Abstract
We investigate the large time behavior of the global weak entropy solutions to the symmetric Keyfitz-Kranzer system with linear damping. It is proved that as the entropy solutions tend to zero in the norm.
1. Introduction
In this paper, we consider the Cauchy problem to the symmetric system of Keyfitz-Kranzer type with linear damping with initial data This system models the propagation models of propagation of forward longitudinal and transverse waves of elastic string which moves in a plane; see [1, 2]. General source term for the system (1) was considered in [3]. The damping term in the system (1) represents external forces proportional to velocity, and this term can produce loss of total energy of system. Consider the scalar case; for example, From the integral representation of (3), it is easy to find the following solution: In this case, the solution of (3) tends to zero when . In Figure 1, we show the graph of solution for the advection equation with initial data For more general case, the behavior of solutions and its computation can be more complicated; for example, we consider Burger’s equation with a particular initial data and linear damping; this equation models the component of the velocity in one-dimensional flow with intial data where is a constant. By an application to the characteristics method, we have that solutions emanating from are given by In general, in systems with source term, the characteristics are nonlinear functions and they could have asymptotic behavior; for example, the characteristics solutions for (6), (7) are asymptotic to the lines It is easy to see that the shock occurs when . In Figures 2, 3, and 4, we show the characteristics solutions for several values of .
We are looking for conditions under which the terms , have a dissipative effect in the solutions of (1).
Let , and we are going to show the following main theorem.
Theorem 1. If the initial data , then the Cauchy problems (1) and (2) have a weak entropy solutions satisfying Moreover, converges to zero in with exponential time decay; that is,
2. Preliminaries
We start with some preliminaries about the general systems of conservation laws; see [4] chapter 5. Let be a smooth vector field. Consider Cauchy problem for the system When , the system (12) is called homogeneous system of conservation laws, if , the system (12) is called inhomogeneous system or balance system of conservation laws. We will work also with the parabolic perturbation to the system (12); namely, Denote by the Jacobian matrix of partial derivatives of .
Definition 2. The system (12) is strictly hyperbolic if, for every , the matrix has real distinct eigenvalues .
Let be the corresponding eigenvector to . Then, one can see the following.
Definition 3. One says that the th characteristic field is genuinely nonlinear if If instead we say that the th characteristic field is linearly degenerate.
For the following definitions, see [5, 6].
Definition 4. A -Riemann invariant is a smooth function , such that
Definition 5. A pair of function is called a entropy-entropy flux pair if it satisfies if is a convex function, then the pair is called convex entropy-entropy flux pair.
Definition 6. A bounded measurable function is an entropy (or admissible) solution for the Cauchy problem (12) if it satisfies the following inequality: in the distributional sense, where is any convex entropy-entropy flux pair.
We consider the general system of Keyfitz-Kranzer type to get some general observations about this type of system. Let in (19), and we have that the eigenvalues and eigenvector of the Jacobian matrix are given by From (20) and (21), we have that and , where , and then the Riemann invariants are given by
Lemma 7. The system (1) is always linear degenerate in the first characteristic field. If then the system (1) is strictly hyperbolic and nonlinear degenerate in the second characteristic field. Moreover, where represents the Hessian matrix.
Lemma 8. Let be a Lipschitz function in a neighborhood of the origin, and let be a function, such that satisfies Then, the pair is a entropy-entropy flux pair for the system (1). Moreover, if is a convex function, then the pair (26) is a convex entropy-entropy flux pair.
3. Global Existence of Weak Entropy Solutions and Asymptotic Behavior
We consider the parabolic regularization of the system (1). Namely, with initial data where is a mollifier. In this case, , with . By (20), the eigenvectors and eigenvalues are given by The following conditions will be necessary in our next discussion:(), as , ;(). The condition () guarantees the strict hyperbolicity to the system (27) according to Lemma 7, while condition () ensures the existence of a positive invariant region. Now, we consider the following subset of : We affirm that is an invariant region. Let , if , where is the level curve of . We have that and if , where is the level curve of , we have that By Theorem 14.7 of [5], is an invariant region for the system (27). It is easy to verify that satisfies the condition in [3]; thus, we have the following Lemma.
Proposition 9. If and the condition holds, then the Cauchy problems (27), (28) have a global weak entropy solution.
Now, for the global behavior of solutions, using ideas of the author in [7], we construct the following entropy-entropy flux pairs as follows: From (25), we have Integrating by parts, we have that Let . Then, we have that Multiplying (1) by , we have that Now, we choose as a function, such as , , and for and set . Then, . Multiplying by in (37) and integrating over , we have By the inequality (36), we have If , we have By Gronwall’s inequality, we have Thus, we have Passing to limit , in (42), we have the inequality (10).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank Professor Laurent Gosse for his suggestions and review, Professor Juan Galvis for his many valuable observations, Professor Yun-Guang Lu for his suggestion for this problem, and the reviewer for his many valuable suggestions.