Under the assumption (1.6) on f(u), Le Floch [3] proved the uniqueness of the solution of (1.1)-(1.5) and derived a formula, which contains a solution of a varia-.
DUKE MATHEMATICAL JOURNAL (C)
Vol. 62, No. 2
March 1991
EXPLICIT FORMULA FOR THE SOLUTION OF CONVEX CONSERVATION LAWS WITH BOUNDARY CONDITION K. T. JOSEPH AND G. D. VEERAPPA GOWDA 1. Introduction. We consider the mixed initial boundary value problem for strictly convex conservation laws
u, + f(u)x
0
(1.1)
u(x, O)= Uo(X).
(1.2)
in x > O, > O, with initial condition
The boundary condition is prescribed in the sense of Bardos, Leroux and Nedelec I-1]. Let Ub(t) be a given bounded function, then this condition requires u(0, t) to satisfy the following:
{Sgn(u(O, t)
sup
k)(f(u(O, t))
f(k))}
0 a.e. > O,
k l(u(O, t), ut,(t))
where
I(u(O, t), u(t))
[Min(u(O, t), ub(t)), Max{u(O, t), ub(t))].
When f(u) is strictly convex, i.e., when f"(u) > 0, this condition is equivalent to (see Le Floch [3]) saying: either
or
!’u(O, t)
ff(t) t)) f(b(t))
}
(1.3)
where
(t)
Max{u(t), }
(1.4)
and 2 is the unique point where f’(u) changes sign. Because of the strict convexity of f, f attains its minimum at 2, i.e, f(2) inf f(u). Received 6 April 1990.
401
402
JOSEPH AND GOWDA
The solution is also required to satisfy the entropy condition:
O, t) > u(x + O, t),
u(x
x>O,t>O.
(1.5)
In this paper we assume either
f"(u)>0 and
lim
f(u)
=
(1.6)
or
f(u)
log[ae
+ be-U].
(1.7)
Under the assumption (1.6) on f(u), Le Floch [3] proved the uniqueness of the solution of (1.1)-(1.5) and derived a formula, which contains a solution of a variational inequality. This variational inequality may not be solvable explicitly. In this paper we derive an explicit formula for the solution of (1.1)-(1.5) when f satisfies either (1.6) or (1.7). The uniqueness result gives us that our formula and Le Floch’s formula are the same, but we cannot yet directly verify this. The present work is motivated by our previous works in the case of special nonlinearities f(u) (see [4-1 and [5]). In fact, in [5-1 an approximate solution ua(x, t) defined by Lax’s difference scheme for a conservation law (1.1) with f(u) given by (1.7) is considered in x > 0, > 0 and an explicit formula for U(x, t)= lima oo[- ua(y, t)dy] is derived. However we did not prove there that this formula satisfies the conservation law. This fact will follow as a corollary of the main result of this paper. Before the statement of our main theorem, we introduce some notations. For each fixed (x, y, t), x > O, y > O, > 0 and a > O, C(x, y, t) denotes the following class of paths fl in the quarter plane
o
z
O,s
o).
Each path connected the point (y, 0) to (x, t) and is of the form
fl(s)
z
where fl is a piecewise linear function with three lines or one straight line of the possible shapes shown in Figure 1, where the absolute value of slope of each straight line is
0,
> 0, define
U(x, t) Min # C (x, y, t) y>O
[-y?u(z)dz-I{s.#(s)=o
f(g(s)) ds
+
f{:a()eo} (1.8)
U(x, t) can also the written as
U(x, t) Min # C(x,y,t) y>O
[-y;Uo(z)dz-f{s:#(s)=o (1.8)’
because f*(0)
-Min f(u)
-f(2). Note that f(fib(s))- f(2)
> O.
,
A classical argument of Conway and Hopf [2] and Lax [6] can be used to show that U(x, t) is Lipschitz continuous and U(x, t) exists a.e. We denote
u(x, t)
0, U(x, t).
(1.9)
404
JOSEPH AND GOWDA
For easier use later we introduce
f(b(S)) ds +
f{,’
(1.10) a()o)
and
H(x, t, fl)
Uo(z) dz + J(fl)
(1.11)
H(x, t, fl).
(1.12)
We have
U(x, t)
Min # C (x, r, t) y >O
For each fixed (x, y, t), x
> 0, y > 0,
> 0, define
Q(x, y, t)
(1.13)
J(fl)
Min I C (x, y, t)
and
C(x, y, t)\{flo}
Ca(x, y, t)
(1.14)
where flo is a straightline path joining (y, 0) to (x, t). Let
A(x, y, t)=
tf*(x-y)t
B(x,y,t)
(1.15)
{J(fl)}.
Min
(1.16)
# C (x, y, 0
From (1.13), (1.15) and (1.16) it follows that Q(x, y, t)
Min[A(x, y, t), B(x, y, t)].
Note that B(x, y, t)
Min
(1.16)’
[J(x, y, t, tl, t2) ]
Yfl Ot, X/(t t2)
where t2
J(x, y, t, tt, t2)
f(fib(s)) ds
+ tl f*
+ (t- t2)f*
,
(1.17)
405
CONVEX CONSERVATION LAWS
For each fixed y > O, B(x, y, t) is Lipschitz continuous and hence Q(x, y, t) is also Lipschitz continuous. Let Q1 (x, y, t)
dxQ(x, y, t).
(1.18)
Also we will show that for a.e. (x, t) there exists only one yo(x, t) which minimises Min [Q(x, y, t)
Uo(Z) dz]
(1.19)
y>0
see Lemma 2.6. Following Lax 16] and Conway and Hopf [2], we shall prove the following theorem.
MAIN THEOREM. Let u(x, t) be defined by (1.9), then
(i) u(x, t) Q (x, yo(x, t), t) a.e. (x, t), where yo(x, t) minimises (1.19) and Q is given by (1.18). Also u(x, t) satisfies ut in the sense
+ f(u)x
0
of distribution.
(ii) For each fixed > 0 and x > O, u(x +_ O, t) exists and satisfies the entropy condition (1.5). u(x, t) satisfies the initial condition (1.2) and. u(O +, t) exists a.e. > 0 and satisfies the boundary condition (1.3). In the next section we shall prove this theorem. 2. Proof of Main Theorem. Proof of the main theorem is broken up into several steps formulated as lemmas. First we introduce some notations. For each fixed (x, y, t), x > 0, y > 0, > 0, let (t2(x, y, t), t (x, y, t)) denote a value (t2, t), 0 < t < t2 < t, for which J(x, y, t, tl, t2) attains its minimum under the constraints as in (1.16)’. Note that there may be several pairs (t2, t) for which this happens. Define
t-(x, y, t)
Max{t2(x, y, t)}
t] (x, y, t)
Min{tz(x, y, t)}
t- (x, y, t)
Max {t (x, y, t)}
t-(x, y, t)
Min {t (x, y,
t)}.
Let yo(x, t) denote a value y > 0 for which (1.19) achieves its minimum. Let
y (x, t)
Max { yo(x, t)}
y (x, t)
Min { yo(x,
t)}.
406
JOSEPH AND GOWDA
Here we remark that
Uo(Z) dz + O(x, yo(X, t), t).
U(x, t)
(2.1)
o(X,t)
First we shall prove the following lemma.
LEMMA2.1. Let x
>0
and let
fl
be a path which is a minimiser
for
Minac(x,r, oJ(fl) and let (x*, t*), t* < be a point on this path, then fl restricted to [0, t*] is a minimiser for Minljc,tx,,r,t,)J(fl ). This lemma is an easy consequence of Jensen’s inequality. Let fl be the lemma and suppose fl restricted to [0, t*] is not a minimiser for the in path Minactx,,r,t,)J(fl). Then there exists a path fl* on [0, t*] a minimiser such that
Proof.
J(fl*) < J(fl). In (2.2) fl means fl restricted to [0, t*]. Now define a path fll on [0, t] by
/
*
on [0, t*] on It*, t].
can be of the following shapes; see Figure 2. Now define a path/ connecting (y*, 0) to (x, t) in the following way.
Case 1. on [0, t] /?: (Straight line joining (0, t) to (x, t) on [t, t].
Case 2. Straight line joining (y*, 0) to (x, t).
Case 3. line joining (y*, 0) to (0, t’) on [0, t’] / Straight on (fl, [t’, t].
Case 4 and Case 5.
fl is the same as in the figure.
By Jensen’s inequality and (2.2) we easily get
J() < J()
(2.2)
407
CONVEX CONSERVATION LAWS s
Case i
s
Case 2
Case 3
(x’t) (X,t) (O,t 2
*
(O,t I
s
(O,t*1 -z
(y*,0)
(y*,0)
t*)
(O,t)
s
Case 4
(X ,t
z
(y*,O)
Case 5
(x,t)
(x,t)
,
(O,t
(O,t 2
2
(x ,t (0 t * 1
(y * ,o)
7-,
Z
y* ,o)
FIGUI 2
which contradicts the fact that fl is a minimiser for Minac, tx, r,oJ(fl). The proof is valid even when (x, t) (0, t), a point on axis. The proof of the lemma is complete. As an immediate corollary of this lemma we deduce
Let fit, i= 1, 2, be minimisers for MintEC,tx,,r,,,)J(fl) or MinaE,tx,,r,t, J(fl), then fll and fiE cannot cross with different slopes in the interior of D. Proof. Let fll and f12 cross each other with different slopes. Then, as in the proof of the previous lemma, one can easily construct another path fl C,(x, y, q) for LEMMA2.2.
1 or 2 such that
< which contradicts the fact that fit is minimiser for Minac,tx,,y,t,)J(fl). The proof of the other part is similar.
In the next lemma, we obtain some properties of t and t using Lemma 2.2.
408
JOSEPH AND GOWDA
LEMMA 2.3. (i) For each fixed > 0, y > 0, t (’, y, t) and t (., y, t) are nonincreasin9 functions of x. t(’, y, t) is right continuous and t (’, y, t) is left continuous. The two
functions have the same set of points of discontinuity which is a countable subset of [0, oo) and, except on this countable set, t-(., y, t) and t (’, y, t) are equal. Moreover
t (x, y, t) t (x + 0, y, t) t (x, y, t)
t (x
0, y, t)
t1 (x + 0, y, t), Vx > 0 t-(x 0, y, t), Vx > 0.
(ii) For each fixed x > 0, y > 0, t (x, y, ") and t (x, y, ") are nondecreasinff functions of t. t’ (x, y, ") is left continuous and t (x, y, ") is right continuous. The two
functions have the same set of points of discontinuity which is a countable set of [0, oo) and, except on this countable set, t (x, y, ") and t (x, y, ") are equal. Moreover
t (x, y, t) t (x, y,
0) t] (x, y, + 0)
t] (x, y, t)
0), Vt > 0 ], t (x, y, + 0), ’t > 0.
t (x, y,
(2.3)t
(iii) For y > O, fixed
t (x, y, t) Proof. From the definition of cannot cross, we get, if xx < xz, t] (x2, Y, t)
t] (x, y, t)
a.e. (x, t).
t (x, y, t) and using the fact that two minimisers
< t (x, y, t) < t] (x, y, t) < t-(x, y, t)
(2.4)
and (i) follows from (2.4). Proof of (ii) is similar and (iii) follows from (i) and (ii). Proof of the lemma is complete.
Next we compute left and right derivatives of B(x, y, t) with respect to x and t, for each fixed y > 0. Denote
OB-+
y, t) Ox (x,
lim
OB + (x, y, t) Ot
lim
We shall prove the following.
hO
h’0
[B(x +_ h, y, t)
B(x, y, t)]
h
[B(x, y, + h) h
B(x, y, t)]
409
CONVEX CONSERVATION LAWS
LEMMA 2.4. cB / (x, y, t) (i)
(f*)’
(ii)
(x, y, t)
(f*)’
(iii)
(x, y, t)
--x
W(x, y, 0
()t t t) x
(x, y,
t)
f f * )’
t
-f
Proof. By the definition of B(x, y, t) Y
f(fib(s)) ds + tl(x, y, t)f*
B(x + h, y, t)
(f*)’
)
0 < r/(h) < h.
0,
(2.6)
t-t(x,y,t)
Combining (2.5) and (2.6), we get (i). In the same way, one can show (ii) also. Next we shall prove (iii). In the same procedure as before, we get
B(x, y, + h)
B(x, y, t)
(
< (t + h t (x, y, t))f* + h (t
Letting h
t (x’ y’ t))f*
(
x t] (x, y, t)
x t- t] (x,
y,t)
)
)
0, we get
cB +
O, Qt + f(Qx)
0 a.e. (x, t).
Proof. From Lemma 2.4 and Lemma 2.3 (iii), it follows that OB/Ox and exists a.e. and
Bt + f(B,)
0
a.e. (x, t).
Also, it follows from Lemma 2.4’ that
A
+ f(A)= O.
Now the lemma follows from a result of Conway and Hopf I-2] which says if V1 and
V2 are solutions of
V + f(V) O, then so does Min(V1, V2). The proof of the lemma is complete. Letting y (x, t) and y (x, t) be defined as before, we have the following.
LEMMA 2.6. Let > 0 be fixed. (i) y (’, t) and y (’, t) are nondecreasin9 functions of x. y (’, t) is rioht continuous and y (’, t) is left continuous. The two functions have the same points of discontinuity and, except at these countably many points, the two functions are equal. Moreover
y (x, t)= y (x + o, t)= y (x + o, t) y (x, t) y (x o, t) y (x o, t). (ii) Suppose the minimum in (1.12) for H(x,t, fl) is attained for some fl C(x, yo(x, t), t). Let x* < x and let fl* attain minimum in (1.12) for H(x*, t, fl) then fl* C(x*, yo(x*, t), t). Moreover
t- (x, y (x, t), t)
? (x*, y (x*, t), t)
412
JOSEPH AND GOWDA
and
y (x, t) y (x, t) y (x*, t) y (x*, t). Proof of (i) is exactly the same as the proof of Lemma 2.3. We shall prove Since (ii). fl and fl* cannot intersect, we get fl*e C(x, yo(x*, t), t). For the same reason, it follows that
Proof.
t (x, y (x, t), t) t (x*, y (x*, t), t)
and
y (x, t) y (x*, t).
But the second equality implies that in [0, x], yg(., t) is constant. In particular, every point of [0, x] is a point of continuity of y (., t). Then apply (i) to get
y (x, t) y (x, t) y (x*, t) y (x*, t). The proof of the lemma is complete.
Now we shall prove the main theorem. Following Lax [6], we introduce
Proof of the main theorem. u(x, t)
A(x, t)
f
Q (x, y, t)e-St-i; uot,+qt,y,ol dy
ff e-t-
f
u(z)dz+Q(x’y’t)]
dy
f(Qx(x, y, t))e-m-I"t)nz+Qt"r")l dy
e-Vt-7 u(z)dz+Q(x’y’t)] dy Vs(x, t) and
Us(x, t)
It is clear that
ff
e-St-"(z)a:+o(:’"t) dy
1
(2.11)
log Vs
lim us(x, t)
Q (x, yo(x, t), t)
(2.12)
f[Q (x, yo(x, t), t)]
(2.13)
N-oo
lim f(x, t) Noo
CONVEX CONSERVATION LAWS
413
where yo(x, t) minimises (1.19) and lim UN(X, t)
U(x, t).
(2.14)
N-oo
Also
(UN),.
uN(x, t)
(2.15)
So it follows from (2.12), (2.14) and (2.15) that,
(x, t)
O.(x, yo(x, t), t).
Next we shall show that
(UN),
(2.16)
--fN"
To do this we consider
(/;N),
-
1 (VN),
(x, y, t)e-Nt-I uot)az+O.t,,y,,) dy
f
f
e -Nt- u(z)dz+Qtx’y’t)] dy
f(Q x(x, y, t))e -Nt-I u(z)dz+Q(x’y’t)] dy
f
e-Nt- uo(z)dz+Q(x,y,t)] dy
In the last equality we used Lemma 2.5. Now (2.16) follows from the definition offN. From (2.15) and (2.16) we get
(u,,), + (TN) Hence for all test functions qg(x, t)
ff
0.
C ((0, ) x (0, )),
(UN(Dt "1- fN(49x) dx dt
Now part (i) of the theorem follows as N
.
O.
414
JOSEPH AND GOWDA
Now we shall show u(x, t) satisfies the initial condition. First we note that a minimiser fl for H(x, t, ) cannot have any of its straight lines parallel to the x-axis. In fact, because of the assumption on f(u) and the initial and boundary data, slopes of each straight line piece of are uniformly bounded. Hence, given e > 0, there exists 6 > 0 such that for all x > e, < 6
u(x, t)=
(f,),(x yo(x,t t).)
where yo(x, t) minimises Miny>o [- Uo(Z) dz argument [6-1 can be used to show that lim u(x, t)
Uo(X)
+ tf*((x y)/t)]. But
a.e. x
t-O
then Lax’s
> e.
Since e is arbitrary, it follows that lim_,o u(x, t) Uo(X) a.e. x > 0. Next we shall show that u(x, t) satisfies the entropy condition (1.5). Because of Lemma 2.4 and Lemma 2.4’ and the definition of Q , it is clear that u(x +_ O, t) exists. In fact (2.17) u(x +_ O, t) Q (x, y- (x, t), t).
Entropy condition (1.5) follows from the increasing nature of (f*)’ and (2.17). Lastly, we show that u(x, t) satisfies the boundary condition (1.3). Because of Lemma 2.6 (ii) and the properties we proved for t (x, y, t) and yo(x, t), lim(f*)’
u(0+,t)=
t
t(x,t)
(2.18)
or
lirno (f,),(.x--y(x,
t))
(2.19)
for a.e. > 0. Here we used t (x, t) t (x, y (x, t), t). Suppose u(0 +, t) is given by (2.18), then, since t- t (x, t) should be a minimum point for J(x, y, t, tl, t2), we get from
OJ
=0 at
Ot2
i.e.,
t2=t,
f((f’)’(.t x_ t))
f((t)).
415
CONVEX CONSERVATION LAWS
Letting x
0, we get, from (2.18),
f(u(O+, t)) If f’(u(O +, t))
f((t)).
> O, then by definition of fib(t) u(O +, t)
b(t).
(2.20)
f(b(t)).
(2.20)’
Now iff’(u(0+, t)) < 0, then
f(u(O +, t)
Let us take the case when (2.19) occurs. Again
u(O+ t)=
(f,),(-y(O, t))
In this case
Since the path joining (0, t) to
(y (0, t), t) is a minimiser for H(x, t, fl) in (1.12) we
have,
o,t)
Uo(Z) dz
