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: ๐๐ก + (๐ (๐ข โ ๐))๐ฅ = 0,
(1)
(๐๐ข)๐ก + (๐๐ข (๐ข โ ๐))๐ฅ = 0,
๐๐ก + (๐๐ (๐, ๐ข1 , ๐ข2 , . . . , ๐ข๐ ))๐ฅ = 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 and ๐ข are the density and the velocity of cars on the roadway, and the function ๐ is smooth and strictly increasing and it satisfies ๐๐๓ธ ๓ธ (๐) + 2๐๓ธ (๐) > 0
๐ (0) = 0,
lim ๐๐๓ธ (๐) = 0,
๐โ0
๐๐๓ธ ๓ธ (๐) + 2๐๓ธ (๐) > 0 for ๐ > 0, lim ๐๐ (๐) = 0,
where ๐ = ๐(๐) with ๐ โฅ 0. Model (1) belongs to the nonsymmetric system of Keyfitz-Kranzer type [1, 2] as
(๐๐ข๐ )๐ก + (๐๐ข๐ ๐ (๐, ๐ข1 , ๐ข2 , . . . , ๐ข๐ ))๐ฅ = 0,
global bounded weak solutions of the Cauchy problem under the following two assumptions on ๐(๐), respectively:
for ๐ > 0.
(3)
Model (1) was also studied by Lu [2]. By using the compensated compactness method, he established the existence of
๐โ0
๓ธ
lim (๐๐ (๐)) โฅ ๐ด,
(4)
๐โโ
๐๐๓ธ ๓ธ (๐) + 2๐๓ธ (๐) > 0 for ๐ > 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). 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 = ๐ (๐) = โ , ๐
(5)
which is the prototype function satisfying ๐๐๓ธ ๓ธ (๐) + 2๐๓ธ (๐) โก 0 for ๐ > 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 [4โ24] and the references therein. Putting (5) into (1), we obtain the equivalent system ๐๐ก + (๐๐ข)๐ฅ = 0, (๐๐ข)๐ก + (๐๐ข2 + ๐ข)๐ฅ = 0.
๐ 1 = ๐ข,
๐2 = ๐ข +
1 ๐
(10)
1 ๐ ) ๐2
(11)
with associated right eigenvectors ๐1โ = (1, 0)๐ ,
๐2โ = (1,
satisfying
๐ฅ < 0, 0 < ๐ฅ < ๐,
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 ๐ข+ +(1/๐+ ) > ๐ขโ , we can construct Riemann solution only involving two different (or just one) contact discontinuities. However, for the case ๐ข+ + (1/๐+ ) โค ๐ขโ , 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 ๐ข 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 (๐ขโ , ๐โ ) , { { { { (๐ข, ๐) (๐ฅ, ๐ก = 0) = {(๐ข๐ , ๐๐ ) , { { { {(๐ข+ , ๐+ ) ,
2. Preliminaries and Solutions Involving Contact Discontinuities
(7)
Firstly, we consider the Riemann problem for (7) with initial data {(๐ขโ , ๐โ ) , ๐ฅ < 0, (๐ข, ๐) (๐ฅ, ๐ก = 0) = { (๐ข , ๐ ) , ๐ฅ > 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.
โ๐ ๐ โ
๐๐โ โก 0,
where ๐ > 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 ๐ฅ (๐ข, ๐) (๐ก, ๐ฅ) = (๐ข, ๐) (๐) , ๐ = , (13) ๐ก for which system (7) becomes โ ๐๐๐ + (๐๐ข)๐ = 0, โ ๐(๐๐ข)๐ + (๐๐ข2 + ๐ข)๐ = 0,
(14)
and initial data (8) changes to the boundary condition (๐ข, ๐) (ยฑโ) = (๐ขยฑ , ๐ยฑ ) .
(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)
๐ฅ > ๐,
๐ = 1, 2.
๐ขโ๐ (
0
๐
๐๐ ) ( ) = 0. 1 ๐๐ข ๐ข+ โ๐ ๐
(16)
It provides either the general solutions (constant states) (๐ข, ๐) = const.
(๐ > 0)
(17)
Advances in Mathematical Physics
3 โ โ ๐ฝ ๐ฝ
๐
or singular solutions
โ ๐ฝ
III
๐ = ๐ 1 = ๐ข, 1 ๐ = ๐1 = ๐ข + , ๐
๐๐ข = 0, 1 ๐ (๐ข + ) = 0. ๐
(18)
V (๐ขโ โ
1 ,๐ ) ๐โ โ
IV II
Integrating (18) from (๐ขโ , ๐โ ) to (๐ข, ๐), respectively, one can get that
1 ๐ขโ โ ๐โ
๐ขโ
๐ขโ +
๐ข = ๐ขโ , 1 ๐=๐ข+ , ๐
(19)
๐ข
Then the phase plane is divided into five regions as follows (shown in Figure 1): I = { (๐ข, ๐) | ๐ขโ < ๐ข < ๐ขโ +
1 1 ๐ข + = ๐ขโ + . ๐ ๐โ
0 0, which is monotonically increasing and has the lines ๐ = 0 and ๐ข = ๐ขโ + (1/๐โ ) as its two asymptotic lines. Also, from the point (๐ขโ โ (1/๐โ ), ๐โ ), we draw the curve ๐ข + (1/๐) = ๐ขโ .
1 }, ๐ขโ + ๐โโ1 โ ๐ข
1 < ๐ < +โ} . ๐ขโ โ ๐ข
(23)
By the analysis method in the phase plane, when (๐ข+ , ๐+ ) โ I, II, III, IV, namely, ๐ข+ + (1/๐+ ) > ๐ขโ , one can easily construct the solution consisting of two different (or just one) contact discontinuities (see Figure 2), in which, โ ๓ณจ ๐ฝ : ๐ฅ = ๐ขโ ๐ก,
๐ฝ โ : ๐ฅ = (๐ข+ +
1 ) ๐ก, ๐+
(24)
the intermediate states (๐ขโ , ๐โ ) connecting two contact discontinuities satisfy ๐ขโ = ๐ขโ ,
๐ขโ +
1 1 = ๐ข+ + . ๐โ ๐+
(25)
4
Advances in Mathematical Physics ๐ก ๐ฅ/๐ก = ๐ข+ + โ ๐ฝ
โ ๐ฝ
๐ก
1 ๐+
๐ฅ/๐ก = ๐ขโ
๐ฅ/๐ก = ๐ข+
(๐ขโ , ๐โ ) (๐ขโ , ๐โ )
(๐ข+ , ๐+ )
ฮฉ
๐ฅ/๐ก = ๐ขโ +
(๐ขโ , ๐โ )
๐ฅ
๐
๐ฟ : ๐ฅ/๐ก = ๐ข๐ฟ
(๐ข+ , ๐+ ) ๐
Figure 2: Riemann solution involving two contact discontinuities.
1 ๐โ
๐ฅ
Figure 3: Riemann solution containing delta shock wave.
3. Delta Shock Solution In this section, let us in detail discuss the last case when (๐ข+ , ๐+ ) โ V (๐ขโ , ๐โ ); that is, ๐ข+ + (1/๐+ ) โค ๐ขโ . The characteristic lines from initial data will overlap in a domain ฮฉ = {(๐ฅ, ๐ก) : ๐ข+ + (1/๐+ ) < ๐ฅ/๐ก < ๐ขโ } 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 ๐ต๐(๐
) the space of bounded Borel measures on ๐
, and then the definition of a measure solution of (7) in ๐ต๐(๐
) can be given as follows (see [16, 22, 24]). Definition 1. A pair (๐ข, ๐) is called a measure solution to (7) if there hold ๐ โ ๐ฟโ ((0, โ) , ๐ต๐ (๐
)) โฉ ๐ถ ((0, โ) , ๐ปโ๐ (๐
)) , ๐ข โ ๐ฟโ ((0, โ) , ๐ฟโ (๐
)) โฉ ๐ถ ((0, โ) , ๐ปโ๐ (๐
)) ,
Let us consider the discontinuity of (7) of the form (๐ข, ๐) (๐ฅ, ๐ก) ๐ฅ < ๐ฅ (๐ก) , (๐ข๐ , ๐๐ ) (๐ฅ, ๐ก) , { { { { = {(๐ข๐ฟ (๐ก) , ๐ค (๐ก) ๐ฟ (๐ฅ โ ๐ฅ (๐ก))) , ๐ฅ = ๐ฅ (๐ก) , { { { ๐ฅ > ๐ฅ (๐ก) , {(๐ข๐ , ๐๐ ) (๐ฅ, ๐ก) ,
in which (๐ข๐ , ๐๐ )(๐ฅ, ๐ก) and (๐ข๐ , ๐๐ )(๐ฅ, ๐ก) are piecewise smooth bounded solutions of (7), for which we also set ๐๐ฅ(๐ก)/๐๐ก = ๐ข๐ฟ (๐ก) for the concentration in ๐ 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,
๐๐ฅ (๐ก) = ๐ข๐ฟ (๐ก) , ๐๐ก
๐ข is measurable with respect to ๐
๐๐ค (๐ก) = โ๐ข๐ฟ (๐ก) [๐] + [๐๐ข] , ๐๐ก
at almost for all ๐ก โ (0, โ) , (26)
๐ผ2 = โฌ
๐
+ ร๐
๐
+ ร๐
(๐๐ก + ๐ข๐๐ฅ ) ๐๐ ๐๐ก = 0,
๐ข (๐๐ก + ๐ข๐๐ฅ ) ๐๐ ๐๐ก + โฌ
๐
+ ร๐
is satisfied. In fact, for any ๐ โ ๐ถ0โ (๐
+ ร ๐
), with Greenโs formula and taking (7) in mind, one can calculate
๐ข๐๐ฅ ๐๐ฅ ๐๐ก = 0 (27)
๐ผ2 = โฌ
for all test function ๐ โ ๐ถ0โ (๐
+ ร ๐
).
๐
+ ร๐
Definition 2. A two-dimensional weighted delta function ๐ค(๐ )๐ฟ๐ฟ supported on a smooth curve ๐ฟ parameterized as ๐ก = ๐ก(๐ ), ๐ฅ = ๐ฅ(๐ ) (๐ โค ๐ โค ๐) is defined by ๐
โจ๐ค (๐ ) ๐ฟ๐ฟ , ๐ (๐ก, ๐ฅ)โฉ = โซ ๐ค (๐ ) ๐ (๐ก (๐ ) , ๐ฅ (๐ )) ๐๐ ๐
for all ๐ โ
๐ถ0โ (๐
2 ).
(30)
๐๐ค (๐ก) ๐ข๐ฟ (๐ก) = โ๐ข๐ฟ (๐ก) [๐๐ข] + [๐๐ข2 + ๐ข] ๐๐ก
and (7) is satisfied in measured and distributional senses; that is, ๐ผ1 = โฌ
(29)
(28)
+โ
=โซ
0
โซ
๐ข (๐๐ก + ๐ข๐๐ฅ ) ๐๐ ๐๐ก + โฌ
๐ฅ(๐ก)
โโ
+โซ
+โ
0
+โซ
+โ
0
โซ
๐ข๐๐ฅ ๐๐ฅ ๐๐ก
(๐๐ ๐ข๐ ๐๐ก + ๐๐ ๐ข๐2 ๐๐ฅ ) ๐๐ฅ ๐๐ก
๐ฅ(๐ก)
โโ
โซ
๐
+ ร๐
+โ
๐ฅ(๐ก)
๐ข๐ ๐๐ฅ ๐๐ฅ ๐๐ก (๐๐ ๐ข๐ ๐๐ก + ๐๐ ๐ข๐2 ๐๐ฅ ) ๐๐ฅ ๐๐ก
Advances in Mathematical Physics +โซ
+โ
โซ
0
+โซ
+โ
+โ
0
โซ
{๐ค (๐ก) ๐ข๐ฟ (๐ก) (๐๐ก + ๐ข๐ฟ (๐ก) ๐๐ฅ )} ๐๐ก
๐ฅ(๐ก)
โโ
โโซ
+โ
โซ
0
+โซ
โซ
+โ
โซ
+โ
+โ
0
โซ
+โ
โซ
0
+โซ
๐ฮฉโ
+โฎ โโซ
+โ
0
(โ๐๐ ๐ข๐
+โ
=โซ
+โ
0
๐ข+ < ๐ข+ +
๐ฅ < ๐ฅ (๐ก) , (๐ขโ , ๐โ ) , { { { { = {(๐ข๐ฟ (๐ก) , ๐ค (๐ก) ๐ฟ (๐ฅ โ ๐ฅ (๐ก))) , ๐ฅ = ๐ฅ (๐ก) , { { { ๐ฅ > ๐ฅ (๐ก) . {(๐ข+ , ๐+ ) ,
๐๐ฅ โ (๐๐ ๐ข๐2 + ๐ข๐ )) ๐ ๐๐ก ๐๐ก
=โซ
+โ
๐ก = 0 : ๐ฅ (0) = 0, ๐๐ค (๐ก) ๐ข๐ฟ (๐ก) } ๐ ๐๐ก, ๐๐ก (31)
0
(๐๐ก + ๐ข๐๐ฅ ) ๐๐ ๐๐ก
{โ๐ข๐ฟ (๐ก) [๐] + [๐๐ข] โ
๐๐ค (๐ก) } ๐ ๐๐ก. ๐๐ก
With the above results, we can reach the assertion easily.
(35)
To determine ๐ฅ(๐ก), ๐ค(๐ก), and ๐ข๐ฟ (๐ก) uniquely, we solve (30) with initial data
where ฮฉโ = {(๐ฅ, ๐ก) : ๐ก โ ๐
+ , โโ < ๐ฅ < ๐ฅ(๐ก)}, ฮฉ+ = {(๐ฅ, ๐ก) : ๐ก โ ๐
+ , ๐ฅ(๐ก) < ๐ฅ < +โ}, ๐ฮฉยฑ is the boundaries of ฮฉยฑ , and โฎ๐ฮฉ are the line integral on boundaries ๐ฮฉยฑ . ยฑ Similarly, we have ๐
+ ร๐
(34๓ธ )
(๐ข, ๐) (๐ฅ, ๐ก)
๐๐ฅ + (๐๐ ๐ข๐2 + ๐ข๐ ) ๐๐ก
{โ๐ข๐ฟ (๐ก) [๐๐ข] + [๐๐ข2 + ๐ข] โ
๐ผ1 = โฌ
1 1 โค ๐ขโ < ๐ขโ + . ๐+ ๐โ
At this moment, the Riemann solution is a delta shock wave besides two constant states with the form
๐๐ค (๐ก) ๐ข๐ฟ (๐ก) ๐ ๐๐ก ๐๐ก
0
(34)
that is,
๐๐ ๐๐ก ๐๐ก
๐๐ค (๐ก) ๐ข๐ฟ (๐ก) ๐ ๐๐ก ๐๐ก
+ ๐๐ ๐ข๐ โโซ
๐ 1 (๐ข+ , ๐+ ) < ๐ 2 (๐ข+ , ๐+ ) โค ๐ 1 (๐ขโ , ๐โ ) < ๐ 2 (๐ขโ , ๐โ ) ,
{(๐๐ ๐ข๐ ๐)๐ก + ((๐๐ ๐ข๐2 + ๐ข๐ ) ๐)๐ฅ } ๐๐ฅ ๐๐ก
โ ๐๐ ๐ข๐ ๐ ๐๐ฅ + (๐๐ ๐ข๐2 + ๐ข๐ ) ๐ ๐๐ก
๐ฮฉ+
+โ
In what follows, the generalized Rankine-Hugoniot relation will in particular be applied to the Riemann problem (7)(8) for the case (๐ข+ , ๐+ ) โ V(๐ขโ , ๐โ ), which is equivalent to the inequality
โ ๐๐ ๐ข๐ ๐ ๐๐ฅ + (๐๐ ๐ข๐2 + ๐ข๐ ) ๐ ๐๐ก
0
=โซ
+โ
๐ค (๐ก) ๐ข๐ฟ (๐ก)
0
๐๐ ๐๐ก ๐๐ก
{(๐๐ ๐ข๐ ๐)๐ก + ((๐๐ ๐ข๐2 + ๐ข๐ ) ๐)๐ฅ } ๐๐ฅ ๐๐ก
๐ฅ(๐ก)
+โ
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 ๐ฟ.
{(๐๐ ๐ข๐ )๐ก + (๐๐ ๐ข๐2 + ๐ข๐ )๐ฅ } ๐ ๐๐ฅ ๐๐ก
๐ค (๐ก) ๐ข๐ฟ (๐ก)
(33)
which means that all characteristics on both sides of the discontinuity are not outcoming and guarantees uniqueness of solution.
{(๐๐ ๐ข๐ ๐)๐ก + ((๐๐ ๐ข๐2 + ๐ข๐ ) ๐)๐ฅ } ๐๐ฅ ๐๐ก
+โ
โโ
+โซ
=โฎ
+โ
๐ฅ(๐ก)
๐ 2 (๐๐ , ๐ข๐ ) โค ๐ข๐ฟ (๐ก) โค ๐ 1 (๐๐ , ๐ข๐ ) ,
{(๐๐ ๐ข๐ )๐ก + (๐๐ ๐ข๐2 + ๐ข๐ )๐ฅ } ๐ ๐๐ฅ ๐๐ก
๐ฅ(๐ก)
0
=โซ
๐ฅ(๐ก)
๐ฅ(๐ก)
0
+โซ
{(๐๐ ๐ข๐ ๐)๐ก + ((๐๐ ๐ข๐2 + ๐ข๐ ) ๐)๐ฅ } ๐๐ฅ ๐๐ก
โโ
+โ
0
โโซ
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
๐ข๐ ๐๐ฅ ๐๐ฅ ๐๐ก
๐ฅ(๐ก)
0
=โซ
+โ
5
(32)
๐ค (0) = 0
(36)
under entropy condition (33) which is, at this moment, ๐ข+ +
1 โค ๐ข๐ฟ (๐ก) โค ๐ขโ . ๐+
(33๓ธ )
From (30) and (36), it follows that ๐ค (๐ก) = โ [๐] ๐ฅ (๐ก) + [๐๐ข] ๐ก, ๐ค (๐ก) ๐ข๐ฟ (๐ก) = โ [๐๐ข] ๐ฅ (๐ก) + [๐๐ข2 + ๐ข] ๐ก.
(37)
Multiplying the first equation in (37) by ๐ข๐ฟ (๐ก) and then subtracting it from the second one, we obtain that [๐] ๐ฅ (๐ก) ๐ข๐ฟ (๐ก) โ [๐๐ข] ๐ข๐ฟ (๐ก) ๐ก โ [๐๐ข] ๐ฅ (๐ก) + [๐๐ข2 + ๐ข] ๐ก = 0, (38)
6
Advances in Mathematical Physics
or [๐๐ข2 + ๐ข] 2 ๐ [๐] 2 ( ๐ฅ (๐ก) โ [๐๐ข] ๐ฅ (๐ก) ๐ก + ๐ก ) = 0. ๐๐ก 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 โ [๐] ๐ 1 (๐ขโ , ๐โ ) + [๐๐ข] = ๐+ (๐ขโ โ ๐ข+ ) > 0,
It gives [๐] ๐ฅ2 (๐ก) โ 2 [๐๐ข] ๐ฅ (๐ก) ๐ก + [๐๐ข2 + ๐ข] ๐ก2 = 0.
(40)
From (40), we can find that ๐ฅ๓ธ (๐ก) = ๐ข๐ฟ (๐ก) := ๐ข๐ฟ is a constant and ๐ฅ(๐ก) = ๐ข๐ฟ ๐ก. Then (40) can be rewritten into ๐ป (๐ข๐ฟ ) :=
[๐] ๐ข๐ฟ2
2
โ 2 [๐๐ข] ๐ข๐ฟ + [๐๐ข + ๐ข] = 0.
๐ฅ (๐ก) =
2
= โ๐+ (๐ขโ โ ๐ข+ ) (๐ขโ โ ๐ข+ โ 2
(๐ขโ + ๐ข+ + (1/๐โ )) , 2
Then, for solution (44), we have ๐ข๐ฟ โ ๐ 2 (๐ข+ , ๐+ )
๓ธ
which satisfies the entropy condition (33 ) obviously. When ๐โ =ฬธ ๐+ , (41) is just a quadratic equation with respect to ๐ข๐ฟ , and the discriminant is
2
=
2
ฮ = 4[๐๐ข] โ 4 [๐] [๐๐ข2 + ๐ข] 1 1 โ ๐ข+ โ ) > 0 ๐โ ๐+
(43)
=
[๐]
=
2
(44)
[๐] [๐๐ข] + โ[๐๐ข] โ
๐ก,
+ ๐ข]
[๐]
[๐]
โ ๐ 1 (๐ขโ , ๐โ )
[๐] (๐ 1 (๐ขโ , ๐โ )) โ 2 [๐๐ข] ๐ 1 (๐ขโ , ๐โ ) + [๐๐ข2 + ๐ข] 2
(โ [๐] ๐ 1 (๐ขโ , ๐โ ) + [๐๐ข]) + โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข]
(47)
,
2
(45)
2
[๐๐ข] + โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข] [๐]
[๐๐ข] โ โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข]
โค 0,
๐ค (๐ก) = โโ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข] ๐ก, ๐ฅ (๐ก) =
2
(โ [๐] ๐ 2 (๐ข+ , ๐+ ) + [๐๐ข]) + โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข]
2
=
2
[๐๐ข] โ โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข]
[๐] [๐๐ข2
[๐] (๐ 2 (๐ข+ , ๐+ )) โ 2 [๐๐ข] ๐ 2 (๐ข+ , ๐+ ) + [๐๐ข2 + ๐ข]
2
,
๐ค (๐ก) = โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข] ๐ก,
๐ข๐ฟ =
โ ๐ 2 (๐ข+ , ๐+ )
๐ข๐ฟ โ ๐ 1 (๐ขโ , ๐โ )
[๐๐ข] โ โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข]
2
[๐]
โฅ 0,
2
๐ฅ (๐ก) =
[๐๐ข] โ โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข]
2
by virtue of (34), and then we can find
๐ข๐ฟ =
1 1 1 ) (๐ขโ + โ ๐ข+ โ ) โฅ 0. ๐+ ๐โ ๐+ (46)
(42)
๐ค (๐ก) = ๐โ (๐ขโ โ ๐ข+ ) ๐ก,
= 4๐โ ๐+ (๐ขโ โ ๐ข+ ) (๐ขโ +
1 ) โค 0, ๐+
[๐] (๐ 2 (๐ข+ , ๐+ )) โ 2 [๐๐ข] ๐ 2 (๐ข+ , ๐+ ) + [๐๐ข2 + ๐ข] = ๐โ (๐ขโ โ ๐ข+ โ
(๐ขโ + ๐ข+ + (1/๐โ )) ๐ก , 2
1 1 โ ๐ข+ โ ) > 0, ๐โ ๐+
[๐] (๐ 1 (๐ขโ , ๐โ )) โ 2 [๐๐ข] ๐ 1 (๐ขโ , ๐โ ) + [๐๐ข2 + ๐ข]
(41)
When ๐โ = ๐+ , (41) is just a linear equation of ๐ข๐ฟ , and then we have ๐ข๐ฟ =
โ [๐] ๐ 2 (๐ข+ , ๐+ ) + [๐๐ข] = ๐โ (๐ขโ +
๐ก.
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=โ are valid if and only if ๐ 1 (๐ขโ , ๐โ ) = ๐ 2 (๐ข+ , ๐+ ). In this case, we have ๐ข๐ฟ = ๐ 1 (๐ขโ , ๐โ ) = ๐ 2 (๐ข+ , ๐+ ).
Advances in Mathematical Physics
7
For solution (45), when ๐โ < ๐+ , ๐ข๐ฟ โ ๐ 2 (๐ข+ , ๐+ ) 2
=
[๐๐ข] + โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข] [๐]
โ ๐ 2 (๐ข+ , ๐+ ) 2
=
โ [๐] ๐ 2 (๐ข+ , ๐+ ) + [๐๐ข] + โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข] [๐]
< 0, (48) and when ๐โ > ๐+ ,
2
[๐๐ข] + โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข] [๐]
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.
๐ข๐ฟ โ ๐ 1 (๐ขโ , ๐โ ) =
the obtained Riemann solutions in two cases ๐ขโ < ๐ข+ +(1/๐+ ) and ๐ขโ โฅ ๐ข+ + (1/๐+ ). We take (๐ขโ , ๐โ ) = (2, 1). For the case ๐ขโ < ๐ข+ + (1/๐+ ), we take (๐ข+ , ๐+ ) = (4, 2), and from (24), we can calculate (๐ขโ , ๐โ ) = (2, 0.4). For the case ๐ขโ โฅ ๐ข+ + (1/๐+ ), we take (๐ข+ , ๐+ ) = (โ1, 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.
โ ๐ 1 (๐ขโ , ๐โ ) 2
=
โ [๐] ๐ 1 (๐ขโ , ๐โ ) + [๐๐ข] + โ[๐๐ข] โ [๐] [๐๐ข2 + ๐ข] [๐]
Case (i). Assume that the initial data in (9) satisfy ๐ขโ +
> 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 (๐ข+ , ๐+ ) โ ๐ (๐ขโ , ๐โ ). Then Riemann problem (8) for (7) admits one and only one entropy solution in the sense of measures of the form (๐ข, ๐) (๐ฅ, ๐ก) (๐ข , ๐ ) , { { โ โ = {(๐ข๐ฟ , ๐ค (๐ก) ๐ฟ (๐ฅ โ ๐ข๐ฟ ๐ก)) , { {(๐ข+ , ๐+ ) ,
๐ฅ < ๐ข๐ฟ ๐ก, ๐ฅ = ๐ข๐ฟ ๐ก, ๐ฅ > ๐ข๐ฟ ๐ก,
(50)
where ๐ข๐ฟ and ๐ค(๐ก) are shown in (42) for ๐โ = ๐+ or (44) for ๐โ =ฬธ ๐+ . 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 ๐ขโ < ๐ข+ + (1/๐+ ) and a delta shock wave when ๐ขโ โฅ ๐ข+ + (1/๐+ ).
4. Numerical Simulations for Riemann Solutions In this section, by employing the Nessyahu-Tadmor scheme [25] with 300 ร 300 cells and ๐ถ๐น๐ฟ = 0.475, we simulate
1 1 = ๐ข๐ + , ๐โ ๐๐
๐ข๐ = ๐ข+ .
(51)
At this moment, a contact discontinuity ๐ฝ โ emits from (0, 0) โ ๓ณจ with a speed ๐ข๐ + (1/๐๐ ) and a contact discontinuity ๐ฝ emits from (๐, 0) with a speed ๐ข๐ , respectively. From ๐ข๐ + (1/๐๐ ) > โ ๓ณจ ๐ข๐ , it is known that ๐ฝ โ will overtake ๐ฝ at a point (๐ฅ0 , ๐ก0 ), which can be calculated to be (๐ฅ0 , ๐ก0 ) = (๐(๐๐ ๐ข๐ + 1), ๐๐๐ ). At the moment ๐ก = ๐ก0 , a new Riemann problem is formed with (๐ขโ , ๐โ ) and (๐ข+ , ๐+ ) on both sides. We consider the following situation for ๐: 1 1 1 โ โฅ , ๐๐ ๐โ ๐+
(52)
which means that from (51) ๐ขโ โฅ ๐ข+ +
1 . ๐+
(53)
Then from Theorem 5, it can be concluded that a delta shock wave ๐ฟ emits from the moment ๐ก = ๐ก0 , whose speed, location, and weight can be obtained by solving (30) under (33) with initial data ๐ก = ๐ก0 : ๐ฅ (๐ก0 ) = ๐ฅ0 ,
๐ค (๐ก0 ) = 0.
(54)
See Figure 6(a), in which m = (๐ข๐ , ๐๐ ) (๐ = โ, ๐, +). Case (ii). Suppose the initial data (9) in satisfy ๐ขโ โฅ ๐ข๐ +
1 , ๐๐
๐ข๐ = ๐ข+ .
(55)
8
Advances in Mathematical Physics 2.5
4.5 4
2
3.5 1.5 ๐
๐ข
3
1 2.5 0.5
0 โ1.5
2
โ1
0
โ0.5
0.5
1
1.5
1.5 โ1.5
โ1
0
โ0.5
๐ฅ
0.5
1
1.5
0.5
1
1.5
๐ฅ
(a)
(b)
Figure 4: Numerical results for the case ๐ขโ < ๐ข+ + (1/๐+ ) at ๐ก = 0.2.
140
2.5
120
2
100
1.5
80
1
๐ 60
๐ข 0.5
40
0
20
โ0.5
0
โ1
โ20 โ1.5
โ1
0
โ0.5
0.5
1
1.5
โ1.5 โ1.5
โ1
0
โ0.5
๐ฅ
๐ฅ
(a)
(b)
Figure 5: Numerical results for the case ๐ขโ โฅ ๐ข+ + (1/๐+ ) at ๐ก = 0.5.
At this time, a delta shock wave ๐ฟ emits from (0, 0) with the speed ๐ satisfying ๐ขโ โฅ ๐ โฅ ๐ข๐ + (1/๐๐ ) and a contact โ ๓ณจ discontinuity ๐ฝ emits from (๐, 0) with the speed ๐ข๐ . Since โ ๓ณจ ๐ โฅ ๐ข๐ + (1/๐๐ ) > ๐ข๐ , ๐ฟ will interact with ๐ฝ at a point (๐ฅ0 , ๐ก0 ) = (๐๐/(๐ โ ๐ข+ ), ๐/(๐ โ ๐ข+ )). As before, at the moment ๐ก = ๐ก0 , a new Riemann problem is formed with (๐ขโ , ๐โ ) and (๐ข+ , ๐+ ) on both sides. Let us suppose that the initial data furthermore satisfy 1 ๐ขโ < ๐ข+ + , ๐+
(56)
then from Theorem 5, we can conclude that two contact discontinuities emit form ๐ก = ๐ก0 as follows: โ ๓ณจ ๐๐ฅ ๐ฝ : = ๐ขโ , ๐๐ก
๐ฝโ :
๐๐ฅ 1 = ๐ข+ + , ๐๐ก ๐+
(57)
and the intermediate states (๐ขโ , ๐โ ) connecting two contact discontinuities satisfy ๐ขโ = ๐ขโ ,
๐ขโ +
1 1 = ๐ข+ + . ๐โ ๐+
See Figure 6(b), in which โ = (๐ขโ , ๐โ ).
(58)
Advances in Mathematical Physics
9 โ ๐ฝ
๐ฟ ๐ก
๐ก
โ
โ ๐ฝ
โ
โ
๐ก0
๐ก0 โ ๐ฝ
โ ๐ฝ
โ
๐ 0
โ ๐ฝ
๐ฟ
๐
๐ฅ
(a)
โ
๐ 0
๐ฅ
๐
(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, โA system of nonstrictly hyperbolic conservation laws arising in elasticity theory,โ Archive for Rational Mechanics and Analysis, vol. 72, no. 3, pp. 219โ241, 1980. [2] Y. Lu, โExistence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type,โ Journal of Functional Analysis, vol. 261, no. 10, pp. 2797โ2815, 2011. [3] A. Aw and M. Rascle, โResurrection of โsecond orderโ models of traffic flow,โ SIAM Journal on Applied Mathematics, vol. 60, no. 3, pp. 916โ938, 2000. [4] F. Bouchut, โOn zero pressure gas dynamics,โ in Advances in Kinetic Theory and Computing: Selected Papers, B. Perthame, Ed., vol. 22, pp. 171โ190, World Scientific, Singapore, 1994. [5] G.-Q. Chen and H. Liu, โFormation of ๐ฟ-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. 925โ938, 2003. [6] G.-Q. Chen and H. Liu, โConcentration 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. 141โ165, 2004. [7] H. Cheng, H. Yang, and Y. Zhang, โRiemann problem for the Chaplygin Euler equations of compressible fluid flow,โ International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, no. 11, pp. 985โ992, 2010. [8] H. Cheng and H. Yang, โRiemann problem for the relativistic Chaplygin Euler equations,โ Journal of Mathematical Analysis and Applications, vol. 381, no. 1, pp. 17โ26, 2011. [9] H. Cheng and H. Yang, โDelta shock waves in chromatography equations,โ Journal of Mathematical Analysis and Applications, vol. 380, no. 2, pp. 475โ485, 2011. [10] H. Cheng and H. Yang, โDelta shock waves as limits of vanishing viscosity for 2-D steady pressureless isentropic flow,โ Acta Applicandae Mathematicae, vol. 113, no. 3, pp. 323โ348, 2011.
[11] V. G. Danilov and D. Mitrovic, โDelta shock wave formation in the case of triangular hyperbolic system of conservation laws,โ Journal of Differential Equations, vol. 245, no. 12, pp. 3704โ3734, 2008. [12] V. G. Danilov and V. M. Shelkovich, โDynamics of propagation and interaction of ๐ฟ-shock waves in conservation law systems,โ Journal of Differential Equations, vol. 211, no. 2, pp. 333โ381, 2005. [13] L. Guo, W. Sheng, and T. Zhang, โThe two-dimensional Riemann problem for isentropic Chaplygin gas dynamic system,โ Communications on Pure and Applied Analysis, vol. 9, no. 2, pp. 431โ458, 2010. [14] K. T. Joseph, โA Riemann problem whose viscosity solutions contain ๐ฟ-measures,โ Asymptotic Analysis, vol. 7, no. 2, pp. 105โ 120, 1993. [15] B. L. Keyfitz and H. C. Kranzer, โSpaces of weighted measures for conservation laws with singular shock solutions,โ Journal of Differential Equations, vol. 118, no. 2, pp. 420โ451, 1995. [16] J. Li and T. Zhang, โGeneralized Rankine-Hugoniot relations of delta-shocks in solutions of transportation equations,โ in NonlInear PDE and Related Areas, G. Q. Chen, Ed., pp. 219โ232, World Scientific, Singapore, 1998. ๓ธ
[17] E. Yu. Panov and V. M. Shelkovich, โ๐ฟ -shock waves as a new type of solutions to systems of conservation laws,โ Journal of Differential Equations, vol. 228, no. 1, pp. 49โ86, 2006. [18] V. M. Shelkovich, โThe Riemann problem admitting ๐ฟ-, ๐ฟ๓ธ shocks, and vacuum states (the vanishing viscosity approach),โ Journal of Differential Equations, vol. 231, no. 2, pp. 459โ500, 2006. [19] W. Sheng and T. Zhang, โThe 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, โTwo-dimensional Riemann problem for a hyperbolic system of nonlinear conservation laws. I. Four๐ฝ cases,โ Journal of Differential Equations, vol. 111, no. 2, pp. 203โ 254, 1994. [21] D. Tan, T. Zhang, and Y. Zheng, โDelta shock waves as limits of vanishing viscosity for hyperbolic systems of conversation laws,โ Journal of Differential Equations, vol. 112, no. 1, pp. 1โ32, 2004. [22] H. Yang, โRiemann problems for a class of coupled hyperbolic systems of conservation laws,โ Journal of Differential Equations, vol. 159, no. 2, pp. 447โ484, 1999.
10 [23] H. Yang and Y. Zhang, โNew developments of delta shock waves and its applications in systems of conservation laws,โ Journal of Differential Equations, vol. 252, no. 11, pp. 5951โ5993, 2012. [24] Y. Zheng, โSystems 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. 399โ426, Word Scientific, Singapore, 1998. [25] H. Nessyahu and E. Tadmor, โNon-oscillatory central differencing for hyperbolic conservation laws,โ Journal of Computational Physics, vol. 87, no. 2, pp. 408โ463, 1990.
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