (i.e. n = 1) and systems in diagonal form (n > 2). In§5, we will study the general system. In §6, we will investigate a system with the form ofconservation laws and give a clear structure of singularities of its C1 solution. Finally, asan appendix, using one of our results we shall partly solve an open problem put forwrd byProfessor Li Ta-tsien.

§2. SINGULARITY THEORY ON SMOOTH MAPPING

Let us review the theory of singularities of mapping in Whitney [10] published in 1955.Since we are only interested in mappings from the plane into the plane, we restrict ourmapping II : U —> R2, where U is an open set in R2. Setting U(x,y) = (u,v) for any(x,y) G U, we introduce the Jacobian matrix

and J(x,y) = uxvy — uyvx its Jacobian.

Definition 1. A point p in U is called a regular point of the mapping U if rank (A) = 2 atp; otherwise, p is called a singular point ofU.

Obviously, a point p in U is said to be singular if and only if the Jacobian J vanishes atp. We shall further restrict the mapping II to be good in the following sense:

Definition 2. Let U be a C2 function. A point p in U is called a good point if eitherJ(p) / 0 or Jx{p) + Jy(p) 7 0. We say H is good if every point in U is good.

According to a Lemma stated in Lu [11], we have

Lemma 1. IfUis good in U C R2, then singular points ofU form a smooth curve in U. Inother words, if U is good, then J(x,y) = 0 defines a smooth curve in U. Let us parametrizethe curve J(x,y) = 0 by T(t) = (t,~f(t)).

Definition 3. Let p be a singular point of II and T(t) = (t,j(t)) be the parametrizingequation for J(x, y) = 0, such that F(0) = p. p is called a fold point of H, if ^(FI o F)(0) ^

(0, 0) , and p is called a cusp point of II, if ^(FI o F)(0) = (0, 0) but ^ ( 1 1 o F)(0) ^ (0, 0).

A typical example for fold point and cusp point is given by the mapping

u = xy — x ,

v = y.

In this case, J(x, y) = y — 3x2, DJ(x, y) = (—6x, 1) ^ (0, 0). By Definition 2, we know thatFI is good. Using Lemma 1, we can see that the singular points of FI form the parabolay = 3x2. Define T(t) = (t,3t2), then (II o T)(t) = (2t3,3t2) and ^(FI o F) = (6t2,6t). Thus,for any t ^ 0, the point F(t) is a fold point. However, for t = 0, ^(11 o F)(0) = (0,0) and^ - ( n o F)(0) = (0,6) ^ (0,0). By Definition 3, we know that (0,0) is a cusp point.

§3. SINGLE DIFFERENTIAL EQUATION

Now we return to the Cauchy problem (l)-(2) with n = 1. We rewrite it as

ut + a(u)ux = 0 (3)

with the initial conditionU(Q x\ = cp(x). (4)

In the region where the classical solution exists, we define the characteristic curve of (3),denoted by x = x(t), which satisfies

dr- = a(u(t,x(t))). (5)

From (5) and equation (3), we find that the solution of (3) is constant along characteristiccurves. If a characteristic curve starts from (0, a) then it will have the following form

x = a + a(ip((r))t (6)

and, the solution u of (3) is constant along this characteristic curve:

u(t, x) = (f(cr). (7)

It can easily be shown that the first derivatives of u can be writen as

u* = T-, , , w ,, X." ( 8)

Equation (8) shows that if -j^a(cp(a)) < 0, then ux will tend to infinity at a finite time alongthe characteristic (6). If we consider a coordinate transformation

' I + ^ ^ (9)— ^" i

then the corresponding characteristics in the (r, a) plane are a = const., and solution u, asexpressed in (7), is smooth in the (r, a) plane. Therefore, we may think that the solutionin the (t, x) plane has singularities only because the mapping given by (9) has singularities.Comparing (9) to (8), we see that ux tends to infinity at the points whose pre-images aresingular points of the smooth mapping II. The Jacobian of U is given by J(T, a) = Xa =1 + au(cp(a))cp'(a)r. Since -^a(cp(a)) < 0 in the region of our consideration, and henceJT z£ Q tne mapping U defined by (9) is good. Therefore, the curve J(r, a) = Xa = 0 is asmooth curve in the (r, a) plane. It is clear that J(r, a) = 0 can be solved for r as

(10)

The minimum of such r is attained at a. In order that this minimum point is a stable point(see the definition listed in Lu [11], pp. 14-20), we make a further assumption:

d3

a(ip(a)) > 0

at <7. Therefore we request that at a = a:

^ ^ ^ 0. (11)

With this a and f such that J(f , a ) = 0, the image point (t , x) under the mapping Ugives the starting point of a shock wave of u if (3) has a form of conservation law (see [12]

or §6 for details). If we parametrize the curve J(r, a) = 0 by T(a) = ( — _j_ * , ,, , a j , thend<T

( n o r ) ( < 7 ) = -

da

It shows that, for a ^ a, £ ( I I o T)(a) ^ (0, 0) and £ ( I I o T)(a) = (0, 0) only if a = a. The

third condition in (11) gives that -^^(TL o T)(a) ^ (0,0) at a = a. Thus, we conclude that

Theorem 1. Cauchy problem (3)-(4) possesses a unique global smooth solution on t > 0 ifand only if the following formula holds

-fa(ip(a)) > 0, Va G R. (12)da

Further, if for some a £ R such that -j^a(cp(a)) < 0 hold, then the smooth solution of (3)-(4) must blow up in finite time, and the first blow-up point must be the lowest point of theenvelope of characteristics. Furthermore, under the hypothesis (11), all singular points of(9) are fold points except (f,a) where the singular point is a cusp point (its image point(i, x) under the mapping II is the starting point of a shock wave of u if (3) has a form ofconservation law (see [12] or §6 for details)).

§4. QUASILINEAR HYPERBOLIC SYSTEM IN DIAGONAL FORM

In this section, we will consider Cauchy problem (l)-(2) with diagonal matrix a(u). Werewrite it as*

dt J dx (13)j- — Q . u . — m-(x).

We define i-th characteristic curves of (13) in the region where the classical solution of (13)exists. Denote the i-th characteristic curve starting from (0,a) by x = Xi(t,a), it satisfies

—l— = Xi(U'(t,Xi(t,a)),Ui(t,Xi(t,a)),U"(t,Xi(t, a))), ,dt (14)

t = 0 : x% = a.

where U' = (u\, • • • , Ui-i), U" = (^i+i, • • • , un).Because Ui is i-th Riemann invariant, along i-th characteristic x = Xi(t,a), we have

/J. /J. W / \ /-I f\

Ui(t,Xi(t,a)) = Lpi(a). {*-£>)

Thus, (14) can be rewriten as

X ^ a' = Xi(U'(t, Xi(t, a)), <pi(a), U"(t, xz(t, a))),

t = 0 : x% = a.

From (16), we get the following formula by simple calculation

dxi(t, a)da

= exp(F(t,a)) • A(t,a),

*In particular, if the system (1) contains only two independent unknown functions i.e., with n = 2, thecorresponding Riemann invariants exist (see Lax [3] for details). Therefore, a system of differential equations(1) can be always transformed into the form

ut + \(u, v)ux = 0,

vt + fi(u, v)vx = 0.

This is the special case of our consideration.

where

A(t,a) = l + ip'i(a) I ^-(£/'(r,a;i(r)),v?i(a),?7"(r,a;i(r,a)))exp(-F(r,

-(s , xi(s, a))ds.ox

We have (cf. [13]).

Lemma 2. There exists a constant CQ only depending on <pi and Xi such that the followingestimation holds

\F(t,a)\<Cot. (18)

Because the Riemann invariant Ui is constant along z-characteristic curve, the classicalsolution u of (13) is always bounded in the region where the solution exists (here we assumethat H^llc1 is bounded). If the first derivative of the solution u is also bounded, then theclassical solution can be solved further. So, the first derivative of solution must tend toinfinity as (t,x) tends to the first blow-up point.

Proposition 1. Along the i-th characteristic curve x = Xi(t,a), when t /* some to

dui(t,x)— >oo, (19)

if and only if

dXl^ ") —y 0. (20)

Proof. Differentiation (15) in a gives

dui(t,Xi(t,a)) dxi(t,a) ,

dx da

Thus, if u'g'x —> oo along the i-th characteristic curve x = Xi(t,a) as t —> to, then,from (21) we can easily get (21). Conversely, from (17)-(18), we can get ^p'i(a) ^ 0 . If (20)holds, from (21), we easily get (19). •

Proposition 2. Along x = Xi(t,a) (20) holds if and only if the point (to,Xi(to,a)) lies atthe envelope curve of i-th family characteristic curves.

Proof. According to the definition of envelope, the envelope curve of i-th family character-istic curves is given by the following parametric equations

x = Xi(t,a), Xt[ 'a' = 0, (22)da

where a is a parameter. From (22) we can easily get our proposition. •

Associating Proposition 1 with Proposition 2, we easily get

Theorem 2. The first blow-up point of the classical solutions of Cauchy problem (13) mustbe the lowest point of the envelope curve of same family characteristic curves.

Now, we return to study the types of singularities of the classical solution for Cauchyproblem (13). We restrict our attention to a neighborhood where system (13) forms a A -shock. More precisely we assume that -^f tends first to infinity at (i,x), here i is a fixednumber. We consider the coordinate transformation

{ x = Xi(r, a),

t = r ( 2 3 )

Hereafter, we may restrict our attention to a with -^ • tp'i(a) < 0- Equation (21) says

that -r f tends to infinity at those points where ga = 0 which are nothing but the

singularities of mapping (23) since J(T^O) = x'^T'a>. On the curve J(T^O) = 0, we have

dJ(r,a) d dxi(r,a) d , „— ) h ~ = a 7 — d a ~ ^ = ~da^ ^ (T>xi(T>a)>(Pi(a)>u ( r ' x « ( r ' a ) ) )

( A dU1 A dU N

+ ^v^' A, • ^ + V^'A, • ^ J J(r, a)

Therefore, the mapping (23) is good and we can solve r from -^-(r, a) = 0. Let

r = T(a) (24)

such that -r^-(T(a), a) = 0. Equation (24) gives a smooth curve in the (r, a) plane, r = T(a)attains its minimum at the point where T'(a) = 0. Differentiate -^-(T(a),a) = 0 withrespect to a to obtain

u xl

~daY

Since -f Jp- ^ O o n # - ( r , a) = 0, T'(a) = 0 at point where C ^ ( r , a) = 0. Let a be the

point on which

8\ • 32r- <93r-) , a ) > 0 , (26)

r = T(a) attains its minimum. Again the third condition makes the minimum stable.Corresponding to (f,a) in (t,x) plane is the point (f, x) with x = Xi(f,a),i = f.

As we did in §3, we parametrize the curve -^-(r , a) = 0 by F(a) = (T(a ) , a ) with T(o)given by (24), then

(IloT)(a) = (T(a),Xi(T(a),a))

and

da

on -rp- = 0 which does not equal to (0,0) for a ^ a. At ( r , a ) , according to (25) and the

second condition of (26), j-(U o T)(a) = (0,0), but

(II o T)(a) = (V'(a), ^ • T » ) + (0, 0),

since at a = a, T"(a) = -^t > 0. Therefore, we conclude that

Theorem 3. Suppose that (f,a) satisfies (26) and U is defined by (23). Then, in the regiona with -g^ • (p'i(a) < 0 the singular points of II are fold points for a ^ a and (f, a) is a cusppoint.

§5. GENERAL QUASILINEAR HYPERBOLIC SYSTEM

For a general system, we consider the following Cauchy problem

ut + a(u)ux = 0,

u(0, x) = <f(x).

We assume that system (27) is strictly hyperbolic in the sense that a(u) has n real anddistinct eigenvalues Ai, • • • , \ n with

A i < - - - < A n . (28)

Let ri and U denote the corresponding right and left eigenvector of a(u) associated to A .We normalize them in the way that

k r j = 8 i j , rt-rt = l} (29)

where Sij stands for the Kronecker's symbol. The Cauchy problem (27) can be written asthe following "standard form" (cf. [14])

c*r Ox^ r , , 5 ^ ~ x ^ _ n (30)at Ox Ox

t = 0 : v = vQ(x),

where v = (vi, • • • , un_i) f , v = (vi, • • • ,vny, and v is n-Riemann invariant in the sense of[1], vn is a function such that v = v(u) is invertible smooth mapping and its Jacobain doesnot vanish, B is an (n — 1) X (n — 1) square matrix, bn is an n — 1 dimensional row vectorand \ n is an eigenvalue of a(v) = a(u(v)).

Consider n—simple wave, i.e. v = const. In this case, we easily see that Cauchy problem(30) can be reduced to the Cauchy problem of single equation, this case will not be consideredhere (see §3 for details).

Now we turn to consider the singularities of the solution of Cauchy problem (27) withsmall initial data. We introduce the "invariants" (cf. [5])

for i = 1,- • • , n,

then, from (29) we have

(31)

(32)

Define the i-th characteristic field to be the solution of

Differentiating each Wi along i-th characteristic field, John [5] has shown the importantequations

dwi ^ - ^—j— = / HkmyUjW kW m, (33)

* k,m=l

where 7 ^ = 0 for k ^ i, Under certain assumptions, John proved, by using the relation(33), that at least one of Wi will become unbounded at a finite time.

Hereafter, we require that the system (27) is genuinely nonlinear in the sense of [1], thederivative of each A in the r direction does not vanish, i.e., ^m ^ 0 (i = 1, • • • , n), andassume that <p(x) is a small initial data with compact support. For the sake of simplicity,we assume that wj(t, x) (I £ {1, • • • , n}) tends first to infinity sooner than all the other Wj(j ^ / ) . In particular, if (i,x) is the first singular point, John [5] shows that all the otherWi will remain bounded in a neighborhood of (i,x).

As in the previous section, we introduce a coordinate transformation

x = X{T,OL)

where X(r,a) satisfiesO y

— = \I(U(T,X)), X(0,a)=a.

<34»

(35)

Under the same conditions of [5], we can show that WJ(T, X(T, a))Xa is a regular functionup to i. Furthermore, WJ(T, X(T, a))Xa is bounded and different from zero.

In fact, for the sake of simplicity, we assume that the initial data has the following form

where & is a nontrival smooth function with compact support.Without loss of generality, we will discuss in the normcoordinates (cf. [7]). From [7] or

[15], we know that if (i,x) is the first blow-up point, then a of its pre-image point (f,d)satisfies

wA0,a) = sA > 0,

where we assume that 7/// > 0, and here A is a constant.As [5] or [7] shows, wj must tend to infinity as t —> i fixed, futhermore there exist two

constants K\,K2 > 0 such that i £ [K\e~1, A^e"1]-Along the /-th characteristic issued from (0,a) we have (cf. [5] or [7])

d(wiXa) -A— = 2_^ Lijk{u)WjWkXa, (36)

J,k=l

where Tjjj = 0, for all j .For fixed constant to (cf. [5] or [7]), we have

\wi(t0) - 10/(0)1 < Ce2, |Xa(t0) - 1| < Ce.

From the above formulas, we have

\Ae < Wl(t0) < \Ae, \Ae < wT (t0) Xa (t0) < \Ae.4 6 4 6

By [5] or [7], we know that wj(t) > \Ae (t > to) is an increasing function, and

wk(t,X(t,a))\ < Ce2 < wi(t), for K2e~x >t > t0, k ^ I.

Integrating (36), we easily get

-As < WlXa (t) < -As, for t0 <t< I ^ e " 1 .

Thus, wj tends to infinity at the points where Xa tends to zero. This shows that the firstblow-up point of the classical solution of Cauchy problem (27) must be the lowest point ofthe envelope curve of characteristics in the same family.

Xa = 0 is the set of singular points of the mapping U defined by (34). Differentiate Xa

with respect to r to yield

cIImwm,O

where c//m depends on u and CJJJ = 7///. Since 7/// and wjXa are different from zero onXa = 0, we have

-—?- = -~fniwiXa. (37)

Following definition 2, the mapping U of (34) is good, Xa = 0 defines a smooth curve ina neighborhood of (f,d) which is the pre-image of (i,x) under II. Relation (37) allows usto solve Xa(r, a) = 0 for r. Let

r = T(a) (38)

10

be its solution. Differentiating Xa(r,a) = 0 with respect to a and using (38), we have

Xaa+XaTT'(a) = 0,

on Xa = 0, XaT ^ 0, T'(a) = 0 if and only if Xaa = 0. Therefore, a which makes f = T(d)minimized satisfies

W'j ± 0, Xaa(T(a), a) = 0, Xaaa{T{a), a) ^ 0. (39)

Again, the third condition in (39) makes the minimum stable.If we parametrize Xa = 0 by T(a) = (T(a), a) with T(a) given by (38), then we have

) = (T(a) ,X(T(a) ,a)) .

The same calculation shows that on Xa = 0 with a ^ a,

and at (f, a)

^ ) = (o,o), but

Theorem 4. Under the same conditions discussed in [5], if u = u(t,x) is the solution of(27) with "small" initial data with compact support, then ux must blow up at finite time, andthe first blow-up point must be the lowest point of envelope curve of same family characteristiccurves. Futhermore, there exists a smooth coordinate transformation U defined by (34) suchthat the singular points of U are the singular points of u, and singular points of U form thecurve Xa = 0 where all points are fold points except (f,a) where (f,a) is a cusp point.

Remark. For the Cauchy problems discussed in [6], [7]-[9], using the same method, we canget the similar results.

§6. SYSTEM WITH A FORM OF CONSERVATION LAWS

6.1 Single conservation lawFirst, consider the single convex equation of conservation law

where f(u) £ C3 and satisfiesf"(u) > 0. (41)

Leta(u) = f'(u). (42)

(42) means equation (40) is genuinely nonlinear in the sence of Lax [1].

11

Suppose that x = x(t) is a shock wave, then on x = x(t) we have Rankine-Hugniotcondition

[u]dx - [f(u)]dt = 0,

namely,

dx \f(u)] f(u+) - f(u-)-jr = r i = r z — = 3\t, xyt)),dt [u\ u+ — u

and the entropy condition

i - w d i ^ i +\a(u ) > —— > a[u J,

where u± = u±(t, x(t)) = u(t, x(t)) ± 0). Since a'(u) = f"(u) > 0, then we have

u~ > u+. D

Consider the following Cauchy problem

du df(u) =

dt dx ' (43)t = 0 : u = 4>(x),

where (f)(x)is a C3 function with bounded C1norm. Let

$(x) = a((j)(x)). (44)

Similar to (11), we suppose that

<j>\x) < 0, Vx £ R (45)

and there is only one point XQ £ R such that

$"(x0) = 0 and $'"(x0) > 0. (46)

Noting (41)-(42) and (44), (45) is obviously equivalent to

while, noting the bounded C1 norm of 4>(x), we know that there are limits as x —> ±oo,denoted by

(f>+ = l im (f>(x) a n d (f>~ = l im (f>(x).x >-+oo x > — oo

Then we have (cf. [16])

12

Theorem 5. Under the assumptions mentioned above, the C1 solution of the Cauchy prob-lem (43) must blow up at the time

, - 1t0 = - {a (4>(XQ)(J)'(XQ)) ,

and for any given point (t, x) in a neighbourhood of (to, XQ) we have

u(t, x) — u(t0, x0) \< Co l(t — t0) + (x — x0)

Du(t,x) |< Co ((t-t0)3 + (x-x0)

2 t0,

, x)\<Co[{t- to

and

where

x) <Co((t-

xo

(X — XQ

for i / i 0 , t ^ t0,

for i / i o , t ^ t0,

D = dt + \(<f>(x))dx and D1- = -\{(j){x))dt + dx,

and Co is a constant independent of (t, x). Furthermore, there exist two envelope curvesx = Xi(t) (i=l,2) on t > to formed by the characteristics x = a + \((j)(a))t (a > XQ) andx = (3 + \(4>(j3))t (J3 < xo) respectively; at the same time, there is one and only one C1

shock wave x = £(t), starting from the point (to, XQ), satisfying

Xl(t) < x(t) < x2{t), t>t0.

Moreover,, the shock x = x(t) tends asympotically to the shock in the similar solution of thefollowing Riemann problem

( du df(u) =

dt dxU = (f>+ , X > Xo,

= t0:U = <f) , X < X0,

as t —y 00.

6.2 System with a form of conservation lawsNow we turn to consider the following model arising from the nonlinearly elastic string

theory and the oil recovery problem in one space dimension:

ut + (g(r)u)x = 0,

vt + (g(r)v)x = 0,(47;

where (u, v) is the unknown function, r = yu2 + v2, g is a given smooth function of r.

13

It is easy to see that the characteristic speeds are

Ai = g(r) and A2 = g(r) + rg'(r).

We assume that

flf'(r) > 0 and (rg(r))rr > 0 for r > 0. (48)

Then, when r > 0, it is easy to know that the system (47) is strictly hyperbolic, namely

Ai(r) <A2(r)

and the eigenvalue Ai(r) is linearly degenerate, while, A2(r) is genuinely nonlinear in thesense of Lax [1].

Consider the following initial data

t = 0: (u,v) = (uo(x), vo(x)), (49)

where UQ(X) and VQ(X) are smooth functions of x with bounded C1 norm.We assume that

furthermore, we assume thatr'o(x)<0, V x G i ? (51)

and there is one and only one point x* G R such that

G"(x+) = 0 and G'"(x±) > 0, (52)

where

G(x)=g(ro(x))+ro(x)g'(ro(x)).

Introducer = v u2 + v2 and 6 = arctan( —),

vthen system (47) can be rewritten as follows

<9r d(rg(r)) _

dt dx ' ^ )

here we assume that u ^ 0.By (50) and the first equation of (53), on the existence domain of the C1 solution we have

r(t,x) > 0.

Under the assumptions of (48), (50)-(52), by Theorem 5, we have

14

Lemma 3. The C1 solution of Cauchy problem for the first equation of (53) with the initialdata (50) must blow up at the time

- i

and for any given point (t, x) in a neighbourhood of (£*, x*) we have

| r(t, x) - r(U, x*) \< Cr ((* - Uf + {x-x*

n r(f T\ < n I (f _ f Y _i_ (T _ T s

x+, t

and

where

t,x) \<Cr((t-U

Dr = dt + G(x±)dx and D^r = -G{x+)dt + dx,

and Cr is a constant independent of (t, x). Furthermore, there is one and only one C1 shockwave x = x(t), starting from the point (£*, x±), for t > t*.

Similarly, we introduce

= dt + g(ro(x±))dx and DJ- = -g(ro(x±))dt + dx.

Then we have

and

Noting

where a = a(t,x) is defined by

l+g(ro(x,))G(x,)U8 = -, , s-<o , ^ Dr —

r0 (x+) g'

9' ^ D _L1 + G2 (

g (r0 (a;*)) G (x+]

(54)

(55)

(56)

6(t,x)=60(a(t,x)),

d= a and £(t) = x.

By (58) we have

a(t,x) = x - (59)

15

Then it follows from (54), (57) and (59) that

De6(t, x) = e'0(a(t, x)) [g (r0 (x*)) - g(r(t, x))} , (60)

andD$-6(t, x) = e'0(a(t, x))[l+g ( r 0 (x*)) g(r(t, x))} . (61)

Noting9 (ro (a;*)) - g(r(t, x)) = g (r (t+, x+)) - g(r(t, x))

= g'(-)(r(U,xir)-r(t,x))

and Lemma 3, it follows from (60) that

I D e 0 ( t , x ) \ < C 0 ({t - Uf + ( x - x ^ 6 (62)

and

\DJ-6(t,x)\<Ce, (63)

hereafter Cg is a constant independent of (t, x). Furthermore,

DeDe6(t, x) = 6o'(a(t, x)) [g ( r 0 (x*)) - g(r(t, x))}2 - 60(a(t, x))Deg(r(t, x))

= e'0'(a(t, x)) [g ( r 0 (x+)) - g(r(t, x))]2 - 6'0(a(t, x))g'(r(t, x))D0r.

Using (55)-(56) and Lemma 3, we have

\D2e6(t,x) \<Cg({t-t+f+ {x-x+)2^ 3 iovx^x^t^U. (64)

Similarly, we get

\DeLDe

L8(t}x)\<Ce((t-U)3 + ( x - x ^ * for x ^ x+, t ^ t+. (65)

Thus, (62)-(65) give the estimates of singularities of 6 = 9(t,x) at the point (t*,x+). Obvi-ously, the singularity of 6 = 9(t, x) at the point (£*, x*) is weaker than that of r = r(t, x) atthe point (t+, x*).

Now we turn to consider the following initial data problem

X (t*) = X*.

Because the function g = g(t,x) = g (r (t,x(t))) is continuous at the point (£*, x±), but isnot Lipschitz continuous, the existence of C1 solution for (66) has been proved in G.Jennings[17]. Our aim is to show the uniqueness of solutions.

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We first show the function g(t,x) is decreasing with respect to x in a neighbourhood of(£*, x*). Noting the first inequality of (48) we have

—g(r(t,x)) =g'(r(t,x))rx(t,x)<0,Ox

where we use the following fact that in a neighborhood of (£*, x*), rx(t, x) has the same signof r'0(x+).

Let x = x\(t) < X2(t) be two solutions of (66), then

-x^t) =g(r (t, Xl(t))) >g(r (t, x2{t))) = -x2(t) for t > t*.

Hence we get x\(t) > x2{t) for t > t*. This contradicts the hypothesis. Thus, we have

Theorem 6. Under the assumption of (48), (50)-(52), the C1 solution of Cauchy problem(53) and (49) must blow up at the time t± given by Lemma 3, and the estimates (62)-(65).Furthermore, there exists a unique C1 shock wave x = x(t; £*, x+) and a unique C1 weakdiscontinuous x = x(t; £*, x+) starting from the point (£*, x±) for t > t+.

APPENDIX

In this appendix, we will give an application of the above results. Consider Cauchyproblem (13), where the initial data <p(x) is smooth and satisfies

\<p(x)\ci = sup \<p(x)\ + sup \<p'(x)\ < oo.

We have (cf. [13])

Theorem A. / /

}(fn(an)) .> 0, for i = 1, • • • , n; a.\ < , • • • , < an,

then Cauchy problem (13) has a unique global smooth solution on t > 0.

In particular, for strictly hyperbolic system (13), in [13] it has been pointed out that theCauchy problem (13) always possesses a unique global smooth solution for arbitrary smoothinitial data if and only if the system (13) is linearly degenerate in the sense that

—- - = 0, for i = 1, • • • ,n.

Furthermore, if the system (13) is genuinely nonlinear in the sense that

dX* -L n f - 1—— T= I), tor i = 1, • • • , n ,

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without loss of generality, we may assume

^ > 0 , (Al)

if

ip'i(x) > 0 , f o r i = I , - - - , n , (A2)

then Cauchy problem (13) has a unique global smooth solution on t > 0. Conversely, iscondition (50) a necessary condition which guarantees a global smooth solution for Cauchyproblem (13) ? This is an open problem put forward by Professor Li ta-tsien. In the followingargument, using the results of §4, we solve partly this question. We have

Theorem B. Assume that system (13) is genuinely nonlinear in the sense of (Al) and theinitial data <p(x) has limits as x —> ±oo. / / Cauchy problem (13) possesses a unique globalsmooth solution in the upper half space, then (A2) must hold.

Proof. We first observe that

lim u(t,x) = lim <f(x), for fixed t > 0.x—>-±oo x—>-±oo

Denote j-th characteristic curves by Xj(t), we compute limt-too u(t,Xj(t)) as follows: Letyj(t) be the point at which the j-th characteristic through (t, Xi(t)) intersects the line t = 0.Then

where

= sup

Thus if j < i,

as t —> oo, by the hypothesis that A^^ > A^ax. Now, since Uj(t,Xi(t)) = <pj(yj(t)), we havethat

Uj(t,Xi(t))-

as t —> oo. Similarly,

Uj(t,Xi(t))

when j > i. Thus

lim u(t,Xi(t)) = $(x),

where3), 3 < h

• o o ) , j > i .

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Evidently the i—th characteristic through (0, Xi(0)) is asymptotically a line with speed -^- =\i(<&(x)). Since different i-th characteristics do not intersect (see Theorem 2), it must bethat Ai($(x)) is an increasing function of x: x\ < x2 implies that

0 < A,nl d

/o d6

= (ifi(x2) — (fi(xi)) / —— (ifi(oo), • • • ,6(fi(x2) + (1 — d)ifi(xi), • • • ,<pn( — oo)) d6.Jo ouz

This together with the assumption (Al) implies that fi(x) is increasing. That is to say,(A2) holds. •

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