When ( u − − u d ) ⋅ ( u + − u d ) ≥ 0 , (7) is degenerated into a corresponding problem with f ∈ C 2 (see [27] ). We now construct the weak entropy solution of (7) only for the case of ( u − − u d ) ⋅ ( u + − u d ) < 0 . We divide our problem into two cases: (1) u − < u d < u + ; (2) u + < u d < u − .

Case (1) u − < u d < u + .

Consider the following Riemann problem corresponding to (7):

{ v t + f ( v ) x = 0 , − ∞ < x < ∞ , t > 0 v ( x , 0 ) = v 0 ( x ) : = { u − , x < 0 u + , x > 0. (8)

In this case, since the flux function has a weak discontinuity point u = u d , the Riemann problem (8) includes only a rarefaction wave R = R ( u − , u d ) ∪ R ( u d , u + ) centered at point ( 0,0 ) of the x − t plane. This rarefaction wave solution can be written as:

v ( x , t ) = { u − ,     x < f ′ ( u − ) t ( f ′ ) − 1 ( x t ) , f ′ ( u − ) t ≤ x < f ′ − ( u d ) t u d ,   f ′ − ( u d ) t ≤ x < f ′ + ( u d ) t ( f ′ ) − 1 ( x t ) , f ′ + ( u d ) t ≤ x < f ′ ( u + ) t u + ,     x ≥ f ′ ( u + ) t

Let u ( x , t ) = v ( x , t ) | x , t > 0   , then u ( 0 + , t ) = m i n { u + ,0 } , hence, it holds the boundary entropy condition:

f ( u ( 0 + , t ) ) − f ( k ) u ( 0 + , t ) − k ≤ 0     ( ∀ k ∈ [ u − , u ( 0 + , t ) ] ,   k ≠ u ( 0 + , t ) ) .

It is easy to verify that u ( x , t ) also satisfies all other conditions in Lemma 2. Therefore, by Lemma 2, u ( x , t ) is the global weak entropy solution of (7).

Case (2) u + < u d < u − .

In this case, (8) includes only a shock wave S ( u − , u + ) at point ( 0,0 ) in the x − t plane. This shock wave solution can be expressed as follows:

v ( x , t ) = { u − , x < s ( u − , u + ) t u + , x > s ( u − , u + ) t

where s ( u − , u + ) is the speed of the shock S ( u − , u + ) . Let u ( x , t ) = v ( x , t ) | x , t > 0   , then

u ( 0 + , t ) = { u + , as s ( u − , u + ) ≤ 0 u − , as s ( u − , u + ) > 0

From Lemma 2, we can easily verify that u ( x , t ) is the global weak entropy solution of (7).

Consider the following initial value problem corresponding to (6):

{ v t + f ( v ) x = 0 , − ∞ < x < ∞ , t > 0 v ( x , 0 ) = v 0 ( x ) : = { u − , x < 0 u m , 0 < x < a u + , x > a . (9)

According to the solution structure of (9), we construct the global weak entropy solution of (6) with u m ≠ u + by dividing our problem into five cases: (1) u − = u m ≠ u + ; (2) u − < u m < u + ; (3) u + < u m < u − ; (4) u − , u + < u m ; (5) u m < u − , u + .

Case (1) u − = u m ≠ u + .

In fact, when u − = u m , ( u − − u d ) ⋅ ( u + − u d ) ≥ 0 , (6) becomes a problem with f ∈ C 2 , which was discussed in [27] . We now investigate the case of ( u − − u d ) ⋅ ( u + − u d ) < 0 . (9) is degenerated into a Riemann problem.

If u − = u m < u + , only a rarefaction wave R = R ( u − , u d ) ∪ R ( u d , u + ) , centered at ( a ,0 ) of the x − t plane, appears in the weak entropy solution of (9). This rarefaction wave solution of (9) can be written as:

v ( x , t ) = { u − , x < a + f ′ ( u − ) t ( f ′ ) − 1 ( x − a t ) , a + f ′ ( u − ) t ≤ x < a + f ′ − ( u d ) t u d , a + f ′ − ( u d ) t ≤ x < a + f ′ + ( u d ) t ( f ′ ) − 1 ( x − a t ) , a + f ′ + ( u d ) t ≤ x < a + f ′ ( u + ) t u + , x ≥ a + f ′ ( u + ) t .

Let u ( x , t ) = v ( x , t ) | x , t > 0 , where v ( x , t ) is the weak entropy solution of (9). It is easy to verify u ( x , t ) satisfies all conditions in Lemma 2, thus u ( x , t ) is the global weak entropy solution of (6). It includes only a rarefaction wave R | x , t > 0   , which will interact with the boundary x = 0 and be completely absorbed (if u + ≤ 0 ) (see Figure 1(a) and Figure 1(b)) or partially absorbed (if u + > 0 ) (see Figure 1(c))by the boundary.

If u − = u m > u + , the weak entropy solution v ( x , t ) of (9) includes only a shock wave emanating at point ( a ,0 ) of the x − t plane, which can be expressed as follow:

v ( x , t ) = { u − , x < a + s ( u − , u + ) t u + , x > a + s ( u − , u + ) t

Let u ( x , t ) = v ( x , t ) | x , t > 0 , then by Lemma 2, it is also easy to verify that u ( x , t ) is the global weak entropy solution of (6). It includes only a shock wave S ( u − , u + ; ( a ,0 ) ) , which will interact with the boundary x = 0 and be absorbed (if s ( u − , u + ) ≤ 0 ) (see Figure 2(a)) or be far away from the boundary (if s ( u − , u + ) > 0 ) (see Figure 2(b) and Figure 2(c)).

Case (2) u − < u m < u + .

If u ± , u m ≥ u d , or u ± , u m ≤ u d , (6) becomes a problem with f ∈ C 2 , see [27] . We now consider the following three cases: u − < u m < u d < u + , u − < u d < u m < u + , and u − < u m = u d < u + .

When u − < u m < u d < u + , two rarefaction waves R ( u − , u m ) and

R 1 = R ( u m , u d ) ∪ R ( u d , u + ) , centered at point ( 0,0 ) and ( a ,0 ) , respectively, appear in the weak entropy solution v ( x , t ) of (9); when u − < u d < u m < u + ,

two rarefaction waves R 2 = R ( u − , u d ) ∪ R ( u d , u m ) and R ( u m , u + ) , centered at point ( 0,0 ) and ( a ,0 ) , respectively, appear in the weak entropy solution v ( x , t ) of (9); when u − < u m = u d < u + , two centered rarefaction waves R ( u − , u d ) and R ( u d , u + ) , centered at point ( 0,0 ) and ( a ,0 ) , respectively, appear in the weak entropy solution v ( x , t ) of (9). The two rarefaction waves in v ( x , t ) , centered at point ( 0,0 ) and ( a ,0 ) , respectively, will not overtake each other since the propagating speed of the wave front in the first wave is not greater than that of the wave back in the second wave. Let u ( x , t ) = v ( x , t ) | x , t > 0 , from Lemma 2, we can easily verify that u ( x , t ) is the global weak entropy solution of (6).

When u − < u m < u d < u + , u ( x , t ) includes only a rarefaction wave R | x , t > 0   , which will interact with the boundary x = 0 and be partially absorbed (if u + > 0 ) or absorbed (if u + ≤ 0 ) by the boundary.

When u − < u d < u m < u + , if f ′ ( u m ) = 0 , u ( x , t ) includes only the central rarefaction wave R ( u m , u + ) far away from the boundary x = 0 ; if f ′ ( u m ) > 0 , u ( x , t ) includes two central rarefaction waves R 2 | x , t > 0 and R ( u m , u + ) far away from the boundary; if f ′ ( u m ) < 0 , u ( x , t ) includes only the central rarefaction wave R ( u m , u + ) | x , t > 0 , which will interact with the boundary and be partially absorbed (if u + > 0 ) or completely absorbed (if u + ≤ 0 ) by the boundary.

When u − < u m = u d < u + , u ( x , t ) includes only the central rarefaction wave R ( u d , u + ) | x , t > 0 , which will interact with the boundary and be partially absorbed (if u + > 0 ) or completely absorbed (if u + ≤ 0 ) by the boundary.

Case (3) u + < u m < u − .

The discussion for this case is the same as that of the corresponding case in [27] .

Case (4) u − , u + < u m .

When u − , u + ≥ u d , or u m ≤ u d , then (6) is degenerated into the problem with f ∈ C 2 . When u − ≥ u d , then the discussion on this problem is the same as that of the case f ∈ C 2 . Hence, we only investigate the case of u − < u d < u m , u + < u m .

In this case, an initial rarefaction wave R = R ( u − , u d ) ∪ R ( u d , u m ) centered at point ( 0,0 ) and an initial shock wave S ( u m , u + ) starting at point ( a ,0 ) appear in the weak entropy solution v ( x , t ) of (9). In what follows, similar to [33] , we give the statement of interaction of the initial rarefaction wave R and the initial shock wave S ( u m , u + ) . The rarefaction wave R interacts with the shock wave S ( u m , u + ) lying on its right at some finite time t = t 1 , and the shock S ( u m , u + ) will cross R after t = t 1 . The initial shock wave curve is denoted as x = x ( t ) , and the resulting shock after the interaction of R and S ( u m , u + ) is still denoted as x = x ( t ) , which is regarded as an extension of the original shock x = x ( t ) . The right state of the resulting shock is a constant u + . If u + < u − , the shock x = x ( t ) will cross the whole of the initial rarefaction wave R completely at some finite time; if u + = u − , the shock x = x ( t ) is able to cross the whole of R completely only when t → ∞ ; if u + > u − , it is impossible for this shock wave to cross the whole of R completely, but it is able to cross the right part of R: R ( u + , u m ) (if u + ≥ u d ) or R ( u + , u d ) ∪ R ( u d , u m ) (if u − < u + < u d ) when t → ∞ . The shock x = x ( t )   ( t > 0 ) is a piecewise smooth curve. First, the shock wave x = x ( t )   ( t > 0 ) cross the right part R ( u d , u m ) of R with a varying speed of propagation during the penetration. If it is able to cross the whole of R ( u d , u m ) completely at some finite time, then it crosses the domain of constant state u = u d with a constant speed of propagation. When the shock x = x ( t )   ( t > 0 ) encounters the rightmost characteristic line of the rarefaction wave R ( u − , u d ) , it begins to cross R ( u − , u d ) with a varying speed of propagation again. For the position of the shock x = x ( t )   ( t > 0 ) , we have one of the following cases: 1) the shock x = x ( t ) is located in the first quadrant of the x − t plane; 2) the shock x = x ( t ) enters the second quadrant from the first quadrant including the t-axis at some finite time and then keeps staying in the second quadrant. Let u ( x , t ) = v ( x , t ) | x , t > 0 , then by Lemma 2, u ( x , t ) is the global weak entropy solution of (6).

We now state the interaction of the elementary and the boundary x = 0 for the global weak entropy solution of (6). When the shock x = x ( t ) in v ( x , t ) is in the first quadrant of the x − t plane, the elementary wave in the solution u ( x , t ) of (6) does not interact with the boundary x = 0 ; when the shock wave x = x ( t ) of v ( x , t ) enters the second quadrant from the first quadrant including the t-axis and then keeps staying in the second quadrant, the shock wave x = x ( t ) in u ( x , t ) interacts with the initial rarefaction wave R | x , t > 0 on its right at t = t 1 , and then crosses R | x , t > 0 at its right at t > t 1 , finally it collides with the boundary x = 0 and is absorbed by the boundary (see Figure 3(a) and Figure 3(b)).

Case (5) u m < u − , u +

If u − , u + ≤ u d , or u m ≥ u d , then (6) becomes a problem with f ∈ C 2 (see [27] ). If u + ≤ u d < u − , the discussion of the problem is the same as that of the case f ∈ C 2 . We only consider the case of u m < u d < u + in the following discussion.

In this case, an initial shock wave S ( u − , u m ) starting at point ( 0,0 ) and an initial rarefaction wave R ( u m , u d ) ∪ R ( u d , u + ) centered at point ( a ,0 ) in the

x − t plane appear in the weak entropy solution v ( x , t ) of (9). We denote this initial shock wave curve by x = x ( t ) . As in [33] , the shock S ( u − , u m ) interacts with the rarefaction wave R on its right at some time t = t 1 and at t > t 1 , it will cross R with a varying speed of propagation during the penetration. We denote the generating shock wave still by x = x ( t ) , whose left state is a constant u − . If u − > u + , the shock wave x = x ( t ) will completely penetrate the initial rarefaction wave R at a finite time; if u + = u − , the shock x = x ( t ) is able to cross the whole of R completely only when t → ∞ ; if u − < u + , it is impossible for this shock wave to cross the whole of R completely, but it is able to cross the left part of R: R ( u m , u − ) (if u − ≤ u d ) or R ( u m , u d ) ∪ R ( u d , u − ) (if u − > u d ) when t → ∞ . After t = t 1 , x = x ( t ) will cross the rarefaction waves on its right with a non-decreasing speed. The shock x = x ( t )   ( t > 0 ) is a piecewise smooth curve. During the process of penetrating R = R ( u m , u d ) ∪ R ( u d , u + ) , it first crosses the leftmost part R ( u m , u d ) of R with a varying speed, and then crosses the constant state u = u d with constant speed. When the shock wave x = x ( t )   ( t > 0 ) encounters the characteristic line of the leftmost characteristic line of the rarefaction wave R ( u d , u + ) , it again begins to cross the rarefaction wave R ( u d , u + ) with a varying speed. For the shock x = x ( t )   ( t > 0 ) , it holds one of the following three cases: (a) when the initial shock speed s ( u − , u m ) ≥ 0 , the shock x = x ( t ) interacts with the initial rarefaction wave R in the first quadrant including the t-axis and keeps staying in the first quadrant after interaction (see Figure 4(a)); (b) when s ( u − , u m ) < 0 , u + < 0 < u − and f ( u − ) ≤ f ( u + ) , the shock x = x ( t ) interacts with R in the second quadrant and remains in the second quadrant after interaction (see Figure 4(b)); (c) when s ( u − , u m ) < 0 , if 0 ≤ u + ≤ u − , u − ≠ 0 or 0 < u − ≤ u + or u + < 0 < u − , f ( u − ) > f ( u + ) , the shock x = x ( t ) crosses the t-axis from the second quadrant at some finite time greater than t 1 , and then enters the first quadrant, and keeps staying in the first quadrant after that finite time (see Figure 4(c) and Figure 4(d)).

In sub-case (a) and (b), let u ( x , t ) = v ( x , t ) | x , t > 0 , then from Lemma 2, we can verify that u ( x , t ) is the global weak entropy solution of (6). The interaction of the elementary wave and the boundary x = 0 in the solution u ( x , t ) of (6) is stated as follows: For sub-case (a), when s ( u − , u m ) > 0 , the weak entropy

solution of (6) does not include the interaction of elementary waves and the boundary x = 0 ; when s ( u − , u m ) = 0 , the rarefaction wave R = R ( u m , u d ) ∪ R ( u d , u + ) collides with the boundary x = 0 at time t = t 1 , and the boundary x = 0 reflects a new shock wave tangent to the boundary itself at point ( 0, t 1 ) , which is just the restriction of x = x ( t ) at t > t 1 and will penetrate R after t = t 1 . For sub-case (b), the weak entropy solution of (6) only includes the rarefaction wave R = R ( u m , u d ) ∪ R ( u d , u + ) | x , t > 0 , which interacts with the boundary x = 0 at some time and is absorbed completely by the boundary.

For sub-case (c), there exists u * ∈ ( u m ,0 ) such that f ( u * ) = f ( u − ) . Furthermore, there is t * > t 1 such that

x ′ ( t ) = s ( u − , u ( x ( t ) + 0 , t ) ) { < 0 , 0 < t < t * = 0 , t = t * > 0 , t > t *

for u * ≠ u d and there are exist t * , t * ¯ > t 1 ( t * ¯ < t * ) such that

x ′ ( t ) = s ( u − , u ( x ( t ) + 0 , t ) ) { < 0 , 0 < t < t * ¯ = 0 , t * ¯ ≤ t ≤ t * > 0 , t > t *

for u * = u d , where x ′ ( t ) is the speed function of the shock wave x = x ( t ) in the weak entropy solution of (9). If we construct the solution of (6) by taking u ( x , t ) = v ( x , t ) | x , t > 0 as in sub-cases (a), (b), then this u ( x , t ) does not satisfy the boundary entropy condition (4) for t > t 2 , where t 2 = − a / f ′ + ( u * ) ( < t * ) is the time at which the characteristic line with speed f ′ + ( u ( x ( t * ) + 0, t * ) ) from the point ( x ( t * ) , t * ) backward to x-axis intersects the t-axis (see Figure 4(c) and Figure 4(d)). Thus, by virtue of Lemma 2, it is not the weak entropy solution of (6). We need to reconstruct the solution of (6). Take

v ( x , t 2 ) = { u − ,       x < 0 v ( x , t 2 − 0 ) , x > 0 (10)

as the new initial value of (9)₁, then the solution v ¯ ( x , t ) of the initial value problem (9)₁, (10) in ( 0, ∞ ) × ( t 2 , ∞ ) includes only a new shock wave x = x + ( t ) starting at point ( 0, t 2 ) , whose original speed is zero and the left state is u − . When t > t 2 , this shock crosses the rarefaction wave R |   ( x − a ) / t     > f ′ + ( u * ) on its right with a varying positive speed of propagation in the first quadrant. Let

V ( x , t ) = { v ( x , t ) , 0 < t < t 2 v ¯ ( x , t ) , t ≥ t 2 ,

then, from Lemma 2, this u ( x , t ) = v ¯ ( x , t ) | x > 0 ,     t > 0 is the global weak entropy solution of (6). Now we give the statement of the interaction of the elementary and the boundary x = 0 in the solution u ( x , t ) of (6) (see Figure 4(c) and Figure 4(d)). For the problem (6), an initial rarefaction wave R = R ( u m , u d ) ∪ R ( u d , u + ) emanates from the point ( a ,0 ) on the x-axis. One part of R collides with the boundary x = 0 , and then the boundary x = 0 reflects a new shock wave x = x + ( t ) at t = t 2 with zero original speed, which will penetrate another part R |   ( x − a ) / t     > f ′ + ( u * ) of R with a varying positive speed of propagation in the first quadrant. This shock is just that one in v ¯ ( x , t ) .

We now construct the global weak entropy solution of (7) under the condition that u d 1 , u d 2 are located between u − and u + . If not so, see [27] or sub-Section 3.1.1.

Case (1) u − < u d 1 , u d 2 < u + .

In this case, since the flux function f has two weak discontinuous points u d 1 , u d 2 , (8) includes only a rarefaction wave R = R ( u − , u d 1 ) ∪ R ( u d 1 , u d 2 ) ∪ R ( u d 2 , u + ) centered at point ( 0,0 ) of the x − t plane. We can express this rarefaction wave solution as:

v ( x , t ) = { u − ,   x < f ′ ( u − ) t ( f ′ ) − 1 ( x t ) ,   f ′ ( u − ) t ≤ x < f ′ − ( u d 1 ) t u d 1 ,   f ′ − ( u d 1 ) t ≤ x < f ′ + ( u d 1 ) t ( f ′ ) − 1 ( x t ) , f ′ + ( u d 1 ) t ≤ x < f ′ − ( u d 2 ) t u d 2 ,     f ′ − ( u d 2 ) t ≤ x < f ′ + ( u d 2 ) t ( f ′ ) − 1 ( x t ) , f ′ + ( u d 2 ) t ≤ x < f ′ ( u + ) t u + ,   x ≥ f ′ ( u + ) t .

Let u ( x , t ) = v ( x , t ) | x , t > 0 then u ( 0 + , t ) = m i n { u + ,0 } . It is easy to verify that u ( x , t ) is the global weak entropy solution of (7).

Case (2) u + < u d < u + .

In this case, only a shock wave S ( u − , u + ) starting at point ( 0,0 ) appears in the weak entropy solution of (8). This shock wave solution can be denoted as:

v ( x , t ) = { u − , x < s ( u − , u + ) t u + , x > s ( u − , u + ) t

where s ( u − , u + ) is the speed of the shock wave S ( u − , u + ) . Let u ( x , t ) = v ( x , t ) | x , t > 0 , then

u ( 0 + , t ) = { u + , if   x < s ( u − , u + ) ≤ 0 u − , if   x > s ( u − , u + ) > 0

By Lemma 2, one can verify that u ( x , t ) is the global weak entropy solution of (7).

For the initial boundary value problem (6) with u m ≠ u + , we only investigate the case of u m < u d 1 < u d 2 < 0 < u − , u + , f ( u m ) > f ( u − ) , which is the most typical and complicated case.

In this case, an initial shock wave S ( u − , u m ) , emanating at point ( 0,0 ) , and an initial rarefaction wave R = R ( u m , u d 1 ) ∪ R ( u d 1 , u d 2 ) ∪ R ( u d 2 , u + ) , centered at point ( a ,0 ) , appear in the weak entropy solution v ( x , t ) of the corresponding initial value problem (9). We denote this shock by x = x ( t ) , whose original speed of propagation is negative. The shock x = x ( t ) will interact with the rarefaction wave R on its right at some finite time t = t 1 . This interaction will generate a new shock, still denoted by x = x ( t ) . The left state of the resulting shock wave is u − . If u − > u + , the shock x = x ( t ) is able to cross the whole of R at finite time (see Figure 5(a)); if u − = u + , the shock x = x ( t ) is able to cross the whole of R completely only when t → ∞ ; if u − < u + , it is impossible for the shock to cross the whole of R completely, but it is able to cross the left part R = R ( u m , u d 1 ) ∪ R ( u d 1 , u d 2 ) ∪ R ( u d 2 , u + ) of R at t → ∞ (see Figure 5(b)). After t = t 1 , x = x ( t ) will penetrate the rarefaction wave on its right with a

non-decreasing speed of propagation. The shock x = x ( t )   ( t > 0 ) is a piecewise smooth curve.

During the process of its penetrating the rarefaction wave R, the shock wave x = x ( t )   ( t > 0 ) first crosses the leftmost part R ( u m , u d 1 ) of R with a varying speed of propagation, and then crosses the constant state u = u d 1 with constant speed. When the shock wave x = x ( t ) encounters the characteristic line of the leftmost characteristic line of the rarefaction wave R ( u d 1 , u d 2 ) , it again begins to cross the rarefaction wave R ( u d 1 , u d 2 ) with a varying speed. And then it crosses the constant state u = u d 2 with constant speed. Finally, it crosses the rightmost part R ( u d 2 , u − ) of R with a varying speed of propagation.

In view of u m < u d 1 < u d 2 < 0 < u − , u + and f ( u m ) > f ( u − ) , there exists u * ∈ ( u m ,0 ) such that f ( u * ) = f ( u − ) . Furthermore, there is t * > t 1 such that

x ′ ( t ) = s ( u − , u ( x ( t ) + 0 , t ) ) { < 0 , 0 < t < t * = 0 , t = t * > 0 , t > t *

for u * ≠ u d 1 , u d 2 and there are exist t 0 * ,   t * > t 1   ( t 0 * < t * ) , such that

x ′ ( t ) = s ( u − , u ( x ( t ) + 0 , t ) ) { < 0 , 0 < t < t 0 * = 0 , t 0 * ≤ t ≤ t * > 0 , t > t *

for u * = u d 1 or u d 2 , where x ′ ( t ) is the speed function of the shock wave x = x ( t ) in the weak entropy solution v ( x , t ) of (9). The position of the shock x = x ( t )   ( t > 0 ) is stated as follows: the shock x = x ( t ) lies in the second quadrant of the x − t plane as t ∈ ( 0, t * * ) , and cross the t-axis at t = t * * , and then enter and keep staying in the first quadrant (see Figure 5(a)), where t * * ( > t * ) is the unique time at which the shock x = x ( t ) and the t-axis axes intersect.

In what follows, we construct the global weak entropy solution of (6). Let t 2 denote the intersection time of the t-axis and the characteristic line with speed f ′ + ( u ( x ( t * ) + 0, t * ) ) from the point ( x ( t * ) , t * ) backward to x-axis, namely, t 2 = − a / f ′ + ( u * ) ( < t * ) . First take u ( x , t ) = v ( x , t ) | x , t > 0 , then by lemma 2, this u ( x , t ) is the local weak entropy solution of (6) on ( 0, + ∞ ) × ( 0, t 2 ) . Next we will extend this u ( x , t ) to ( 0, + ∞ ) × ( 0, + ∞ ) . Consider the following Cauchy problem:

{ ( v ¯ ) t + f ( v ¯ ) x = 0 , − ∞ < x < + ∞ ,   t > t 2 v ¯ ( x , t 2 ) = { u − ,         x < 0 v ( x , t 2 − 0 ) , x > 0 (11)

The weak entropy solution v ¯ ( x , t ) of (11) in ( 0, + ∞ ) × ( t 2 , + ∞ ) includes only a shock wave x = x + ( t ) starting at ( 0, t 2 ) , whose original speed of propagation is zero. The shock x = x + ( t ) will cross this part of the rarefaction wave on its right: R |   ( x − a ) / t     > f ′ + ( u * ) with a varying positive speed of propagation during the penetration in the first quadrant as t > t 2 . Then by Lemma 2,

u ( x , t ) : = v ¯ ( x , t ) | x > 0 ,     t > t 2 is the weak entropy solution of (6) on ( 0, + ∞ ) × ( t 2 , + ∞ ) .

Thus we accomplish the construction of the solution to (6) (see Figure 5(b)). The weak entropy solution of (6) has the following geometric structure near the point ( 0, t 2 ) : A part of the rarefaction wave

R = R ( u m , u d 1 ) ∪ R ( u d 1 , u d 2 ) ∪ R ( u d 2 , u + ) centered at point ( a ,0 ) collides with the boundary x = 0 , then the boundary reflects a new shock wave tangent to the boundary x = 0 at time t = t 2 , which will penetrate another part R |   ( x − a ) / t     > f ′ + ( u * ) of R with a varying positive speed of propagation in the first quadrant. This shock is just that one in v ¯ ( x , t ) .