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On a nonsymmetric Keyfitz-Kranzer system of conservation laws with generalized and modified Chaplygin gas pressure law.

1. Introduction Nonlinear hyperbolic conservation laws are a fundamental principle in building mathematical models for many natural processes. For them, there exists an important kind of nonclassical solution, that is, delta shockwave. It is a generalization of an ordinary shock. From the mathematical point of view, it is a kind of discontinuity, on which at least one of the state variables contains Dirac delta function with a shock as its support. From the physical point of view, it represents the process of concentration of the mass. The theory of nonlinear hyperbolic conservation laws admitting delta shock waves is interesting and has been extensively developed in the last several years; see [1-13] and the references cited therein.

Consider the hyperbolic system of conservation laws:

[[rho].sub.t] + [([rho](u - P)).sub.x] = 0, [([rho]u).sub.t] + [([rho]u (m - [rho])).sub.x] = 0, (1)

where p = p([rho]) and [rho] [greater than or equal to] 0. It belongs to the nonsymmetric Keyfitz-Kranzer system (see [14,15]):

[[rho].sub.t] + [([rho][phi] ([rho], [u.sub.1], [u.sub.2], ..., [u.sub.n])).sub.x] = 0, [([rho][u.sub.i]).sub.t] + [([rho][u.sub.i][phi] ([rho], [u.sub.1], [u.sub.2], ..., [u.sub.n])).sub.x] = 0, (2) i = 1, 2, ..., n,

which is of interest because it arises in such areas as elasticity theory, magnetohydrodynamics, and enhanced oil recovery. For delta shock waves, the nonsymmetric form is more convenient than the symmetric form (see [14]). Model (1) is also a transformation of the traffic flow model introduced by Aw and Rascle [16], where [rho] and u > 0 are the density and velocity of cars on the roadway and p is the velocity offset.

Let us recall the linear degeneracy and genuine nonlinearity of characteristic fields for quasilinear hyperbolic systems. Assume that [[lambda].sub.k] is the kth eigenvalue of a quasilinear hyperbolic system and [r.sub.k] is the corresponding eigenvector.

Then the kth characteristic field is said to be genuinely nonlinear if [nabla][[lambda].sub.k] x [r.sub.k] [not equal to] 0, while if, on the other hand, [nabla][[lambda].sub.k] x [r.sub.k] [equivalent to] 0, we call it linearly degenerate (see [17,18]). For (1), one can easily calculate that the eigenvalues are

[[lambda].sub.1] = u - p, [[lambda].sub.2] = u - p - [rho]p' (3)

and the corresponding right eigenvectors are

[r.sub.1] = (1, p'), [r.sub.2] = (1, 0). (4)

Then

[nabla][[lambda].sub.1] x [r.sub.1] [equivalent to] 0, [nabla][[lambda].sub.2] x [r.sub.2] = -([rho]p" + 2p'). (5)

Therefore the first characteristic field is always linearly degenerate and the second one is linearly degenerate or genuinely nonlinear depending on the behaviors of p([rho]).

In general, a solution to system (1) strongly depends on the function p([rho]). In 15], it is required that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)

A prototype function satisfying (6) or (7) is p = [[rho].sup.y], [gamma] > 0, with which the second characteristic [[lambda].sub.2] is genuinely nonlinear. Recently, in [19], Cheng introduced the function

p = -B/[rho], B > 0, (8)

which is the prototype function satisfying

[rho]p" ([rho]) + 2p' ([rho]) = 0 for [rho] > 0. (9)

At this moment, both eigenvalues are linearly degenerate; that is, this is a fully linearly degenerate system. The overlapping of linear degenerate characteristics results in the formation of delta shock waves.

In the present paper, we continue to study the system (1).

We extend (8) to the function

P = -B/[[rho].sup.[alpha]], 0 < [alpha] < 1, (10)

which does not satisfy (9). For the Riemann problem with initial data

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

with the analysis method in phase plane, we construct solutions which are R + J when [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) [greater than or equal to] [u.sub.-] + (B/[[rho].sup.[alpha].sub.-]) and S + J when [u.sub.-] < [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) < [u.sub.-] + (B/[[rho].sup.[alpha].sub.-]). However, for the rest case [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) [less than or equal to] [u.sub.-], the solution cannot be constructed by classical waves and delta shock waves should occur. Both existence and uniqueness of solutions involving delta shock waves are obtained by solving the generalized Rankine-Hugoniot relation under entropy condition.

We also introduce the function

p = A[rho] - [B/[[rho].sup.[alpha]]], A > 0. (12)

For the Riemann problem with initial data (11), by the analysis in phase plane, only two kinds of classical solutions R + J and S + J are constructed.

Noticing that (12) formally becomes (10) as A [right arrow] 0, we are interested in the topic that the limits of Riemann solutions to (1) and (12) are whether or not the Riemann solutions to (1) and (10). It is rigorously shown that, when A [right arrow] 0, some Riemann solutions containing S and J tend to a delta shock solution. However, its weight and speed are different from those of delta shock solution to (1), (10) with the same initial data (11). Besides, it is also proven that the rest Riemann solutions tend to just the ones of (1), (10) with the same initial data (11).

To confirm the theoretical analysis, we simulate the Riemann solutions and the formation process of delta shock waves by using the Nessyahu-Tadmor scheme [20]. The numerical results coincide completely with the theoretical analysis.

We remark that the functions (8), (10), and (12) are also the equations of state characterizing the standard, generalized, and modified Chaplygin gas, respectively, where p is the pressure and [rho] the density. As everyone knows, the universe is experiencing an accelerated expansion. Usually, it is thought that the source of this acceleration is attributed to an exotic type of fluid called dark energy. Then people began to search for different candidates of the dark energy. Up to now, some theoretical models have been proposed. Among them, the standard, generalized, and modified Chaplygin gas are plausible (see [21-25]).

This paper is organized as follows. In Sections 2 and 3, we solve the Riemann problem for (1) with (10) and (12), respectively. In Section 4, we investigate the limits of solutions to (1), (12), and (11) as A [right arrow] 0. In Section 5, some numerical results are presented. Section 6 gives a conclusion.

2. Riemann Problem for (1) with (10)

In this section, we construct the solutions to system (1), (10) with initial data (11). The considered system has two eigenvalues:

[[lambda].sub.1] = u + [B/[[rho].sup.[alpha]]], [[lambda].sub.2] = u + [B(1 - [alpha])/[[rho].sup.[alpha]]] (13)

with corresponding right eigenvectors:

[r.sub.1] = [(1, B[alpha]/[[rho].sup.[alpha]+1]).sup.T], [r.sub.2] = [(1,0).sup.T] (14)

satisfying, respectively,

[nabla][[lambda].sub.1] x [r.sub.1] [equivalent to] 0, [nabla][[lambda].sub.2] x [r.sub.2] = B[alpha](1 - [alpha])/[[rho].sup.[alpha]+1]. (15)

Therefore the system is strictly hyperbolic. The first wave family is linearly degenerate and the second one is genuinely nonlinear.

As usual, we seek the self-similar solution:

(u, p)(t, x) [equivalent to] (u, [rho])([xi]), [xi] = x/t, (16)

for which (1) with 10) becomes

-[xi][[rho].sub.[xi]] + ([rho](u + [B/[[rho].sup.[alpha]]])) = 0, -[xi][([rho]u).sub.[xi]] + [([rho]u(u + [B/[[rho].sup.[alpha]]])).sub.[xi]] = 0 (17)

and (11) changes into the infinity boundary condition

(u, [rho])([+ or -][infinity]) = ([u.sub.[+ or -]], [[rho].sub.[+ or -]]). (18)

For any smooth solutions, (17) is equivalent to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII, (19)

which provides either the general solution (constant state)

(u, [rho]) = Const. ([rho] > 0) (20)

or singular solution, which is a wave of the first characteristic family,

[xi] = [[lambda].sub.1] = u + [B/[[rho].sup.[alpha]]], d(u + [B/[[rho].sup.[alpha]]]) = 0, (21)

or rarefaction wave, which is a wave of the second characteristic family,

[xi] = [[lambda].sub.2] = u + B(1 - [alpha])/[[rho].sup.[alpha]], du = 0, (22)

Integrate (21) to obtain

[xi] = u + [B/[[rho].sup.[alpha]]], u + [B/[[rho].sup.[alpha]]] = [u.sub.-] + [B/[[rho].sup.[alpha].sub.-]], (23)

which is actually a contact discontinuity (see (26)). We also integrate (22) and take the requirement [[lambda].sub.2]([u.sub.-], [[rho].sub.-]) < [[lambda].sub.2] (u, [rho]) into account to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (24)

For a bounded discontinuity at [xi] = [sigma], the Rankine-Hugoniot relation

-[sigma][[rho]] + [[rho] (u + [B/[[rho].sup.[alpha]]])] = 0, -[sigma][[rho]u]+ [[rho]u(u + [B/[[rho].sup.[alpha]] ]) = 0 (25)

holds, where and in what follows [a] = [a.sub.-] - a denotes the jump of of a across the discontinuity. By solving (25), we obtain contact discontinuity, which is a wave of the first characteristic family,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (26)

or shock wave, which is a wave of the second characteristic family,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

The stability condition (entropy condition) for shocks can be defined as "three incoming, one outgoing," which means that three of the characteristic lines on both sides of shock, one [[lambda].sub.1] and two [[lambda].sub.2], are incoming with respect to the shock, while the remaining one, [[lambda].sub.1], is outgoing. The stability condition is equivalent to

[[rho].sub.-] < [rho]. (28)

The curves in (u, [rho])-plane expressed by the second equation in (24), (26), and (27) are called as the rarefaction wave curve, contact discontinuity curve, and shock wave curve, respectively. For the contact discontinuity curve, we have [u.sub.[rho]] = (B[alpha])/[[rho].sup.[alpha]+1] > 0 and [u.sub.pp] = -B[alpha]([alpha] + 1)/[[rho].sup.[alpha]+2] < 0, which mean that the curve is monotonic increasing and concave; moreover, it can be proved that [lim.sub.p[right arrow]0] u = -[infinity] and [lim.sub.[rho][right arrow]+[infinity]] u = [u.sub.-] + (B/[[rho].sup.[alpha].sub.-]), which show that the curve has [rho] = 0 and u = [u.sub.-] + (B/[[rho].sup.[alpha].sub.-]) as its two asymptotes.

Fixing a left state ([u.sub.-], [[rho].sub.-]), we draw the rarefaction wave curve, contact discontinuity curve, and shock wave curve. Also, we draw the curve u + (B/[[rho].sup.[alpha]]) = [u.sub.-]. Then the phase plane is divided into three domains I([u.sub.-], [[rho].sub.-]), II([u.sub.-], [[rho].sub.-]) and III([u.sub.-], [[rho].sub.-]) by the curves u + (B/[[rho].sup.[alpha]]) = [u.sub.-] + (B/[[rho].sup.[alpha].sub.-]), and u + (B/[[rho].sup.[alpha]]) = [u.sub.-] (see Figure 1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Then by the analysis method in phase plane, using the classical elementary waves R, S, and I, one can construct the solutions of Riemann problem in the following cases:

(1) when ([u.sub.+], [p.sub.+]) [member of] I([u.sub.-], [[rho].sub.-]), the solution is R + J;

(2) when ([u.sub.+], [[rho].sub.+]) [member of] II([u.sub.-], [[rho].sub.-]), the solution is S + J.

In these solutions, the intermediate state ([u.sub.*], [[rho].sub.*]) satisfies

[u.sub.*] = [u.sub.-], [u.sub.*] + [B/[[rho].sup.[alpha]]] = [u.sub.+] + [B/[[rho].sup.[alpha].sub.+]]. (30)

However, for the case ([u.sub.+], [[rho].sub.+]) [member of] III([u.sub.-], [[rho].sub.-]), one can check that the Riemann solution cannot be constructed by the classical elementary waves. In fact, at this moment, the characteristic lines from initial data will overlap in a domain. So singularity must happen. It is easy to know that the singularity is impossible to be a jump with finite amplitude because the Rankine-Hugoniot relation is not satisfied on the bounded jump. In other words, there is no solution which is piecewise smooth and bounded.

We define a two-dimensional weighted delta function w(s)[[delta].sub.L] supported on a smooth curve L parameterized as t = t(s), and x = x(s) (c [less than or equal to] s [less than or equal to] d) by

<w (s) [[delta].sub.L], [phi](x, t)) = [[integral].sup.d.sub.c] w (s) [phi] (x (s), t(s)) ds (31)

for all [phi] [member of] [C.sup.[infinity].sub.0] ([R.sup.2]).

With this definition, motivated by [4-6, 10, 12, 19], it is well known that the Riemann solution should be the delta shock solution of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (32)

with 1/[rho] defined by (see 10,12, 26])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (33

which satisfies the generalized Rankine-Hugoniot relation

dw (t)/dt = -[sigma][[rho]]+ [[rho] (u + [B/[[rho].sup.[alpha]]]), dw(t)[sigma]/dt = -[sigma][[rho]u] + [[rho]u(u + [B/[[rho].sup.[alpha]])] (34)

the entropy condition

[[lambda].sub.1] ([[rho].sub.+], [u.sub.+]) [less than or equal to] [sigma] [less than or equal to] [[lambda].sub.2] ([[rho].sub.-], [u.sub.-]), (35)

where w(t) and a are the weight and velocity of delta shock wave, respectively.

In what follows, we establish the existence and uniqueness of solution to the generalized Rankine-Hugoniot relation (34) under the entropy condition (35).

From (34) with initial data w(0) = 0, we obtain the algebra equations:

w(t) = -[sigma][[rho]]t + [[rho](u + [B/[[rho].sup.[alpha]]])] t, w(t)[sigma] = -[sigma][[rho]u]t + [[rho]u(u + [B/[[rho].sup.[alpha]]])] t, (36)

from which it follows that

[[rho]] [[sigma].sup.2] - ([[rho](u + [B/[[rho].sup.[alpha]]] + [[rho]u])[sigma] + [[rho]u (u + [B/[[rho].sup.[alpha]])] = 0. (37)

Entropy condition (35) also reads

[u.sub.+] + [B/[[rho].sup.[alpha].sub.+]] [less than or equal to] [sigma] [less than or equal to] [u.sub.-] + [B(1 - [alpha])/[[rho].sup.[alpha].sub.-]]. (38)

Set

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (40)

Denote

[E.sub.1] = [[u.sub.+] + [B/[[rho].sup.[alpha].sub.+]], [u.sub.-]], [E.sub.2] = ([u.sub.-], [u.sub.-] + [B(1 - [alpha])/[[rho].sup.[alpha].sub.-]]. (41)

If [sigma] [member of] [E.sub.2], we have F([sigma]) < 0, which means that (37) has no solution in [E.sub.2]. If [sigma] [member of] [E.sub.1], we have

F ([u.sub.+] + [B/[[rho].sup.[alpha].sub.+]]) > 0, F ([u.sub.-]) < 0, F'([sigma]) < 0, (42)

which can show that (37) has a unique solution in [E.sub.1] by zero point theorem in mathematical analysis. In summary, we can conclude that (37) has a unique solution under (38).

Returning to (36), we can obtain w(t) uniquely.

Theorem 1. For the Riemann problem (1), (10), and (11), there exists a unique entropy solution, which consists of a rarefaction wave and a contact discontinuity when [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) [greater than or equal to] [u.sub.-] + (B/[[rho].sup.[alpha].sub.-]), a shock and a contact discontinuity when [u.sub.+] < [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) < [u.sub.-] + (B/[[rho].sup.[alpha].sub.-]), and a delta shock wave when [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) [less than or equal to] [u.sub.-].

3. Riemann Problem for (1) with (12)

In this section, we briefly solve Riemann problem (1), (12) and (11). The eigenvalues are

[[lambda].sub.1] = u - A[rho] + [B/[[rho].sup.[alpha]]], [[lambda].sub.2] = u - 2A[rho] + [B(1 - [alpha])/[[rho].sup.[alpha]]]. (43)

The corresponding right eigenvectors are [r.sub.1] = (1, A + [B[alpha]/[[rho].sup.[alpha]+1]]), [r.sub.2] = [(1,0).sup.T]. (44)

Moreover,

[nabla][[lambda].sub.1] x [r.sub.1] [equivalent to] 0, [nabla][[lambda].sub.2] x [r.sub.2] = -2A - [B[alpha](1 - [alpha])/[[rho].sup.[alpha]+1]]. (45)

Thus the system is strictly hyperbolic. The first wave family is linearly degenerate and the second one is genuinely nonlinear.

One can check that, besides the constant state

(u, [rho]) = Const., (46)

the classical elementary waves consist of rarefaction waves

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (47)

contact discontinuities

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (48)

and shocks

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (49)

In (u, [rho])-plane, from (48), for the contact discontinuity curve u = A[rho] - (B/[[rho].sup.[alpha]]) + [u.sub.-] - A[[rho].sub.-] + (B/[[rho].sup.[alpha].sub.-]), we have [u.sub.[rho]] = A + (B[alpha]/[[rho].sup.[alpha]+1]) > 0, and [u.sub.pp] = -B[alpha]([alpha] + 1)/[[rho].sup.[alpha]+2] < 0, which show that the contact discontinuity curve is monotonic increasing and concave. Moreover, we have [lim.sub.[rho][right arrow]+[infinity]] u = +[infinity] and [lim.sub.p[right arrow]0] u = -[infinity].

For a given state ([u.sub.-], [[rho].sub.-]), the phase plane is divided into two domains RJ([u.sub.-], [[rho].sub.-]) and SJ([u.sub.-], [[rho].sub.-]) by the curves u - A[rho] + (B/[[rho].sup.[alpha]]) = [u.sub.-] - A[[rho].sub.-] + (B[/[rho].sup.[alpha].sub.-]) (see Figure 2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (50)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (51)

Using the classical elementary waves, one can obtain the solutions of Riemann problem as follows:

(a) when ([u.sub.+], [[rho].sub.+]) [member of] RJ([u.sub.-], [[rho].sub.-]), the solution is R + J;

(b) when ([u.sub.+], [[rho].sub.+]) [member of] SJ([u.sub.-], [[rho].sub.-]), the solution is S + J.

In these solutions, the intermediate state ([u.sub.*], [[rho].sub.*]) satisfies

[u.sub.*] = [u.sub.-], [u.sub.*] - A[[rho].sub.*] + [B[/[rho].sup.[alpha].sub.*]] = [u.sub.+] - A[[rho].sub.+] + [B/[[rho].sup.[alpha].sub.-]]. (52)

Theorem 2. For the Riemann problem (1), (12), and (11), there exists a unique entropy solution, which consists of a rarefaction wave and a contact discontinuity when u - A[rho] + (B/[[rho].sup.[alpha]]) [greater than or equal to] [u.sub.-] - A[[rho].sub.-] + (B/[[rho].sup.[alpha].sub.-]) and a shock and a contact discontinuity when u - A[rho] + (B/[[rho].sup.[alpha]]) < [u.sub.-] - A[[rho].sub.-] + (B/[[rho].sup.[alpha].sub.-]).

4. Limits of Riemann Solutions to (1) with (12)

In this section, we consider the limits of the Riemann solutions to (1), (12), and (11) as A [right arrow] 0 case by case.

Case 1. Consider ([u.sup.+], [[rho].sub.+]) [member of] I([u.sub.-], [[rho].sub.-]).

Lemma 3. When ([u.sub.+], [[rho].sub.+]) [member of] I([u.sub.-], [[rho].sub.-]), there exists [A.sub.0] > 0 such that ([u.sub.+], [[rho].sub.+]) [member of] RJ([u.sub.-], [[rho].sub.-]) when 0 < A < [A.sub.0].

Proof. All states (u, [rho]) connected with ([u.sub.-], [[rho].sub.-]) by a contact discontinuity (48) satisfy

u - A[rho] + [B/[[rho].sup.[alpha]]] = [u.sub.-] - A[[rho].sub.-] + [B/[[rho].sup.[alpha].sub.-]]. (53)

If [[rho].sub.+] [less than or equal to] [[rho].sub.-], [A.sub.0] can be taken as arbitrary nonnegative real number. If [[rho].sub.+] > [[rho].sub.-], by taking A0 determined by

[u.sub.+] - [A.sub.0][[rho].sub.+] + [B/[[rho].sup.[alpha].sub.+]] = [u.sub.-] - [A.sub.0][[rho].sub.-] + [B/[[rho].sup.[alpha].sub.-], (54)

we have the conclusion.

For any [A.sub.0] > A > 0, the solution to Riemann problem (1), (11), and (12) is a rarefaction wave R followed by a contact discontinuity J with the intermediate state ([u.sup.A.sub.*], [[rho].sup.A.sub.*]) besides two constant states ([u.sub.-], [[rho].sub.-]) and ([u.sub.+], [[rho].sub.+]). They have the following relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (55)

Denote ([u.sub.*], [[rho].sub.*]) = [lim.sub.A[right arrow]0] ([u.sup.A.sub.*], [[rho].sup.A.sub.*]). Letting A [right arrow] 0 in (55), then R turns into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (56)

and J becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (57)

One can see that the limit in this situation is the same as the solution to (1), (10) with the same initial data (11).

Case 2. Consider ([u.sub.+], [[rho].sub.+]) [member of] II([u.sub.-], [[rho].sub.-]).

Lemma 4. When ([u.sub.+], [[rho].sub.+]) [member of] II([u.sub.-], [[rho].sub.-]), there exists [A.sub.0] > 0 such that ([u.sub.+], [[rho].sub.+]) [member of] SJ([u.sub.-], [[rho].sub.-]) when 0 < A < [A.sub.0].

This lemma can be proved in a way completely similar to that of Lemma 3.

For any [A.sub.0] > A > 0, the solution of Riemann problem (1), (11), and (12) is a shock S followed by a contact discontinuity J with the intermediate state ([u.sup.A.sub.*], [[rho].sup.A.sub.*]) besides two constant states ([u.sub.-], [[rho].sub.-]) and ([u.sub.+], [[rho].sub.+]). The solution can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (58)

where [[sigma].sup.A] and [[tau].sup.A] are the propagation speeds of S and J, respectively.

Denote ([u.sub.*], [[rho].sub.*]) = [lim.sub.A[right arrow]0] ([u.sup.A.sub.*], [[rho].sup.A.sub.*]). We have [[rho].sub.*] [not equal to] +[infinity]. In fact, if [[rho].sub.*] * +[infinity], from 58), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (59)

which contradicts with ([u.sub.+], [p.sub.+]) [member of] II([u.sub.-], [[rho].sub.-]), which is [u.sub.-] < [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) < [u.sub.-] + (B/[[rho].sup.[alpha].sub.-]).

Letting A [right arrow] 0 in (58), then S turns into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (60)

and J tends to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (61)

One can observe that the limit in this case is also the same as that in the solution to (1), (10) with the same initial data (11).

Case 3. Consider ([u.sub.+], [[rho].sub.+]) [member of] III ([u.sub.-], [[rho].sub.-]).

For this case, analogous to Lemma 3, we have the following.

Lemma 5. When ([u.sub.+], [[rho].sub.+]) [member of] III ([u.sub.-], [[rho].sub.-]), there exists [A.sub.0] > 0 such that ([u.sub.+], [[rho].sub].+]) [member of] SJ([u.sub.-], [[rho].sub.-]) when 0 < A < [A.sub.0].

When 0 < A < [A.sub.0], the solution of Riemann problem (1), (11), and (12) is a shock S followed by a contact discontinuity J with the intermediate state ([u.sup.A.sub.*], [[rho].sup.A.sub.*]) besides two constant states ([u.sub.-], [[rho].sub.-]) and ([u.sub.+], [[rho].sub.+]). They have the following relations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (62)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (63)

where [[sigma].sup.A] and [[tau].sup.A] are the propagation speeds of S and J, respectively. Then we have the following lemmas.

Lemma 6. One has [lim.sub.A[right arrow]0] [[rho].sup.A.sub.*] = +[infinity].

Proof. From the second equations in (62) and 63), we easily get

[u.sub.-] - A[[rho].sup.A.sub.*] + [B/([[rho].sup.A.sub.*])] = [u.sub.+] - A[[rho].sub.+] + [B/[rho][alpha]], [[rho].sup.A.sub.*] > [[rho].sub.-]. (64)

If [lim.sub.A[right arrow]0] [[rho].sup.A.sub.*] = k [not equal to] + [infinity], we have

[u.sub.-] + [B/[k.sup.[alpha]]] = [u.sub.+] + [B/[[rho].sup.[alpha].sub.+]], (65)

which is impossible since ([u.sub.+], [[rho].sub.+]) [member of] III([u.sub.-], [[rho].sub.-]); that is, [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) [less than or equal to] [u.sub.-]. Therefore, [lim.sub.A[right arrow]0] [[rho].sup.A- sub.*] = +[infinity].

Lemma 7. One has [lim.sub.A[right arrow]0] [[sigma].sup.A] = [lim.sub.A[right] arrow]0] [[tau].sup.A] = [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) := [[sigma].sub.0].

Proof. From (62) and 63), we conclude that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (66)

The proof is complete.

Lemmas 6-7 shows that when A drops to zero, S and J coincide, and the intermediate density [[rho].sup.A.sub.*] becomes singular.

Lemma 8. One has

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (67)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (68)

Proof. The Rankine-Hugoniot relations for S and J read, respectively,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (69)

from which and Lemma 7 we can easily get (68) and (70).

In what follows, we show the conclusion characterizing the limit of the solution to (1), (11) with initial data (12) for the case ([u.sub.+], [[rho].sub.+]) [member of] III([u.sub.-], [[rho].sub.-]) as A [right arrow] 0.

Theorem 9. Let ([u.sup.+], [[rho].sub.+]) e III([u.sub.-], [[rho].sub.-]) and assume that (uA, pA)(t, x) is the solution S + J to Riemann problem (1), (11), and (12) for 0 < A < A0 constructed in Section 3. Then

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (70)

and [[rho].sup.A] and [[rho].sup.A][u.sup.A] converge in the sense of distributions, and the limit functions [rho] and [rho]u are the sums of a step function and a Dirac delta function supported on x = [[sigma].sub.0]t with weights

(-[[sigma].sub.0] [[rho]] + [[rho](u + [B/[[rho].sup.[alpha]]]))t, (-[[sigma].sub.0] [[rho]u]+ [[rho]u(u + [B/[[rho].sup.[alpha]]]))t, (71)

respectively.

Proof. (1) Set [xi] = x/t. Then for each A > 0, the Riemann solution S + J can be expressed as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (72)

satisfying weak formulations

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (73)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (74)

for any [phi] [member of] [C.sup.1.sub.0] (-[infinity], +[infinity]). Equation (70) can be obtained easily.

(2) The first integral on the left-hand side of (73) can be decomposed into

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (75)

The sum of the first and last terms on the right-hand side of (75) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

which converges as A [right arrow] 0 to

(-[[sigma].sub.0] [[rho]] + [[rho](u + [B/[[rho].sup.[alpha]]])) [phi]([[sigma].sub.0]) + [[integral].sup.+[infinity].sub.-[infinity]] H([xi] - [[sigma].sub.0]) [phi]([xi])d[xi] (77)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (78)

The second term on the right hand of (75) is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (79)

which converges as A [right arrow] 0 to

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (80)

Returning to (73), we immediately obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (81

Similarly, from (74), we can obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (82)

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (83)

(3) We study the limits of [[rho].sup.A] and [[rho].sup.A][u.sup.A] by tracking the time-dependence of the weights of the d-measure as A [right arrow] 0. Taking (81) into account, we have for any [psi] [member of] [C.sup.[infinity].sub.0] (R x [R.sup.+])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (84)

in which by (31)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (85)

with

[w.sub.1](t) = (-[[sigma].sub.0][[rho]] + [[rho](u + [B/[[rho].sup.[alpha]]]))t. (86)

Similarly, from (82), we have for any [psi] [member of] [C.sup.[infinity].sub.0] (R x [R.sup.+])

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (87)

in which

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (88)

with

[w.sub.2] (t) = (-[[sigma].sub.0] [[rho]u]+ [[rho]u(u + [B/[[rho].sup.[alpha]]]))t. (89)

This completes the proof.

One can observe that when [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) < [u.sub.-], the weight and speed of the delta shock wave in the limit situation are different from the weight and speed of the delta shock wave obtained by solving (1), (10) with the same initial data (11) in Section 2; and only when [u.sub.+] + (B/[[rho].sup.[alpha].sub.+]) = [u.sub.-], they are coincident.

5. Numerical Simulations

In this section, we simulate the Riemann solutions obtained in Sections 2 and 3 and examine the processes of formation of delta shock waves studied in Section 4. To discretize the system, we employ the Nessyahu-Tadmor scheme [20] with 500 x 500 cells and CFL = 0.475. [alpha] = 0.5 and B = 1 are taken.

5.1. Numerical Simulations of Solutions to (1) and (11) with (10)

Casel (R + J). For this case, we take the initial data ([u.sub.-], [p.sub.-]) = (0.5, 1.0) and ([u.sub.+], [[rho].sub.+]) = (2.0, 0.16). The numerical results at t = 0.3 are shown in Figure 3.

Case 2 (S + J). In this case, the initial data are taken as ([u.sub.-], [[rho].sub.-]) = (2.0,0.04) and ([u.sub.+], [[rho].sub.+]) = (0.0,0.04). The numerical results at t = 0.3 are presented in Figure 4.

Case 3 ([delta]). At this moment, the initial data are taken as ([u.sub.-], [[rho].sub.-]) = (6.0,1.0) and ([u.sub.+], [[rho].sub.+]) = (1.0,0.1225). The numerical results at t = 0.2 are given in Figure 5.

One can observe that all the numerical results are in complete agreement with the theoretical analysis.

5.2. Numerical Simulations of Solutions to (1) and (11) with (12)

Case 1 (R + J). For this case, we take the initial data ([u.sub.-], [[rho].sub.-] = (0.5,1.0) and ([u.sub.+], [[rho].sub.+]) = (2.0,0.16). The numerical results with A = 0.5 at t = 0.3 are shown in Figure 6.

Case 2 (S + J). In this case, the initial data are taken as ([u.sub.-], [[rho].sub.-]) = (6.0,1.0) and ([u.sub.+], [[rho].sub.+]) = (0.32,1.0). The numerical results with A = 0.5 at t = 1 are presented in Figure 7.

It can be seen that these numerical results are also in accord with the theoretical analysis.

5.3. Numerical Simulations of Formation Process of Delta Shock Waves. We take the initial data ([u.sub.-], [[rho].sub.-]) = (6.0,1.0) and ([u.sub.+], [[rho].sub.+]) = (1.0, 0.1225). The numerical results corresponding to different A at t = 0.2 are given in Figures 8, 9, and 10.

One can see clearly from these numerical results that when A decreases, the location of the shock and contact discontinuity becomes closer, and the density of the intermediate state increases dramatically.

6. Conclusion

The nonlinear hyperbolic systems of conservation laws of Keyfitz-Kranzer type in the symmetric or nonsymmetric form are important models. This paper is concerned with a special nonsymmetric Keyfitz-Kranzer system (1) with an unknown function p([rho]), which is often encountered in some problems. Generally, its solution depends strongly on the behaviors of p([rho]). It is very necessary to study such a system for different choices of p([rho]).

Previously, the investigations are focused on the polytropic gas pressure law p = [[rho].sup.[gamma]], [gamma] > 0 [15, 16]. Recently, Cheng [19] introduced the Chaplygin gas pressure law (8). In the present paper, we consider the generalized Chaplygin gas pressure law (10) and the modified Chaplygin gas pressure law (12).

It is well known that, for nonlinear conservation laws, the classical waves consist of rarefaction wave (R), shock (S), contact discontinuity (J), and vacuum (Vac), and the nonclassical ones contain delta shockwave ([delta]). Forsystem 1), we find that the waves involved in the Riemann solutions are different for different choices of the function p([rho]). This fact is shown in Table 1, where PG, CG, GCG, and MCG denote the pressure law for polytropic gas, standard Chaplygin gas, generalized Chaplygin gas, and modified Chaplygin gas, respectively; [check] means that the waves will appear in the Riemann solutions, and M is opposite. It is also concluded that only when the domain III([u.sub.-], [[rho].sub.-]) appears in the phase plane, delta shockwave will develop.

It is seen that the modified Chaplygin gas pressure law can be regarded as a linear combination of the polytropic gas pressure law (the case for [gamma] = 1) and the generalized Chaplygin gas pressure law. We are interested in the limits of the Riemann solutions for the modified Chaplygin gas pressure law as parameter A vanishes. As a result, it is in particular proved that, for some initial data, the limit is a delta shockwave. However, it is not coincident with the delta shock solution to the limit system. This is different from the results in [7,8].

http://dx.doi.org/10.1155/2013/187217

Acknowledgments

The authors would like to thank the anonymous referees for providing valuable suggestions and pointing out imprecise expressions. The research Supported by the National Natural Science Foundation of China (11226191, 10961025).

References

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Hongjun Cheng and Hanchun Yang

Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China

Correspondence should be addressed to Hongjun Cheng; hjcheng@ynu.edu.cn

Received 2 June 2013; Revised 20 October 2013; Accepted 25 October 2013

Academic Editor: Stephen C. Anco

TABLE 1: Waves involved in the Riemann solutions
for different p([rho]).

         R         S         J        Vac     [delta]

PG    [check]   [check]   [check]   [check]      X
CG       X         X      [check]      X      [check]
GCG   [check]   [check]   [check]      X      [check]
MCG   [check]   [check]   [check]      X         X
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Title Annotation:Research Article
Author:Cheng, Hongjun; Yang, Hanchun
Publication:Advances in Mathematical Physics
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Date:Jan 1, 2014
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