Delta Shock Waves for a Linearly Degenerate Hyperbolic System of ...

40 downloads 0 Views 2MB Size Report
Mar 18, 2013 - This paper is devoted to the study of delta shock waves for a hyperbolic system ... waves, we refer the readers to the papers [4รขย€ย“24] and the refe-.
Hindawi Publishing Corporation Advances in Mathematical Physics Volume 2013, Article ID 958120, 10 pages http://dx.doi.org/10.1155/2013/958120

Research Article Delta Shock Waves for a Linearly Degenerate Hyperbolic System of Conservation Laws of Keyfitz-Kranzer Type Hongjun Cheng School of Mathematics and Statistics, Yunnan University, Kunming, Yunnan 650091, China Correspondence should be addressed to Hongjun Cheng; [email protected] Received 6 December 2012; Revised 1 March 2013; Accepted 18 March 2013 Academic Editor: Dongho Chae Copyright ยฉ 2013 Hongjun Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper is devoted to the study of delta shock waves for a hyperbolic system of conservation laws of Keyfitz-Kranzer type with two linearly degenerate characteristics. The Riemann problem is solved constructively. The Riemann solutions include exactly two kinds. One consists of two (or just one) contact discontinuities, while the other contains a delta shock wave. Under suitable generalized Rankine-Hugoniot relation and entropy condition, the existence and uniqueness of delta shock solution are established. These analytical results match well the numerical ones. Finally, two kinds of interactions of elementary waves are discussed.

1. Introduction Consider the following hyperbolic system of conservation laws: ๐œŒ๐‘ก + (๐œŒ (๐‘ข โˆ’ ๐‘ƒ))๐‘ฅ = 0,

(1)

(๐œŒ๐‘ข)๐‘ก + (๐œŒ๐‘ข (๐‘ข โˆ’ ๐‘ƒ))๐‘ฅ = 0,

๐œŒ๐‘ก + (๐œŒ๐œ™ (๐œŒ, ๐‘ข1 , ๐‘ข2 , . . . , ๐‘ข๐‘› ))๐‘ฅ = 0, ๐‘– = 1, 2, . . . , ๐‘›. (2)

System (1) was also introduced as a macroscopic model for traffic flow by Aw and Rascle [3], where ๐œŒ and ๐‘ข are the density and the velocity of cars on the roadway, and the function ๐‘ƒ is smooth and strictly increasing and it satisfies ๐œŒ๐‘ƒ๓ธ€ ๓ธ€  (๐œŒ) + 2๐‘ƒ๓ธ€  (๐œŒ) > 0

๐‘ƒ (0) = 0,

lim ๐œŒ๐‘ƒ๓ธ€  (๐œŒ) = 0,

๐œŒโ†’0

๐œŒ๐‘ƒ๓ธ€ ๓ธ€  (๐œŒ) + 2๐‘ƒ๓ธ€  (๐œŒ) > 0 for ๐œŒ > 0, lim ๐œŒ๐‘ƒ (๐œŒ) = 0,

where ๐‘ƒ = ๐‘ƒ(๐œŒ) with ๐œŒ โ‰ฅ 0. Model (1) belongs to the nonsymmetric system of Keyfitz-Kranzer type [1, 2] as

(๐œŒ๐‘ข๐‘– )๐‘ก + (๐œŒ๐‘ข๐‘– ๐œ™ (๐œŒ, ๐‘ข1 , ๐‘ข2 , . . . , ๐‘ข๐‘› ))๐‘ฅ = 0,

global bounded weak solutions of the Cauchy problem under the following two assumptions on ๐‘ƒ(๐œŒ), respectively:

for ๐œŒ > 0.

(3)

Model (1) was also studied by Lu [2]. By using the compensated compactness method, he established the existence of

๐œŒโ†’0

๓ธ€ 

lim (๐œŒ๐‘ƒ (๐œŒ)) โ‰ฅ ๐ด,

(4)

๐œŒโ†’โˆž

๐œŒ๐‘ƒ๓ธ€ ๓ธ€  (๐œŒ) + 2๐‘ƒ๓ธ€  (๐œŒ) > 0 for ๐œŒ > 0. One can find that system (1) has two characteristics; one is always linearly degenerate while the other is linearly degenerate or genuinely nonlinear depending on the behaviors of ๐‘(๐œŒ). In [2, 3], the condition (3) is required, which implies that the second characteristic is genuinely nonlinear. Thus, one interesting topic is the case when both characteristics are linearly degenerate. These motivate us to consider (1) with 1 ๐‘ƒ = ๐‘ƒ (๐œŒ) = โˆ’ , ๐œŒ

(5)

which is the prototype function satisfying ๐œŒ๐‘ƒ๓ธ€ ๓ธ€  (๐œŒ) + 2๐‘ƒ๓ธ€  (๐œŒ) โ‰ก 0 for ๐œŒ > 0.

(6)

2

Advances in Mathematical Physics

For system (1) with (5) or (6), a distinctive feature is that both eigenvalues are linearly degenerate; that is, this is a fully linearly degenerate system. Thus, all the classical elementary waves only consist of contact discontinuities. Moreover, the overlapping of linearly degenerate characteristics may result in the formation of delta shock wave. A delta shock wave is a generalization of an ordinary shock wave. Speaking informally, it is a kind of discontinuity, on which at least one of the state variables may develop an extreme concentration in the form of a Dirac delta function with the discontinuity as its support. It is more compressive than an ordinary shock wave in the sense that more characteristics enter the discontinuity line. From the physical point of view, a delta shock wave represents the process of concentration of the mass and formation of the universe. For related results of delta shock waves, we refer the readers to the papers [4โ€“24] and the references therein. Putting (5) into (1), we obtain the equivalent system ๐œŒ๐‘ก + (๐œŒ๐‘ข)๐‘ฅ = 0, (๐œŒ๐‘ข)๐‘ก + (๐œŒ๐‘ข2 + ๐‘ข)๐‘ฅ = 0.

๐œ† 1 = ๐‘ข,

๐œ†2 = ๐‘ข +

1 ๐œŒ

(10)

1 ๐‘‡ ) ๐œŒ2

(11)

with associated right eigenvectors ๐‘Ÿ1โƒ— = (1, 0)๐‘‡ ,

๐‘Ÿ2โƒ— = (1,

satisfying

๐‘ฅ < 0, 0 < ๐‘ฅ < ๐‘Ž,

In this section, we analyze some basic properties and then solve the Riemann problem (7)-(8) by the analysis in phase plane. The system has two eigenvalues

(8)

By the analysis method in phase plane, when ๐‘ข+ +(1/๐œŒ+ ) > ๐‘ขโˆ’ , we can construct Riemann solution only involving two different (or just one) contact discontinuities. However, for the case ๐‘ข+ + (1/๐œŒ+ ) โ‰ค ๐‘ขโˆ’ , we find that the Riemann solution can not be constructed by these classical contact discontinuities and delta shock wave should occur. For delta shock wave, by a definition of measure solutions to (7), we derive the generalized Rankine-Hugoniot relation which describes the relation among the limit states on both sides of the discontinuity, location, propagation speed, weight, and the assignment of the component ๐‘ข on its discontinuity relative to the delta shock wave. The entropy condition is an overcompressive one and guarantees the uniqueness of solution. With the generalized Rankine-Hugoniot relation and entropy condition, we solve the delta shock solution uniquely. Secondly, we simulate the obtained Riemann solutions by using N-T scheme [25] and get the numerical results coinciding with the analytical ones. Thirdly, we study two kinds of interactions of delta shock waves and contact discontinuities by considering the initialvalue problem with initial data as (๐‘ขโˆ’ , ๐œŒโˆ’ ) , { { { { (๐‘ข, ๐œŒ) (๐‘ฅ, ๐‘ก = 0) = {(๐‘ข๐‘š , ๐œŒ๐‘š ) , { { { {(๐‘ข+ , ๐œŒ+ ) ,

2. Preliminaries and Solutions Involving Contact Discontinuities

(7)

Firstly, we consider the Riemann problem for (7) with initial data {(๐‘ขโˆ’ , ๐œŒโˆ’ ) , ๐‘ฅ < 0, (๐‘ข, ๐œŒ) (๐‘ฅ, ๐‘ก = 0) = { (๐‘ข , ๐œŒ ) , ๐‘ฅ > 0. { + +

encountered in the interactions of elementary waves for some other systems, such as the zero-pressure gas dynamics. This paper is organized as follows. In Section 2, we present some preliminaries and construct the Riemann solution consisting of contact discontinuities by phase plane analysis method. Then, in Section 3, with suitable generalized Rankine-Hugoniot relation and entropy condition, we solve delta shock solution uniquely. Section 4 presents some numerical results of Riemann solutions obtained by using the Nessyahu-Tadmor scheme [25]. Finally, in Section 5, we study two kinds of interactions of delta shock waves and contact discontinuities.

โˆ‡๐œ† ๐‘– โ‹… ๐‘Ÿ๐‘–โƒ— โ‰ก 0,

where ๐‘Ž > 0 is a constant. In the first kind, the interaction of two contact discontinuities leads to a delta shock wave. In the second one, the delta shock wave vanishes after it interacts with a contact discontinuity. Such two cases are not

(12)

Thus, system (7) is fully linearly degenerate. The linear degeneracy also excludes the possibility of rarefaction waves and shock waves. As usual, we seek the self-similar solution ๐‘ฅ (๐‘ข, ๐œŒ) (๐‘ก, ๐‘ฅ) = (๐‘ข, ๐œŒ) (๐œ‰) , ๐œ‰ = , (13) ๐‘ก for which system (7) becomes โˆ’ ๐œ‰๐œŒ๐œ‰ + (๐œŒ๐‘ข)๐œ‰ = 0, โˆ’ ๐œ‰(๐œŒ๐‘ข)๐œ‰ + (๐œŒ๐‘ข2 + ๐‘ข)๐œ‰ = 0,

(14)

and initial data (8) changes to the boundary condition (๐‘ข, ๐œŒ) (ยฑโˆž) = (๐‘ขยฑ , ๐œŒยฑ ) .

(15)

This is a two-point boundary value problem of first-order ordinary differential equations with the boundary values in the infinity. For any smooth solution, (14) turns into

(9)

๐‘ฅ > ๐‘Ž,

๐‘– = 1, 2.

๐‘ขโˆ’๐œ‰ (

0

๐œŒ

๐‘‘๐œŒ ) ( ) = 0. 1 ๐‘‘๐‘ข ๐‘ข+ โˆ’๐œ‰ ๐œŒ

(16)

It provides either the general solutions (constant states) (๐‘ข, ๐œŒ) = const.

(๐œŒ > 0)

(17)

Advances in Mathematical Physics

3 โ†’ โ† ๐ฝ ๐ฝ

๐œŒ

or singular solutions

โ†’ ๐ฝ

III

๐œ‰ = ๐œ† 1 = ๐‘ข, 1 ๐œ‰ = ๐œ†1 = ๐‘ข + , ๐œŒ

๐‘‘๐‘ข = 0, 1 ๐‘‘ (๐‘ข + ) = 0. ๐œŒ

(18)

V (๐‘ขโˆ’ โˆ’

1 ,๐œŒ ) ๐œŒโˆ’ โˆ’

IV II

Integrating (18) from (๐‘ขโˆ’ , ๐œŒโˆ’ ) to (๐‘ข, ๐œŒ), respectively, one can get that

1 ๐‘ขโˆ’ โˆ’ ๐œŒโˆ’

๐‘ขโˆ’

๐‘ขโˆ’ +

๐‘ข = ๐‘ขโˆ’ , 1 ๐œ‰=๐‘ข+ , ๐œŒ

(19)

๐‘ข

Then the phase plane is divided into five regions as follows (shown in Figure 1): I = { (๐‘ข, ๐œŒ) | ๐‘ขโˆ’ < ๐‘ข < ๐‘ขโˆ’ +

1 1 ๐‘ข + = ๐‘ขโˆ’ + . ๐œŒ ๐œŒโˆ’

0 0, which is monotonically increasing and has the lines ๐œŒ = 0 and ๐‘ข = ๐‘ขโˆ’ + (1/๐œŒโˆ’ ) as its two asymptotic lines. Also, from the point (๐‘ขโˆ’ โˆ’ (1/๐œŒโˆ’ ), ๐œŒโˆ’ ), we draw the curve ๐‘ข + (1/๐œŒ) = ๐‘ขโˆ’ .

1 }, ๐‘ขโˆ’ + ๐œŒโˆ’โˆ’1 โˆ’ ๐‘ข

1 < ๐œŒ < +โˆž} . ๐‘ขโˆ’ โˆ’ ๐‘ข

(23)

By the analysis method in the phase plane, when (๐‘ข+ , ๐œŒ+ ) โˆˆ I, II, III, IV, namely, ๐‘ข+ + (1/๐œŒ+ ) > ๐‘ขโˆ’ , one can easily construct the solution consisting of two different (or just one) contact discontinuities (see Figure 2), in which, โ† ๓ณจ€ ๐ฝ : ๐‘ฅ = ๐‘ขโˆ’ ๐‘ก,

๐ฝ โƒ— : ๐‘ฅ = (๐‘ข+ +

1 ) ๐‘ก, ๐œŒ+

(24)

the intermediate states (๐‘ขโˆ— , ๐œŒโˆ— ) connecting two contact discontinuities satisfy ๐‘ขโˆ— = ๐‘ขโˆ’ ,

๐‘ขโˆ— +

1 1 = ๐‘ข+ + . ๐œŒโˆ— ๐œŒ+

(25)

4

Advances in Mathematical Physics ๐‘ก ๐‘ฅ/๐‘ก = ๐‘ข+ + โ† ๐ฝ

โ†’ ๐ฝ

๐‘ก

1 ๐œŒ+

๐‘ฅ/๐‘ก = ๐‘ขโˆ’

๐‘ฅ/๐‘ก = ๐‘ข+

(๐‘ขโˆ— , ๐œŒโˆ— ) (๐‘ขโˆ’ , ๐œŒโˆ’ )

(๐‘ข+ , ๐œŒ+ )

ฮฉ

๐‘ฅ/๐‘ก = ๐‘ขโˆ’ +

(๐‘ขโˆ’ , ๐œŒโˆ’ )

๐‘ฅ

๐‘‚

๐›ฟ : ๐‘ฅ/๐‘ก = ๐‘ข๐›ฟ

(๐‘ข+ , ๐œŒ+ ) ๐‘‚

Figure 2: Riemann solution involving two contact discontinuities.

1 ๐œŒโˆ’

๐‘ฅ

Figure 3: Riemann solution containing delta shock wave.

3. Delta Shock Solution In this section, let us in detail discuss the last case when (๐‘ข+ , ๐œŒ+ ) โˆˆ V (๐‘ขโˆ’ , ๐œŒโˆ’ ); that is, ๐‘ข+ + (1/๐œŒ+ ) โ‰ค ๐‘ขโˆ’ . The characteristic lines from initial data will overlap in a domain ฮฉ = {(๐‘ฅ, ๐‘ก) : ๐‘ข+ + (1/๐œŒ+ ) < ๐‘ฅ/๐‘ก < ๐‘ขโˆ’ } shown in Figure 3. So singularity must happen in ฮฉ. 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. Motivated by [7, 8, 16, 21, 22], we seek solutions with delta distribution at the jump. Denote by ๐ต๐‘€(๐‘…) the space of bounded Borel measures on ๐‘…, and then the definition of a measure solution of (7) in ๐ต๐‘€(๐‘…) can be given as follows (see [16, 22, 24]). Definition 1. A pair (๐‘ข, ๐œŒ) is called a measure solution to (7) if there hold ๐œŒ โˆˆ ๐ฟโˆž ((0, โˆž) , ๐ต๐‘€ (๐‘…)) โˆฉ ๐ถ ((0, โˆž) , ๐ปโˆ’๐‘  (๐‘…)) , ๐‘ข โˆˆ ๐ฟโˆž ((0, โˆž) , ๐ฟโˆž (๐‘…)) โˆฉ ๐ถ ((0, โˆž) , ๐ปโˆ’๐‘  (๐‘…)) ,

Let us consider the discontinuity of (7) of the form (๐‘ข, ๐œŒ) (๐‘ฅ, ๐‘ก) ๐‘ฅ < ๐‘ฅ (๐‘ก) , (๐‘ข๐‘™ , ๐œŒ๐‘™ ) (๐‘ฅ, ๐‘ก) , { { { { = {(๐‘ข๐›ฟ (๐‘ก) , ๐‘ค (๐‘ก) ๐›ฟ (๐‘ฅ โˆ’ ๐‘ฅ (๐‘ก))) , ๐‘ฅ = ๐‘ฅ (๐‘ก) , { { { ๐‘ฅ > ๐‘ฅ (๐‘ก) , {(๐‘ข๐‘Ÿ , ๐œŒ๐‘Ÿ ) (๐‘ฅ, ๐‘ก) ,

in which (๐‘ข๐‘™ , ๐œŒ๐‘™ )(๐‘ฅ, ๐‘ก) and (๐‘ข๐‘Ÿ , ๐œŒ๐‘Ÿ )(๐‘ฅ, ๐‘ก) are piecewise smooth bounded solutions of (7), for which we also set ๐‘‘๐‘ฅ(๐‘ก)/๐‘‘๐‘ก = ๐‘ข๐›ฟ (๐‘ก) for the concentration in ๐œŒ needs to travel at the speed of discontinuity (also see [16, 22, 24]). We assert that the discontinuity (29) is a measure solution to (7) if and only if the following relation

๐‘  > 0,

๐‘‘๐‘ฅ (๐‘ก) = ๐‘ข๐›ฟ (๐‘ก) , ๐‘‘๐‘ก

๐‘ข is measurable with respect to ๐œŒ

๐‘‘๐‘ค (๐‘ก) = โˆ’๐‘ข๐›ฟ (๐‘ก) [๐œŒ] + [๐œŒ๐‘ข] , ๐‘‘๐‘ก

at almost for all ๐‘ก โˆˆ (0, โˆž) , (26)

๐ผ2 = โˆฌ

๐‘…+ ร—๐‘…

๐‘…+ ร—๐‘…

(๐œ™๐‘ก + ๐‘ข๐œ™๐‘ฅ ) ๐‘‘๐œŒ ๐‘‘๐‘ก = 0,

๐‘ข (๐œ™๐‘ก + ๐‘ข๐œ™๐‘ฅ ) ๐‘‘๐œŒ ๐‘‘๐‘ก + โˆฌ

๐‘…+ ร—๐‘…

is satisfied. In fact, for any ๐œ™ โˆˆ ๐ถ0โˆž (๐‘…+ ร— ๐‘…), with Greenโ€™s formula and taking (7) in mind, one can calculate

๐‘ข๐œ™๐‘ฅ ๐‘‘๐‘ฅ ๐‘‘๐‘ก = 0 (27)

๐ผ2 = โˆฌ

for all test function ๐œ™ โˆˆ ๐ถ0โˆž (๐‘…+ ร— ๐‘…).

๐‘…+ ร—๐‘…

Definition 2. A two-dimensional weighted delta function ๐‘ค(๐‘ )๐›ฟ๐ฟ supported on a smooth curve ๐ฟ parameterized as ๐‘ก = ๐‘ก(๐‘ ), ๐‘ฅ = ๐‘ฅ(๐‘ ) (๐‘ โ‰ค ๐‘  โ‰ค ๐‘‘) is defined by ๐‘‘

โŸจ๐‘ค (๐‘ ) ๐›ฟ๐ฟ , ๐œ™ (๐‘ก, ๐‘ฅ)โŸฉ = โˆซ ๐‘ค (๐‘ ) ๐œ™ (๐‘ก (๐‘ ) , ๐‘ฅ (๐‘ )) ๐‘‘๐‘  ๐‘

for all ๐œ™ โˆˆ

๐ถ0โˆž (๐‘…2 ).

(30)

๐‘‘๐‘ค (๐‘ก) ๐‘ข๐›ฟ (๐‘ก) = โˆ’๐‘ข๐›ฟ (๐‘ก) [๐œŒ๐‘ข] + [๐œŒ๐‘ข2 + ๐‘ข] ๐‘‘๐‘ก

and (7) is satisfied in measured and distributional senses; that is, ๐ผ1 = โˆฌ

(29)

(28)

+โˆž

=โˆซ

0

โˆซ

๐‘ข (๐œ™๐‘ก + ๐‘ข๐œ™๐‘ฅ ) ๐‘‘๐œŒ ๐‘‘๐‘ก + โˆฌ

๐‘ฅ(๐‘ก)

โˆ’โˆž

+โˆซ

+โˆž

0

+โˆซ

+โˆž

0

โˆซ

๐‘ข๐œ™๐‘ฅ ๐‘‘๐‘ฅ ๐‘‘๐‘ก

(๐œŒ๐‘™ ๐‘ข๐‘™ ๐œ™๐‘ก + ๐œŒ๐‘™ ๐‘ข๐‘™2 ๐œ™๐‘ฅ ) ๐‘‘๐‘ฅ ๐‘‘๐‘ก

๐‘ฅ(๐‘ก)

โˆ’โˆž

โˆซ

๐‘…+ ร—๐‘…

+โˆž

๐‘ฅ(๐‘ก)

๐‘ข๐‘™ ๐œ™๐‘ฅ ๐‘‘๐‘ฅ ๐‘‘๐‘ก (๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ ๐œ™๐‘ก + ๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ2 ๐œ™๐‘ฅ ) ๐‘‘๐‘ฅ ๐‘‘๐‘ก

Advances in Mathematical Physics +โˆซ

+โˆž

โˆซ

0

+โˆซ

+โˆž

+โˆž

0

โˆซ

{๐‘ค (๐‘ก) ๐‘ข๐›ฟ (๐‘ก) (๐œ™๐‘ก + ๐‘ข๐›ฟ (๐‘ก) ๐œ™๐‘ฅ )} ๐‘‘๐‘ก

๐‘ฅ(๐‘ก)

โˆ’โˆž

โˆ’โˆซ

+โˆž

โˆซ

0

+โˆซ

โˆซ

+โˆž

โˆซ

+โˆž

+โˆž

0

โˆซ

+โˆž

โˆซ

0

+โˆซ

๐œ•ฮฉโˆ’

+โˆฎ โˆ’โˆซ

+โˆž

0

(โˆ’๐œŒ๐‘™ ๐‘ข๐‘™

+โˆž

=โˆซ

+โˆž

0

๐‘ข+ < ๐‘ข+ +

๐‘ฅ < ๐‘ฅ (๐‘ก) , (๐‘ขโˆ’ , ๐œŒโˆ’ ) , { { { { = {(๐‘ข๐›ฟ (๐‘ก) , ๐‘ค (๐‘ก) ๐›ฟ (๐‘ฅ โˆ’ ๐‘ฅ (๐‘ก))) , ๐‘ฅ = ๐‘ฅ (๐‘ก) , { { { ๐‘ฅ > ๐‘ฅ (๐‘ก) . {(๐‘ข+ , ๐œŒ+ ) ,

๐‘‘๐‘ฅ โˆ’ (๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ2 + ๐‘ข๐‘Ÿ )) ๐œ™ ๐‘‘๐‘ก ๐‘‘๐‘ก

=โˆซ

+โˆž

๐‘ก = 0 : ๐‘ฅ (0) = 0, ๐‘‘๐‘ค (๐‘ก) ๐‘ข๐›ฟ (๐‘ก) } ๐œ™ ๐‘‘๐‘ก, ๐‘‘๐‘ก (31)

0

(๐œ™๐‘ก + ๐‘ข๐œ™๐‘ฅ ) ๐‘‘๐œŒ ๐‘‘๐‘ก

{โˆ’๐‘ข๐›ฟ (๐‘ก) [๐œŒ] + [๐œŒ๐‘ข] โˆ’

๐‘‘๐‘ค (๐‘ก) } ๐œ™ ๐‘‘๐‘ก. ๐‘‘๐‘ก

With the above results, we can reach the assertion easily.

(35)

To determine ๐‘ฅ(๐‘ก), ๐‘ค(๐‘ก), and ๐‘ข๐›ฟ (๐‘ก) uniquely, we solve (30) with initial data

where ฮฉโˆ’ = {(๐‘ฅ, ๐‘ก) : ๐‘ก โˆˆ ๐‘…+ , โˆ’โˆž < ๐‘ฅ < ๐‘ฅ(๐‘ก)}, ฮฉ+ = {(๐‘ฅ, ๐‘ก) : ๐‘ก โˆˆ ๐‘…+ , ๐‘ฅ(๐‘ก) < ๐‘ฅ < +โˆž}, ๐œ•ฮฉยฑ is the boundaries of ฮฉยฑ , and โˆฎ๐œ•ฮฉ are the line integral on boundaries ๐œ•ฮฉยฑ . ยฑ Similarly, we have ๐‘…+ ร—๐‘…

(34๓ธ€  )

(๐‘ข, ๐œŒ) (๐‘ฅ, ๐‘ก)

๐‘‘๐‘ฅ + (๐œŒ๐‘™ ๐‘ข๐‘™2 + ๐‘ข๐‘™ ) ๐‘‘๐‘ก

{โˆ’๐‘ข๐›ฟ (๐‘ก) [๐œŒ๐‘ข] + [๐œŒ๐‘ข2 + ๐‘ข] โˆ’

๐ผ1 = โˆฌ

1 1 โ‰ค ๐‘ขโˆ’ < ๐‘ขโˆ’ + . ๐œŒ+ ๐œŒโˆ’

At this moment, the Riemann solution is a delta shock wave besides two constant states with the form

๐‘‘๐‘ค (๐‘ก) ๐‘ข๐›ฟ (๐‘ก) ๐œ™ ๐‘‘๐‘ก ๐‘‘๐‘ก

0

(34)

that is,

๐‘‘๐œ™ ๐‘‘๐‘ก ๐‘‘๐‘ก

๐‘‘๐‘ค (๐‘ก) ๐‘ข๐›ฟ (๐‘ก) ๐œ™ ๐‘‘๐‘ก ๐‘‘๐‘ก

+ ๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ โˆ’โˆซ

๐œ† 1 (๐‘ข+ , ๐œŒ+ ) < ๐œ† 2 (๐‘ข+ , ๐œŒ+ ) โ‰ค ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ ) < ๐œ† 2 (๐‘ขโˆ’ , ๐œŒโˆ’ ) ,

{(๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ ๐œ™)๐‘ก + ((๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ2 + ๐‘ข๐‘Ÿ ) ๐œ™)๐‘ฅ } ๐‘‘๐‘ฅ ๐‘‘๐‘ก

โˆ’ ๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ ๐œ™ ๐‘‘๐‘ฅ + (๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ2 + ๐‘ข๐‘Ÿ ) ๐œ™ ๐‘‘๐‘ก

๐œ•ฮฉ+

+โˆž

In what follows, the generalized Rankine-Hugoniot relation will in particular be applied to the Riemann problem (7)(8) for the case (๐‘ข+ , ๐œŒ+ ) โˆˆ V(๐‘ขโˆ’ , ๐œŒโˆ’ ), which is equivalent to the inequality

โˆ’ ๐œŒ๐‘™ ๐‘ข๐‘™ ๐œ™ ๐‘‘๐‘ฅ + (๐œŒ๐‘™ ๐‘ข๐‘™2 + ๐‘ข๐‘™ ) ๐œ™ ๐‘‘๐‘ก

0

=โˆซ

+โˆž

๐‘ค (๐‘ก) ๐‘ข๐›ฟ (๐‘ก)

0

๐‘‘๐œ™ ๐‘‘๐‘ก ๐‘‘๐‘ก

{(๐œŒ๐‘™ ๐‘ข๐‘™ ๐œ™)๐‘ก + ((๐œŒ๐‘™ ๐‘ข๐‘™2 + ๐‘ข๐‘™ ) ๐œ™)๐‘ฅ } ๐‘‘๐‘ฅ ๐‘‘๐‘ก

๐‘ฅ(๐‘ก)

+โˆž

Definition 3. A discontinuity which is presented in the form (29) and satisfies (30) and (33) will be called a delta shock wave to system (7), symbolized by ๐›ฟ.

{(๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ )๐‘ก + (๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ2 + ๐‘ข๐‘Ÿ )๐‘ฅ } ๐œ™ ๐‘‘๐‘ฅ ๐‘‘๐‘ก

๐‘ค (๐‘ก) ๐‘ข๐›ฟ (๐‘ก)

(33)

which means that all characteristics on both sides of the discontinuity are not outcoming and guarantees uniqueness of solution.

{(๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ ๐œ™)๐‘ก + ((๐œŒ๐‘Ÿ ๐‘ข๐‘Ÿ2 + ๐‘ข๐‘Ÿ ) ๐œ™)๐‘ฅ } ๐‘‘๐‘ฅ ๐‘‘๐‘ก

+โˆž

โˆ’โˆž

+โˆซ

=โˆฎ

+โˆž

๐‘ฅ(๐‘ก)

๐œ† 2 (๐œŒ๐‘Ÿ , ๐‘ข๐‘Ÿ ) โ‰ค ๐‘ข๐›ฟ (๐‘ก) โ‰ค ๐œ† 1 (๐œŒ๐‘™ , ๐‘ข๐‘™ ) ,

{(๐œŒ๐‘™ ๐‘ข๐‘™ )๐‘ก + (๐œŒ๐‘™ ๐‘ข๐‘™2 + ๐‘ข๐‘™ )๐‘ฅ } ๐œ™ ๐‘‘๐‘ฅ ๐‘‘๐‘ก

๐‘ฅ(๐‘ก)

0

=โˆซ

๐‘ฅ(๐‘ก)

๐‘ฅ(๐‘ก)

0

+โˆซ

{(๐œŒ๐‘™ ๐‘ข๐‘™ ๐œ™)๐‘ก + ((๐œŒ๐‘™ ๐‘ข๐‘™2 + ๐‘ข๐‘™ ) ๐œ™)๐‘ฅ } ๐‘‘๐‘ฅ ๐‘‘๐‘ก

โˆ’โˆž

+โˆž

0

โˆ’โˆซ

Relation (30) reflects the exact relationship among the limit states on two sides of the discontinuity, the weight, propagation speed and the location of the discontinuity. We call it the generalized Rankine-Hugoniot relation of the discontinuity. In addition, the admissibility (entropy) condition for the discontinuity (29) is

๐‘ข๐‘Ÿ ๐œ™๐‘ฅ ๐‘‘๐‘ฅ ๐‘‘๐‘ก

๐‘ฅ(๐‘ก)

0

=โˆซ

+โˆž

5

(32)

๐‘ค (0) = 0

(36)

under entropy condition (33) which is, at this moment, ๐‘ข+ +

1 โ‰ค ๐‘ข๐›ฟ (๐‘ก) โ‰ค ๐‘ขโˆ’ . ๐œŒ+

(33๓ธ€  )

From (30) and (36), it follows that ๐‘ค (๐‘ก) = โˆ’ [๐œŒ] ๐‘ฅ (๐‘ก) + [๐œŒ๐‘ข] ๐‘ก, ๐‘ค (๐‘ก) ๐‘ข๐›ฟ (๐‘ก) = โˆ’ [๐œŒ๐‘ข] ๐‘ฅ (๐‘ก) + [๐œŒ๐‘ข2 + ๐‘ข] ๐‘ก.

(37)

Multiplying the first equation in (37) by ๐‘ข๐›ฟ (๐‘ก) and then subtracting it from the second one, we obtain that [๐œŒ] ๐‘ฅ (๐‘ก) ๐‘ข๐›ฟ (๐‘ก) โˆ’ [๐œŒ๐‘ข] ๐‘ข๐›ฟ (๐‘ก) ๐‘ก โˆ’ [๐œŒ๐‘ข] ๐‘ฅ (๐‘ก) + [๐œŒ๐‘ข2 + ๐‘ข] ๐‘ก = 0, (38)

6

Advances in Mathematical Physics

or [๐œŒ๐‘ข2 + ๐‘ข] 2 ๐‘‘ [๐œŒ] 2 ( ๐‘ฅ (๐‘ก) โˆ’ [๐œŒ๐‘ข] ๐‘ฅ (๐‘ก) ๐‘ก + ๐‘ก ) = 0. ๐‘‘๐‘ก 2 2

(39)

Next, with the help of the entropy condition (33๓ธ€  ), we will choose the admissible solution to Riemann problem (7)(8) from (44) and (45). Taking into account (34๓ธ€  ), one can calculate that โˆ’ [๐œŒ] ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ ) + [๐œŒ๐‘ข] = ๐œŒ+ (๐‘ขโˆ’ โˆ’ ๐‘ข+ ) > 0,

It gives [๐œŒ] ๐‘ฅ2 (๐‘ก) โˆ’ 2 [๐œŒ๐‘ข] ๐‘ฅ (๐‘ก) ๐‘ก + [๐œŒ๐‘ข2 + ๐‘ข] ๐‘ก2 = 0.

(40)

From (40), we can find that ๐‘ฅ๓ธ€  (๐‘ก) = ๐‘ข๐›ฟ (๐‘ก) := ๐‘ข๐›ฟ is a constant and ๐‘ฅ(๐‘ก) = ๐‘ข๐›ฟ ๐‘ก. Then (40) can be rewritten into ๐ป (๐‘ข๐›ฟ ) :=

[๐œŒ] ๐‘ข๐›ฟ2

2

โˆ’ 2 [๐œŒ๐‘ข] ๐‘ข๐›ฟ + [๐œŒ๐‘ข + ๐‘ข] = 0.

๐‘ฅ (๐‘ก) =

2

= โˆ’๐œŒ+ (๐‘ขโˆ’ โˆ’ ๐‘ข+ ) (๐‘ขโˆ’ โˆ’ ๐‘ข+ โˆ’ 2

(๐‘ขโˆ’ + ๐‘ข+ + (1/๐œŒโˆ’ )) , 2

Then, for solution (44), we have ๐‘ข๐›ฟ โˆ’ ๐œ† 2 (๐‘ข+ , ๐œŒ+ )

๓ธ€ 

which satisfies the entropy condition (33 ) obviously. When ๐œŒโˆ’ =ฬธ ๐œŒ+ , (41) is just a quadratic equation with respect to ๐‘ข๐›ฟ , and the discriminant is

2

=

2

ฮ” = 4[๐œŒ๐‘ข] โˆ’ 4 [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข] 1 1 โˆ’ ๐‘ข+ โˆ’ ) > 0 ๐œŒโˆ’ ๐œŒ+

(43)

=

[๐œŒ]

=

2

(44)

[๐œŒ] [๐œŒ๐‘ข] + โˆš[๐œŒ๐‘ข] โˆ’

๐‘ก,

+ ๐‘ข]

[๐œŒ]

[๐œŒ]

โˆ’ ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ )

[๐œŒ] (๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ )) โˆ’ 2 [๐œŒ๐‘ข] ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ ) + [๐œŒ๐‘ข2 + ๐‘ข] 2

(โˆ’ [๐œŒ] ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ ) + [๐œŒ๐‘ข]) + โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข]

(47)

,

2

(45)

2

[๐œŒ๐‘ข] + โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข] [๐œŒ]

[๐œŒ๐‘ข] โˆ’ โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข]

โ‰ค 0,

๐‘ค (๐‘ก) = โˆ’โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข] ๐‘ก, ๐‘ฅ (๐‘ก) =

2

(โˆ’ [๐œŒ] ๐œ† 2 (๐‘ข+ , ๐œŒ+ ) + [๐œŒ๐‘ข]) + โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข]

2

=

2

[๐œŒ๐‘ข] โˆ’ โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข]

[๐œŒ] [๐œŒ๐‘ข2

[๐œŒ] (๐œ† 2 (๐‘ข+ , ๐œŒ+ )) โˆ’ 2 [๐œŒ๐‘ข] ๐œ† 2 (๐‘ข+ , ๐œŒ+ ) + [๐œŒ๐‘ข2 + ๐‘ข]

2

,

๐‘ค (๐‘ก) = โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข] ๐‘ก,

๐‘ข๐›ฟ =

โˆ’ ๐œ† 2 (๐‘ข+ , ๐œŒ+ )

๐‘ข๐›ฟ โˆ’ ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ )

[๐œŒ๐‘ข] โˆ’ โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข]

2

[๐œŒ]

โ‰ฅ 0,

2

๐‘ฅ (๐‘ก) =

[๐œŒ๐‘ข] โˆ’ โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข]

2

by virtue of (34), and then we can find

๐‘ข๐›ฟ =

1 1 1 ) (๐‘ขโˆ’ + โˆ’ ๐‘ข+ โˆ’ ) โ‰ฅ 0. ๐œŒ+ ๐œŒโˆ’ ๐œŒ+ (46)

(42)

๐‘ค (๐‘ก) = ๐œŒโˆ’ (๐‘ขโˆ’ โˆ’ ๐‘ข+ ) ๐‘ก,

= 4๐œŒโˆ’ ๐œŒ+ (๐‘ขโˆ’ โˆ’ ๐‘ข+ ) (๐‘ขโˆ’ +

1 ) โ‰ค 0, ๐œŒ+

[๐œŒ] (๐œ† 2 (๐‘ข+ , ๐œŒ+ )) โˆ’ 2 [๐œŒ๐‘ข] ๐œ† 2 (๐‘ข+ , ๐œŒ+ ) + [๐œŒ๐‘ข2 + ๐‘ข] = ๐œŒโˆ’ (๐‘ขโˆ’ โˆ’ ๐‘ข+ โˆ’

(๐‘ขโˆ’ + ๐‘ข+ + (1/๐œŒโˆ’ )) ๐‘ก , 2

1 1 โˆ’ ๐‘ข+ โˆ’ ) > 0, ๐œŒโˆ’ ๐œŒ+

[๐œŒ] (๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ )) โˆ’ 2 [๐œŒ๐‘ข] ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ ) + [๐œŒ๐‘ข2 + ๐‘ข]

(41)

When ๐œŒโˆ’ = ๐œŒ+ , (41) is just a linear equation of ๐‘ข๐›ฟ , and then we have ๐‘ข๐›ฟ =

โˆ’ [๐œŒ] ๐œ† 2 (๐‘ข+ , ๐œŒ+ ) + [๐œŒ๐‘ข] = ๐œŒโˆ’ (๐‘ขโˆ’ +

๐‘ก.

which imply that the entropy condition (33๓ธ€  ) is valid. One can easily observe especially that, in the two inequalities above, both of the signs โ€œ=โ€ are valid if and only if ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ ) = ๐œ† 2 (๐‘ข+ , ๐œŒ+ ). In this case, we have ๐‘ข๐›ฟ = ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ ) = ๐œ† 2 (๐‘ข+ , ๐œŒ+ ).

Advances in Mathematical Physics

7

For solution (45), when ๐œŒโˆ’ < ๐œŒ+ , ๐‘ข๐›ฟ โˆ’ ๐œ† 2 (๐‘ข+ , ๐œŒ+ ) 2

=

[๐œŒ๐‘ข] + โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข] [๐œŒ]

โˆ’ ๐œ† 2 (๐‘ข+ , ๐œŒ+ ) 2

=

โˆ’ [๐œŒ] ๐œ† 2 (๐‘ข+ , ๐œŒ+ ) + [๐œŒ๐‘ข] + โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข] [๐œŒ]

< 0, (48) and when ๐œŒโˆ’ > ๐œŒ+ ,

2

[๐œŒ๐‘ข] + โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข] [๐œŒ]

5. Two kinds of Interactions of Elementary Waves In this section, we investigate two kinds of interesting interactions of elementary waves by considering the initial-value problem (9) for system (7). The first case is that the interaction of two contact discontinuities results in a delta shock wave. The second case is that a delta shock wave vanishes after its interaction with a contact discontinuity.

๐‘ข๐›ฟ โˆ’ ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ ) =

the obtained Riemann solutions in two cases ๐‘ขโˆ’ < ๐‘ข+ +(1/๐œŒ+ ) and ๐‘ขโˆ’ โ‰ฅ ๐‘ข+ + (1/๐œŒ+ ). We take (๐‘ขโˆ’ , ๐œŒโˆ’ ) = (2, 1). For the case ๐‘ขโˆ’ < ๐‘ข+ + (1/๐œŒ+ ), we take (๐‘ข+ , ๐œŒ+ ) = (4, 2), and from (24), we can calculate (๐‘ขโˆ— , ๐œŒโˆ— ) = (2, 0.4). For the case ๐‘ขโˆ’ โ‰ฅ ๐‘ข+ + (1/๐œŒ+ ), we take (๐‘ข+ , ๐œŒ+ ) = (โˆ’1, 3, ). The numerical results are presented in Figures 4 and 5, respectively. It can be clearly observed that the contact discontinuities develop in Figure 4 while a delta shock wave is developed in Figure 5. All the numerical results are in complete agreement with the theoretical analysis.

โˆ’ ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ ) 2

=

โˆ’ [๐œŒ] ๐œ† 1 (๐‘ขโˆ’ , ๐œŒโˆ’ ) + [๐œŒ๐‘ข] + โˆš[๐œŒ๐‘ข] โˆ’ [๐œŒ] [๐œŒ๐‘ข2 + ๐‘ข] [๐œŒ]

Case (i). Assume that the initial data in (9) satisfy ๐‘ขโˆ’ +

> 0. (49) These show that the solution (45) does not satisfy the entropy condition (33๓ธ€  ). Thus we have proved the following result. Theorem 4. Let (๐‘ข+ , ๐œŒ+ ) โˆˆ ๐‘‰ (๐‘ขโˆ’ , ๐œŒโˆ’ ). Then Riemann problem (8) for (7) admits one and only one entropy solution in the sense of measures of the form (๐‘ข, ๐œŒ) (๐‘ฅ, ๐‘ก) (๐‘ข , ๐œŒ ) , { { โˆ’ โˆ’ = {(๐‘ข๐›ฟ , ๐‘ค (๐‘ก) ๐›ฟ (๐‘ฅ โˆ’ ๐‘ข๐›ฟ ๐‘ก)) , { {(๐‘ข+ , ๐œŒ+ ) ,

๐‘ฅ < ๐‘ข๐›ฟ ๐‘ก, ๐‘ฅ = ๐‘ข๐›ฟ ๐‘ก, ๐‘ฅ > ๐‘ข๐›ฟ ๐‘ก,

(50)

where ๐‘ข๐›ฟ and ๐‘ค(๐‘ก) are shown in (42) for ๐œŒโˆ’ = ๐œŒ+ or (44) for ๐œŒโˆ’ =ฬธ ๐œŒ+ . At last, associating with the results in Section 2, we can conclude the following. Theorem 5. For Riemann problem (7)-(8), there exists a unique entropy solution, which contains two different (or just one) contact discontinuities when ๐‘ขโˆ’ < ๐‘ข+ + (1/๐œŒ+ ) and a delta shock wave when ๐‘ขโˆ’ โ‰ฅ ๐‘ข+ + (1/๐œŒ+ ).

4. Numerical Simulations for Riemann Solutions In this section, by employing the Nessyahu-Tadmor scheme [25] with 300 ร— 300 cells and ๐ถ๐น๐ฟ = 0.475, we simulate

1 1 = ๐‘ข๐‘š + , ๐œŒโˆ’ ๐œŒ๐‘š

๐‘ข๐‘š = ๐‘ข+ .

(51)

At this moment, a contact discontinuity ๐ฝ โƒ— emits from (0, 0) โ† ๓ณจ€ with a speed ๐‘ข๐‘š + (1/๐œŒ๐‘š ) and a contact discontinuity ๐ฝ emits from (๐‘Ž, 0) with a speed ๐‘ข๐‘š , respectively. From ๐‘ข๐‘š + (1/๐œŒ๐‘š ) > โ† ๓ณจ€ ๐‘ข๐‘š , it is known that ๐ฝ โƒ— will overtake ๐ฝ at a point (๐‘ฅ0 , ๐‘ก0 ), which can be calculated to be (๐‘ฅ0 , ๐‘ก0 ) = (๐‘Ž(๐œŒ๐‘š ๐‘ข๐‘š + 1), ๐‘Ž๐œŒ๐‘š ). At the moment ๐‘ก = ๐‘ก0 , a new Riemann problem is formed with (๐‘ขโˆ’ , ๐œŒโˆ’ ) and (๐‘ข+ , ๐œŒ+ ) on both sides. We consider the following situation for ๐œŒ: 1 1 1 โˆ’ โ‰ฅ , ๐œŒ๐‘š ๐œŒโˆ’ ๐œŒ+

(52)

which means that from (51) ๐‘ขโˆ’ โ‰ฅ ๐‘ข+ +

1 . ๐œŒ+

(53)

Then from Theorem 5, it can be concluded that a delta shock wave ๐›ฟ emits from the moment ๐‘ก = ๐‘ก0 , whose speed, location, and weight can be obtained by solving (30) under (33) with initial data ๐‘ก = ๐‘ก0 : ๐‘ฅ (๐‘ก0 ) = ๐‘ฅ0 ,

๐‘ค (๐‘ก0 ) = 0.

(54)

See Figure 6(a), in which m = (๐‘ข๐‘– , ๐œŒ๐‘– ) (๐‘– = โˆ’, ๐‘š, +). Case (ii). Suppose the initial data (9) in satisfy ๐‘ขโˆ’ โ‰ฅ ๐‘ข๐‘š +

1 , ๐œŒ๐‘š

๐‘ข๐‘š = ๐‘ข+ .

(55)

8

Advances in Mathematical Physics 2.5

4.5 4

2

3.5 1.5 ๐œŒ

๐‘ข

3

1 2.5 0.5

0 โˆ’1.5

2

โˆ’1

0

โˆ’0.5

0.5

1

1.5

1.5 โˆ’1.5

โˆ’1

0

โˆ’0.5

๐‘ฅ

0.5

1

1.5

0.5

1

1.5

๐‘ฅ

(a)

(b)

Figure 4: Numerical results for the case ๐‘ขโˆ’ < ๐‘ข+ + (1/๐œŒ+ ) at ๐‘ก = 0.2.

140

2.5

120

2

100

1.5

80

1

๐œŒ 60

๐‘ข 0.5

40

0

20

โˆ’0.5

0

โˆ’1

โˆ’20 โˆ’1.5

โˆ’1

0

โˆ’0.5

0.5

1

1.5

โˆ’1.5 โˆ’1.5

โˆ’1

0

โˆ’0.5

๐‘ฅ

๐‘ฅ

(a)

(b)

Figure 5: Numerical results for the case ๐‘ขโˆ’ โ‰ฅ ๐‘ข+ + (1/๐œŒ+ ) at ๐‘ก = 0.5.

At this time, a delta shock wave ๐›ฟ emits from (0, 0) with the speed ๐œŽ satisfying ๐‘ขโˆ’ โ‰ฅ ๐œŽ โ‰ฅ ๐‘ข๐‘š + (1/๐œŒ๐‘š ) and a contact โ† ๓ณจ€ discontinuity ๐ฝ emits from (๐‘Ž, 0) with the speed ๐‘ข๐‘š . Since โ† ๓ณจ€ ๐œŽ โ‰ฅ ๐‘ข๐‘š + (1/๐œŒ๐‘š ) > ๐‘ข๐‘š , ๐›ฟ will interact with ๐ฝ at a point (๐‘ฅ0 , ๐‘ก0 ) = (๐‘Ž๐œŽ/(๐œŽ โˆ’ ๐‘ข+ ), ๐‘Ž/(๐œŽ โˆ’ ๐‘ข+ )). As before, at the moment ๐‘ก = ๐‘ก0 , a new Riemann problem is formed with (๐‘ขโˆ’ , ๐œŒโˆ’ ) and (๐‘ข+ , ๐œŒ+ ) on both sides. Let us suppose that the initial data furthermore satisfy 1 ๐‘ขโˆ’ < ๐‘ข+ + , ๐œŒ+

(56)

then from Theorem 5, we can conclude that two contact discontinuities emit form ๐‘ก = ๐‘ก0 as follows: โ† ๓ณจ€ ๐‘‘๐‘ฅ ๐ฝ : = ๐‘ขโˆ’ , ๐‘‘๐‘ก

๐ฝโƒ— :

๐‘‘๐‘ฅ 1 = ๐‘ข+ + , ๐‘‘๐‘ก ๐œŒ+

(57)

and the intermediate states (๐‘ขโˆ— , ๐œŒโˆ— ) connecting two contact discontinuities satisfy ๐‘ขโˆ— = ๐‘ขโˆ’ ,

๐‘ขโˆ— +

1 1 = ๐‘ข+ + . ๐œŒโˆ— ๐œŒ+

See Figure 6(b), in which โŠ› = (๐‘ขโˆ— , ๐œŒโˆ— ).

(58)

Advances in Mathematical Physics

9 โ† ๐ฝ

๐›ฟ ๐‘ก

๐‘ก

โŠ›

โ†’ ๐ฝ

โŠ•

โŠ–

๐‘ก0

๐‘ก0 โ† ๐ฝ

โ†’ ๐ฝ

โŠ–

๐‘š 0

โ† ๐ฝ

๐›ฟ

๐‘Ž

๐‘ฅ

(a)

โŠ•

๐‘š 0

๐‘ฅ

๐‘Ž

(b)

Figure 6: Two kinds of interactions of elementary waves.

Acknowledgments This paper is partly supported by the National Natural Science Foundation of China (11226191) and the Scientific Research Foundation of Yunnan University (2011YB30). The author would like to thank Professor Hanchun Yang, his advisor, for some valuable and inspiring discussions and suggestions.

References [1] B. L. Keyfitz and H. C. Kranzer, โ€œA system of nonstrictly hyperbolic conservation laws arising in elasticity theory,โ€ Archive for Rational Mechanics and Analysis, vol. 72, no. 3, pp. 219โ€“241, 1980. [2] Y. Lu, โ€œExistence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type,โ€ Journal of Functional Analysis, vol. 261, no. 10, pp. 2797โ€“2815, 2011. [3] A. Aw and M. Rascle, โ€œResurrection of โ€œsecond orderโ€ models of traffic flow,โ€ SIAM Journal on Applied Mathematics, vol. 60, no. 3, pp. 916โ€“938, 2000. [4] F. Bouchut, โ€œOn zero pressure gas dynamics,โ€ in Advances in Kinetic Theory and Computing: Selected Papers, B. Perthame, Ed., vol. 22, pp. 171โ€“190, World Scientific, Singapore, 1994. [5] G.-Q. Chen and H. Liu, โ€œFormation of ๐›ฟ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids,โ€ SIAM Journal on Mathematical Analysis, vol. 34, no. 4, pp. 925โ€“938, 2003. [6] G.-Q. Chen and H. Liu, โ€œConcentration and cavitation in the vanishing pressure limit of solutions to the Euler equations for nonisentropic fluids,โ€ Physica D, vol. 189, no. 1-2, pp. 141โ€“165, 2004. [7] H. Cheng, H. Yang, and Y. Zhang, โ€œRiemann problem for the Chaplygin Euler equations of compressible fluid flow,โ€ International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 11, pp. 985โ€“992, 2010. [8] H. Cheng and H. Yang, โ€œRiemann problem for the relativistic Chaplygin Euler equations,โ€ Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 17โ€“26, 2011. [9] H. Cheng and H. Yang, โ€œDelta shock waves in chromatography equations,โ€ Journal of Mathematical Analysis and Applications, vol. 380, no. 2, pp. 475โ€“485, 2011. [10] H. Cheng and H. Yang, โ€œDelta shock waves as limits of vanishing viscosity for 2-D steady pressureless isentropic flow,โ€ Acta Applicandae Mathematicae, vol. 113, no. 3, pp. 323โ€“348, 2011.

[11] V. G. Danilov and D. Mitrovic, โ€œDelta shock wave formation in the case of triangular hyperbolic system of conservation laws,โ€ Journal of Differential Equations, vol. 245, no. 12, pp. 3704โ€“3734, 2008. [12] V. G. Danilov and V. M. Shelkovich, โ€œDynamics of propagation and interaction of ๐›ฟ-shock waves in conservation law systems,โ€ Journal of Differential Equations, vol. 211, no. 2, pp. 333โ€“381, 2005. [13] L. Guo, W. Sheng, and T. Zhang, โ€œThe two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system,โ€ Communications on Pure and Applied Analysis, vol. 9, no. 2, pp. 431โ€“458, 2010. [14] K. T. Joseph, โ€œA Riemann problem whose viscosity solutions contain ๐›ฟ-measures,โ€ Asymptotic Analysis, vol. 7, no. 2, pp. 105โ€“ 120, 1993. [15] B. L. Keyfitz and H. C. Kranzer, โ€œSpaces of weighted measures for conservation laws with singular shock solutions,โ€ Journal of Differential Equations, vol. 118, no. 2, pp. 420โ€“451, 1995. [16] J. Li and T. Zhang, โ€œGeneralized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations,โ€ in NonlInear PDE and Related Areas, G. Q. Chen, Ed., pp. 219โ€“232, World Scientific, Singapore, 1998. ๓ธ€ 

[17] E. Yu. Panov and V. M. Shelkovich, โ€œ๐›ฟ -shock waves as a new type of solutions to systems of conservation laws,โ€ Journal of Differential Equations, vol. 228, no. 1, pp. 49โ€“86, 2006. [18] V. M. Shelkovich, โ€œThe Riemann problem admitting ๐›ฟ-, ๐›ฟ๓ธ€  shocks, and vacuum states (the vanishing viscosity approach),โ€ Journal of Differential Equations, vol. 231, no. 2, pp. 459โ€“500, 2006. [19] W. Sheng and T. Zhang, โ€œThe Riemann problem for the transportation equations in gas dynamics,โ€ Memoirs of the American Mathematical Society, vol. 137, no. 654, 1999. [20] D. C. Tan and T. Zhang, โ€œTwo-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws. I. Four๐ฝ cases,โ€ Journal of Differential Equations, vol. 111, no. 2, pp. 203โ€“ 254, 1994. [21] D. Tan, T. Zhang, and Y. Zheng, โ€œDelta shock waves as limits of vanishing viscosity for hyperbolic systems of conversation laws,โ€ Journal of Differential Equations, vol. 112, no. 1, pp. 1โ€“32, 2004. [22] H. Yang, โ€œRiemann problems for a class of coupled hyperbolic systems of conservation laws,โ€ Journal of Differential Equations, vol. 159, no. 2, pp. 447โ€“484, 1999.

10 [23] H. Yang and Y. Zhang, โ€œNew developments of delta shock waves and its applications in systems of conservation laws,โ€ Journal of Differential Equations, vol. 252, no. 11, pp. 5951โ€“5993, 2012. [24] Y. Zheng, โ€œSystems of conservation laws with incomplete sets of eigenvectors everywhere,โ€ in Advances in Nonlinear Partial Differential Equations and Related Areas, G. Q. Chen, Ed., pp. 399โ€“426, Word Scientific, Singapore, 1998. [25] H. Nessyahu and E. Tadmor, โ€œNon-oscillatory central differencing for hyperbolic conservation laws,โ€ Journal of Computational Physics, vol. 87, no. 2, pp. 408โ€“463, 1990.

Advances in Mathematical Physics

Advances in

Operations Research Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Advances in

Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Applied Mathematics

Algebra

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Probability and Statistics Volume 2014

The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Differential Equations Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Submit your manuscripts at http://www.hindawi.com International Journal of

Advances in

Combinatorics Hindawi Publishing Corporation http://www.hindawi.com

Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Journal of

Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of Mathematics and Mathematical Sciences

Mathematical Problems in Engineering

Journal of

Mathematics Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Discrete Mathematics

Journal of

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Discrete Dynamics in Nature and Society

Journal of

Function Spaces Hindawi Publishing Corporation http://www.hindawi.com

Abstract and Applied Analysis

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

International Journal of

Journal of

Stochastic Analysis

Optimization

Hindawi Publishing Corporation http://www.hindawi.com

Hindawi Publishing Corporation http://www.hindawi.com

Volume 2014

Volume 2014