5.1. Construction of Solutions for Piecewise Monotone Initial Data
We assume that f”(u) has exactly two zero points a, and a2. For definiteness, we assume that
(A): f”(u) < 0 for u>a2 and u<a,, f”(U) > 0 for a,<u<a,,
f(al+v)~f(a,)+f’(a,)rl+GP, lvll small,
f(az + rl)gf(aJ +f’(ad v - KZqk2,
144 small,
where K, and K, are two positive constants and k, and k, are two odd integers which are greater than three.
In order to consider the most complete case, we also assume that (B): ,‘\rnm f(u)= lim f(u)= --CO.
U’ -cc We need some properties off:
LEMMA 5.1. Assume that f satisfies assumptions (A) and (B). Then there exist 6, and 6, such that
f’(b )=f(b1)-f(b2)=f~(b I )
b, -6, 2
(5.1)
where b, < a, and a2 < b,. Furthermore, 6, and b2 are uniquely determined.
104 KUO-SHUNG CHENG
ProoJ Let
Since
f
is convex on ul, a2, it is easy to see thatHence a, ES. Now let 6, = inf 5’. From assumption (B), we know that h, is finite. Let {c,};= , be a sequence of S such that
lim c,=bl. (5.3)
n-a2
Let {d,);=,, &>a, for n= 1, 2,..., be a sequence corresponding to the sequence {c, ),“= 1 satisfying
f’(c n
) <f(CJ -f(dn)<f’(d
cn - 4
n 9 ) n = 1, 2,... .
From assumption (B), {d,}, = I is a bounded sequence. Hence, by passing to a subsequence if necessary, we can assume that
Iim d,,=b,.
n-30 (5.5)
We shall prove that
f’(b )=f(b+f(b2)=yb ),
1
61-h
2 (5.6)
Assume that
(5.7)
Then from the assumption (A), there exists a sufficiently small E > 0, such that
f’@,) <f’(b, - &I< I-V, -E) -f(U
b,-E-bb3
<fWf(W<f,(b )
b,-b, ’ 2’ (5.8)
This contradicts the definition 6, = inf S. Hence
f,(b ) Jv,) -f(b)
1 b,-bz ’
Now assume that
f(bl)-f(bJ <f,(b2) b, - 6,
(5.9)
(5.10)
Then there exists a sufficiently small E < 0, such that f’(b )<f(b,)-f’(bz)<f’(bl)-f’(b,+&)
I -.
b, -b2 b, - (b, + E)
<f’(b,+~)<f’(bz).
(5.11)But in this case there still exists a sulliciently small 6 > 0, such that
(5.12) This means that (6, -6)~ S, which is a contradiction to b, =inf S. Thus
(5.13) It is not difficult to see that b, and b, are uniquely determined. This
completes the proof of this lemma. Q.E.D.
LEMMA 5.2. For b, < u < a,, there exists unique u*, u* > a,, such that f’(U) J-(u) -f(u*) <f’(u*)
U-Mu* (5.14)
Proof The line passing through (u, f(u)) with slopef’(u) must intersect the graph off at exactly two points (u*,j(u*)) and (U*,f(ii*)), where a, <u* < 6, and U* > b,. It is easy to see that
f(u) -f(u*) <f’(u*)
u-u* and “f(u) -Au*) >s,(il*).
u--ii*
This completes the proof of this lemma. Q.E.D.
106 KUO-SHUNG CHENG
LEMMA 5.3. For a, -C u < bZ, there exists unique u*, u* -C aI, such that f(u*)<f(u)-f(u*Lj-‘(u).
u-u* (5.15)
Proof: The proof is similar to that of Lemma 5.2. We omit it. Q.E.D.
LEMMA 5.4. For a, <u < a,, there exist unique u* and u* such that u* <a, and u* > a,, and
f’(u,) >f(u*)-f(u)=f’(u) =f(u)-f(u*) >f’(u*) . (5 16) .
u*-u U-U*
ProoJ The proof is also similar to that of Lemma 5.2. We omit it.
Q.E.D.
We need some definitions.
DEFINITION 5.5. Let x( .): (a, b) + R be a shock of a generalized solution u(x, t) of (l.l), that is, u(x(t) + 0, t) # u(x(t) - 0, t) for all t E (a, b) and x( . ) is Lipschitz continuous on (a, b). We call this shock
(i) a type-1 shock, if
.7(4x(t) - 0, t)) > x’(t) >f’(u(x(t) + 0, t)) for almost all t E (a, b).
(ii) a type-II-f. shock, if
f’(u(x(t) - 0, t)) > x’(t) =f’(u(x(t) + 0, t))
for almost all t E (a, b) and f’(u(x(t)-0, t)) is a monotone decreasing function of t.
(iii) a type-II-R shock, if
f’(u(x(t)-0, t))=x’(t)>f’(u(x(t)+O, t))
for almost all t E (a, b) and f’(u(x(t) + 0, t)) is a monotone increasing function of t.
(iv) a type-III-L shock, if
j-‘(u(x(t)-0, t))>x’(t)=f’(u(x(t)+O, t))
for almost all t E (a, 6) and f’(u(x(t) -0, t)) is a monotone increasing function of t.
(v) a type-III-R shock, if
f’(u(x(t)-0, t))=x’(t)>f’(u(x(t)+O, t))
for almost all t E (a, b) and S’(u(x(t) + 0, t)) is a monotone decreasing function of t.
(vi) a type-IV shock, if
f’(u(x(t) + 0, t)) =x’(t) =f’(u(x(t) - 0, t)) for all t E (a, b).
We give some examples of these shocks.
EXAMPLE 5.6. Let U,,(X) = U, for x<O and U,(X)= U, for x>O, where b, < U, < u, < 6,. Then the function u defined by
24(x, t) = ll/ if x < dt,
= u, if x > bt, (5.17)
is a generalized solution of (1.1) with these initial data, where CJ = (f(u,) - f(u,))/(u, - u,) and x(t) = at is a type-1 shock.
Proof: It is easy to see that u is a weak solution. From the definition of b, and b2, we easily obtain that
f(u) -f(4) po -“our) ,s(uc) -f(u)
(5.18)u - 24, UI - ur u,-u
for all UE [u,, u,]. Hence u satisfies the entropy condition E and u is a generalized solution of (1.1). Taking the. limits u + uI and u + U, in (5.18), we obtain
f’(# r \ ) <f(“+f’(ur),f’(u,).
u/--u,
(5.19) Since b, -c u, < u, < b2, we have
f’(u
r ) <f(u() -f(h) <f’(u,) u/--u,
Thus x(t) = at is a type-1 shock. Q.E.D.
For type-II-L shock (type-II-R shock is similar), there is already an example in Section 4. So we give a type-III-L shock in the following exam- ple.
108 KUO-SHUNG CHENG
EXAMPLE 5.7. Let u,,(. ) be monotonically increasing for x < 0, a,du,(x)< b2 for X-CO, and q,(x)=u, for x>O, where 6, <u,<al and (u,)* = u,(O - 0) (recall the definition for (u,)* in Lemma 5.2). We shall briefly describe the construction process of the generalized solution of ( 1.1) with these initial data.
Let ui(x, t) be the generalized solution of (1.1) with the initial data 4(X)>
d(x) = kl(x)
if x < 0,=a2 if x> 0.
Since ub(x) E [a,, bJ, ur(x, t) can be obtained from the method in Sec- tion 3. We can divide the x - t half plane t > 0 into three regions D, , D2, and D,,
D, = {(x, t): x < y(t), 0 < t}, D2=((x,t):y(t)<x<f’(a,)t,0<f}, D3= {(x, t):f’(a,)t<x,O<t},
where r(t) is a type-I shock. u,(., t) is monotonically increasing in (- co, y(t)), ur(x, t) = h3(x/t) for (x, t) E D,, and u,(x, t) = a, for (x, t) E: D3, where h3 is the inverse function off’ restricted in [a,, co). Now consider +(x, t) which is defined in DI by
(%(X, t))* =%(x, t), (x, ~)EDI.
Let x = x(t) be a Lipschitz continuous curve satisfying
y =fT(u*(x(t), t)) x(0) = 0.
for almost all 2 E (0, 00 ),
It is easy to see that x(t) is a concave Lipschitz continuous curve in D, and dx( t)/dr is monotonically decreasing. From every point of (x(t), t), we draw all lines L,(U) with speed v between x’(t + 0) and x’(t - 0) in the positive time direction. It is easy to see that all these lines L,(v) cover a fan-like region H without intersection with each other, where
H= {(x’, t’): x(f) <x’<f’(u,) t’, 0< t’}.
Let u(x, t) be defined by
u(x’, t’) = 24*(x’, t’) if x’ < x( t’),
= h,(u) if (x’, t’) E L,(u) c H, (5.20)
= 24, if f’(u,) t’ < x’,
where h, is the inverse function off’ restricted in (-co, a,]. Now the function u defined in (5.20) is a generalized solution of (1.1) and x = x(t) is a type-III-L shock.
EXAMPLE 5.8. Let uO( + ) be a monotonically decreasing function in (- co, co) with u,(O + 0) = b, and ~(0 - 0) = b,. For every ,v < 0, we draw all lines L,(u) from (y, 0) with speed v between f’(uo(y - 0)) and f’(u,(y + 0)). Along the line L,(u) we assign u = h3(o). For every y > 0, we
draw all lines L,(u) from (y, 0) with speed between f’(u,(y - 0)) and f’(u,( y + 0)). Along this line L,(u) we assign u = h,(u). Here h, and h, are inverse functions of j”’ restricted in regions ( - co, ai] and (a,, co ), respec- tively. It is easy to see that these lines L,(u) cover the upper x - t plane except the line x = ot, where r~ = (f(b*)-f(b,))/(b* - b,). The function u defined by the above arguments is continuous except on line x = at. It is easy to see that this u is a generalized solution for (1.1) and x = at is a type-IV shock.
For piecewise monotone and bounded initial data uO(. ), we shall briefly describe the method of construction of the generalized solution for (1.1).
For details we refer to Cheng [5]. For convenience we let a, = -co and a3=oo.
Assume that uO( .) is bounded and piecewise monotone. Furthermore
%(X)E(-% a,1
for x<Oand
for x>O. (5.21)
First we use the construction method of Section 4 to construct the generalized solution of (1.1) with the initial data
&l(x) = %(X)
if x CO,= u2 if x > 0. (5.22)
We obtain a G(x, t) which is Lipschitz continuous and aG(x, t)/dx = ii(x, t) is a generalized solution with initial data (5.22). It is easy to see that ii may
110 KUO-SHUNG CHENG
8GR(x, t)/dx z u~(x, t) is a generalized solution of (1.1) with initial data
5.2. Properties of Solutions for Piecewise Monotone Initial Data From the construction of generalized solutions for ( 1.1) we have which are genuine characteristics, type-1 shocks, type-II-L shocks, type-II-R shocks, type-III-L shocks, type-III-R shocks, or t+vpe-IV shocks.
(ii) u( ., t) is piecewise monotone, especially u(x * 0, t) = u k exists for
112 or
KUO-SHUNG CHENG
u(x(t) + 0, t) = (u(x(t) - 0, t))*.
(ii) Zf x(.): (a, b) + R 1s a type-II-L shock, then x’( .) is Lipschitz continuous, x”(t) < 0 for almost all t E (a, b), and
u(x(t) - 0, t) = (u(x(t) + 0, t))*
u(x(t) - 0, t) = (24(x(t) + 0, t))*.
(iii)
If
x( .): (a, 6) + R is a type-III-L shock, then x( .) is Lipschitz continuous, x’(t) is a decreasing function of t. Furthermore, for fixed toEta, b),x’(t,+ 0) = lim f’(u(x(t) + 0, t)), t-to+
x’(tO-0)= lim f’(u(x(t)+O, t)), , - to-
b,> @(to) - 0, toI > a,, a, > Wto) + 0, to) 2 b,,
lim u(x(t) - 0, t) = ( lim u(x( t) + 0, t))*,
, + to+ r + to+
and
lim u(x(t)-0, t)=( lim u(x(t)+O, t))*.
f + *o- r - to-
(iv) If x(. ): (a, 6) + R is a type-III-R shock, then x(.) is Lipschitz continuous, x’(t) is an increasing function of t. Furthermore, for fixed
to 6 (a, b),
x’(to + 0) = ,lJT+ f’(u(x(t) -0, t)), x’(to - 0) = pm f’(u(x(t) - 0, t)),
6, < u(x(t,) + 0, to) <a,, a2 < u(x(t,) - 0, to) < bz, lim u(x(t)+O, t)=( lim u(x(t)-0, t)),,
t+,0+ t--r@+
and
lim u(x(t) + 0, t) = ( lim 24(x(t) - 0, t))*.
I - 10- r-to-
Proof: (i) From the definition of type-II-R shock, we have
f’(u(x(t)-0, t))=x’(t)>f’(u(x(t)+O, t)) (5.28) for almost all t E (a, 6) and f’(u(x(t) + 0, t)) is monotone increasing. From conditions (R-H) and (E) of Section 1, we have
(5.29)
x,(t) JIW~) + 02 t)) -.f(u(x(t) -09 t)) u(x(t)+O, t)-u(x(t)-0, t) ’
J-(4x(t) +o> cl) -f(u(x(t) - 0, t)) $4 --f(u(x(t) -07 cl)
u(x(t)+O, t)-u(x(t)-0, t) (5 30)
u-u(x(t)-0, t) .
for all U’S between u(x(t) + 0, t) and u(x(t) - 0, t). From (5.28) and (5.29) we have
4x(t) + 0, f) = (4x(t) - 0, f)),
(5.31)if U(X( t) + 0, t) E [a,, a2] and
u(x(t) + 0, 2) = (u(x(t) - 0, I))* (5.32) if U(X( t) + 0, t) E [a,, co). Furthermore, since f’(u(x(t) + 0, t)) is monotone increasing, u(x(t) + 0, t) is Lipschitz continuous. From (5.31) and (5.32), we can regard u(x(t) - 0, t) as a differentiable function of u(x(t) + 0, t).
Hence from (5.29), x’(t) is a Lipschitz continuous function and x”( 2) = d f(u(x(t) + 0, t)) --f(WU - 0, t))
44x(f) + 0, t)) u(x(t)+O, t)-24(x(t)-0, t)
1
du(x( t) + 0, I)
dt ’
For either case (5.31) or (5.32) it is easy to see that x”(r) 20 and x”(t) = 0 only if du(x(r) +O, l)/dt =O. This proves (i). Case (ii) can be similarly proved.
Cases (iii) and (iv) can also be similarly proved. The only difference is that u(x(t) - 0, t) can be discontinuous in case (iii) and u(x(t) + 0, t) can be discontinuous in case (iv). See also Example 5.7 to get a feeling of type-III
shocks. This completes the proof. Q.E.D.
114 KUO-SHUNGCHENG
THEOREM 5.11. Let u(x, t) be the generalized solution in Theorem 5.9. If f’(u( ., to)) is monotonically decreasing or is a constant in the interval (a, b),
then
L,={(x,t):x=a+f’(u(a+O,t,))(t-t,),O<t<t,}
and
L2 = {(x, t): x = b + f ‘(u(b - 0, to))( t - to), 0 < t < to}
are two genuine characteristics, that is,
u(x + 0, t) = u(x - 0, t) = u(a + 0, to) for all (x, t)E L, and
u(x + 0, t) = u(x - 0, t) = u(b - 0, to) for all (x, t) E Lz.
Proof. From our construction of solution U(X, t), each type-II or type-III shock generates a fan with monotonically increasing f’(u( ., t)).
Hence the backward genuine characteristics from (a + 0, to) and (b - 0, to) cannot terminate at a type-II or type-III shock. Obviously they also cannot terminate to a type-1 or type-IV shock (entropy condition (E)). Hence they must extend to t = 0. This completes the proof. Q.E.D.
THEOREM 5.12. Let u(x, t) be the generalized solution in Theorem 5.9.
Let (x0, to) be a point with to > 0. Zf u(x, - 0, to) = b, and u(xO + 0, to) = b,, then u(x - 0, t) = b, and u(x + 0, t) = b, for ah (x, t) E L, where
L = ((x, t): x = x0 + f’(b,)(t - to), 0 < t < to}.
Proof It is obvious that the backward genuine characteristics from (x,-O, to) and (x0 +O, to) are the line L. This line L is a type-IV shock which cannot terminate to any type-II or type-III shock. This completes
the proof. Q.E.D.
THEOREM 5.13. Let u(x, t) be the generalized solution in Theorem 5.9. If f’(u( ., to)) is increasing in the interval (x1, x,), then f’(u( ., t)) is continuous in (x1, x2), where t, > 0. Furthermore, zf there is no type-IV shock passing through the line segment {(x, t,): x1 <x < x2}, then u( ., to) is also con- tinuous in (x1, x2) and
{u(x, t,):x,<x<x,}c(--,a,] or [a,,a,] or [a,, co). (5.33) Proof: Assume that f’(u( ., to) has a jump discontinuity at X~E (xi, x,), then since f’(u( ., to)) is increasing in (x,, x,), we have f’(u(x,- 0, to)) <
f’(u(x, + 0, to)), which contradicts Theorem 5.9(ii). Hence f’(u( ., to)) is
116 KUO-SHUNG CHENG generate new genuine characteristics in the positive time direction besides type-II and type-III shocks is the interactions of different types of shocks.
ficiently small such that there is a type-II-L shock or type-II-R shock .zR(t) passing through points ( yk( tk), tk) and (x,J Tk), Tk) and
It, - Tkl d 6, k = 1, 2 ,..., n, Tkafk-1, tkb Tk-,, k = 1, 2 ,..., n,
where 6 is the constant in Lemma 4.6 which is still valid in thisf. Then there exists a constant C depending only on f and M, such that
04f’(v,)-f’(u,)~~A(c,d; T), (5.35)
where A(c, d; T) is the area of region bounded by C,(c, T) and C,(d, T), t=O and t=T.
Proof The proof is similar to the proof of Lemma 4.7 or the proof of the following Lemma, Lemma 5.17. Since we will give a complete proof for Lemma 5.17, we just omit the proof of this one. Q.E.D.
LEMMA 5.17. The same assumptions as in Lemma 5.16 except that t, > 0 and T, > 0. Assume that (d-c) is sufficiently small such that there is a type-III-L shock or a type-III-R shock passing through (x,(TO), T,,) and (y,,( to), to). We extend C,(c, T) and C,( y, T) to t = 0 by fines
x-l(t)=x,,(To)+f’(u-,)(t- T,), O<t<T,,, Y-l(t)=Yo(to)+f’(v-,)(t-to), o<t<t,, where
u_, = (uo)* or u-1 = (%A
V -, = (vo)* or v1 = (v0)*.
Then the conclusion of Lemma 5.16 in (5.35) is still valid except now A(c, d; T) is the area of the region bounded by Cb(c, T), C,(d, T), x-,(t), y,(t), t=O and t= T.
Proof: From the properties of type-III shock, Theorem 5.10, it is easy to see that f’(u-,) >f’(v-r). It is to be noted that x-,(t) and yeI need not be true backward genuine characteristics from points (xO(TO), T,,) and ( yo(to), to). But A(c, d; T) and A(c’, d’; T) do not have any overlapping if (c, d) and (c’, d) do not overlap. Now if zk(t) is a type-II-L shock, then Tk > t,. We extend xk(t) backward to intersect yk(t) at time t;. If zk(t) is a
118 KUO-SHUNG CHENG
type-II-R shock, then tk > Tk. We extend yk(t) backward to intersect xk(t) at time T;. It is easy to see that
A,+d,-,+ ... +A,+A-,<A(c,d; T), (5.36) where A, is the area of triangle bounded by the three lines
{(x,t):x=c+f’(u,)(t-T),t<T},
{(x, t):x=d+f’(u,)(t- T), td T}, {(x, t): t= T, cdx<d},
A,, k = l,..., (n - l), is the area of triangle bounded by the three lines ((4 t): x=xk+l (Tk+l)+f’(~k)(t-Tk+,), fGc+d,
{(X,t):X=Yk+I(fk+l)+f’(Uk)(t-tk+I),t~tk+l}, {(x, t): t=min{tk+ly Tk+l)), A, is the area of region bounded by the lines
{(x, t):x=x,(T1)+f’(uo)(t- T,), t< T,}, {(X,t):X=Yl(tl)+f’(uo)(t--t,),t~t,},
{(x, t): t=min(t,, T,}}, {(x, t): t=max{h,, To}},
and A -i is the area of triangle bounded by the lines {(x, t):x=s()+f’(u-I)(t-ql), ta,}, {(x, t):x=s,+f’(u_,)(t--to), t<z,}, {(x, t): t=O}.
Here
r,=max{T,, to) and
Thus we have
so = xo(tO) or Yd%).
A, = f(f’(u,) -f’(d)(T- G)* (5.37)
or
A, = fcf’c~n, -f’(%J)(T- KY?
Ak=i(f’(uk)-f’(u,))Cmin{T,+,, fk+lI-G12
or
A,= 4(f’(uk)-f’(Uk))Cmin(T,+ 1, h+ 1> - Gl’,
AoZ~(f’(~o)-S(~o))Cmin{Tl, tll -d*
and
A -,=tcf’c~~,,-f(u-,))z~.
Let
(5.37)’
(5.38)
k= 1, 2 ,..., (n- l), (5.38)’
(5.39)
(5.40)
Cmin{Tk+I,tk+,)-Gl or Cmin{Tk+l,tk+I)-Tbl=Zk+I,
k = 2,..., (n - 1 ),
T,+~=(T-~;) or (T- ZJ
z,=min{T,,t,}-z,.
and
Let
A J’(~n-I)-f’(~n-I)
n f’(%) -S’(wJ ’ A _ /ml-*)-m-*)
n l-f’(u,-l)-f’(u,-l)‘“’
f(uo) -f’(uo) Izl =s’M -f’(U1)
and
1 J’(Q)-m-1)
O f’(uo) -f’bo) . (5.41)
Then from the assumption ) tk - Tkl < 6 and Lemma 4.6, we have z,+T,+ ... +T~+,>+T.
505 61 l-9
120 KUO-SHUNG CHENG Furthermore, from Lemma 4.7, we have
lim A., = (1 + 8,)k1P’ or (1 + SJ+l. (5.42) n+cx
From (5.36k(5.41), we have
0 <f’(u,) -f’(u,) <
2A(c, & T)B(Ao, ~1,...~ A,; 705 ~lY.9 z,+1)
(5.43)
where
2
a-
0 [
4 l+++ n . . . +I nnl...~O ~ -l1
-1 . (5.44)From (5.42), the sum in (5.44) is bounded for all n. Now take C=32 max max l+k+ ... +A ~ 1
(5.45) luo GM ” Iv0 GM n n n-l”’ Lo .
1
We have from (5.43), (5.44), and (5.45)
OGf’(&J -f’(%) +, d; T).
This completes the proof of this lemma. Q.E.D.
Remark. If to = To > 0, x0( To) = yo(to), and there is no type-III shock passing through (x0( To), To), from the construction method, we know that (x0( To), To) is a point of interaction of two shocks. In this case, if u _ I and u-r are both belonging to [a2, co) or (-co, a,] (actually, [a,, b,] or [b,,a,]), then we can still extend CJc, T) and C,(d, T) by x-i(t) and y-i(t). It is easy to see that the above lemma still holds. The point
(xo(To), To) can be regarded as a limiting type-III shock.
LEMMA 5.18. Let u(cf0, T)=u,, u(dkO, T)=u2, d>c, and C,(c, T) and C,(d, T) are two backward genuine characteristics, that is,
C,(c, T)= {(x, t):x=c+f’(u,)(t- T), O< t< T), Cd4 T) = ((x, t): x=d+f’(u,)(t-T),O<t<T}.
Assume that u1 E [b,, a,] and u2 E [a,, b,]; then there exists a constant p
122 KUO-SHUNG CHENG
and t1 is the time when x*(t) and x,(t) meet, that is, co +f’(m) ?I = do + f(e) -f(m) t,
G--m ’
For tl<t<tz,
%n(x, t) = Ul if x<x,(t),
= h,((x - co)lt) if xl(t)<x<x,(t),
= F(x, t) if x,(t)<x<x3(f),
= h((x - do)lt) if x3(t)<x<x4(f),
= u* if x4(t) <x,
where x,(t) is a type-II-L shock satisfying x*(t)--0
t )) ’ t, < t,
x,(t,) = x,(t,) = xdt,)
and
(s, (y-j)*&, (y),
For
t,< t,
%(X, t) = Ul if x<xJt),
= F(x, t) if x6(f)<x<x3(t),
= M(x - do)lt) if x3(t) <x <x4(t),
= u* if x4(t) < x, where
X6(f) = Xl(b) +f’(h)(t- f2)? (zl)*=ul.
Now comparing the initial data of the two generalized solutions u,(x, t) and 5(x, t), we have
L&(x, 0) < qx, 0)
for all x E ( - co, co ). Hence from the ordering principle, Theorem 2.4, we have for all (x, t) E z T
u,(x, t) < qx, t).
Hence from the explicit solution u,(x, t), and (5.47) we have
(5.47)
t,> T. (5.48)
Now from our construction of solution u,(x, t), or from the invariance property of (1.1) under similarity transformation x -+ c(x, t -+ at, there is a constant p depending only on f and A& such that
From our assumptions off, we also have
(5.49)
(5.50) Combining (5.46) (5.48), (5.49), and (5.50), we obtain
(d- cl = (4 - co) + (f ‘(u,) -f ‘(u,)) T
=Bb+(f’(d-f’(~1)) T
>PT.
This completes the proof of Lemma 5.18. Q.E.D.
LEMMA 5.19. Let (c, t) be an interaction point of shocks (u,, u,} and ,u,, u,} and (4 T) b e also a interaction point of shocks {U,, ii,,,} and
f??l, iir}, where c < d and t > 0. Assume that
0) 4, &E (b, 4, (ii) u,, 11, E (a,, &I, (iii) u,, U,E Lb,, a,),
or
(i)’ u/, 4~ (a,, &I,
(ii)’ a,,
2, E (h, a,],
(iii)’
4, K E (a,, 5, I,
where (F,), = b1 and (&)* = b,. Then there exists a constant j? depending only on f and M, such that
(d-c) > fiT.
124 KUO-SHUNG CHENG
Remark. We say that shock {u,, u,} interacts shock {u,, u,} at (c, T).
We mean that there are two shocks x(t), y(t) satisfying (i) x(T)=y(T)=c,
(ii) -44 < ~(0 for t < T, (iii) lim u(x(t)-0, t)=u,,
I-T-O
lim u(x(t)+O, t)=r>Tm_ou(y(t)-O, t)=z4,,
f-T-0
lim u(y(t)+O, t)=u,.
t-T-0
Proof of Lemma 5.19. We only consider the first case. The second case can be similarly treated. From our construction, we can draw a genuine characteristic L, from d with speed f’(ti,), that is,
L,={(x,t):x=d+f’(ii,)(t-T),O<t<T} (5.51)
is a generalized backward characteristic from (d, T). Let
A = {x0: u(x, T) E ( - co, a,] for all xE (c, x0)}. (5.52) Set sup A = e. From our construction method, we can draw a genuine characteristic L, from (e, T) with speed f’(u(e - 0, T)),
L,= {(x, t): x=e+f’(u(e-0, T))(t- T), 0-c t < T}.
If u(e - 0, T) E [b, , a, ), then from Lemma 5.18, we have
(d-c)>(d-e)>,flT. (5.53)
If u(e - 0, T) q! [b,, a,), then u(e - 0, T) < 6,. From our construction method, we can find a point (e’, T), c < e’ < e, such that u(e’ + 0, T) = bl or u(e’-0, T) = b,. It is easy to see that we can draw a genuine backward characteristic L,, from (e’, T) with speedf’(b,). That is,
L,,= {(x, t):x=e’+f(b,)(t- T),O< t< T}. (5.54) Again, using Lemma 5.18, we have
(d-c) 2 (d-e’) 2 PT.
This completes the proof.
Now we are in a position to prove Theorem 5.15.
Q.E.D.
Proof of Theorem 5.15. Let - 00 <a< b< co be fixed. Assume that f’(u(., T)) is monotonically increasing in the interval (c, d) c [a, b]. If
necessary we can divide (c, d) into small intervals. Hence we can assume that C,(c, T) and C,(d, T) are of the type in Lemma 5.16 and Lemma 5.17.
There are several cases that we have to consider.
Case A. t, = T, = 0. Lemma 5.16 can be applied.
Case B. t, > 0, To > 0, To # t,. There must be a type-III shock passing through (x9( T,,), T,) and ( yo(to), to). Lemma 5.17 can be applied.
Case C. to = To > 0. In this case, (x0( T,), T,) = (y,,(t,), to) is an interaction point of two shocks {u,, u,} and {u,, u,}. But either uI, U, E [a,, b2] or u,, U,E [b,, a,]. As remarked after the proof of Lemma 5.17, Lemma 5.17 can be applied.
Case D. to = T,, > 0. In this case, (x0( T,,), T,) = (yo(ro), to) is an interaction point of two shocks {ZQ, u,} and {u,, u,}. But U[E (a,, az) or u,E h a*).
Let
I=((c,d)c(a,b):f’(u(.T))isincreasingin(c,d),(d-c)
is sufficiently small, and (c, d)‘s are disjoint}, II = { (c, d) E I: (c, d) E Case A, Case B, or Case C,
(c, d) E Case D with t, = T,, < T/2},
III = {(c, d) E I: (c, d) E Case D with &, = T,, > T/2}.
Now across a discontinuity of f’(u( ., T)), f’(u( ., T)) is decreasing. Hence we have
v+(f’(4., T)); [a, 61)
= ,c.;EI Cf’(u(d- 0, T)) -f’(u(c + 0, T))l
= ( c+c )
[f’(u(d-0, T))-f’(u(c+O, T))]. (5.55)(c,d)E II (c,d) E III
Now we can apply Lemmas 5.16 and 5.17 to the sum (c, d) E II but from t = T/2 to t = T. This gives us
,c;;E,I [f’(u(d-0, T)) -f’(u(c + o, t))]
G$[:(b-a)+ ~(“1. (5.56)
126 KUO-SHUNG CHENG
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