Asymptotic Behavior of Global Entropy Solutions for Nonstrictly Hyperbolic Systems with Linear Damping

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 t → ∞. 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 A is a constant. By an application to the characteristics method, we have that solutions emanating from x₀ 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 x₀ + A(1 − x₀) ² > 1. In Figures 2, 3, and 4, we show the characteristics solutions for several values of A.

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 g(u) = 0, the system (12) is called homogeneous system of conservation laws, if g(u) ≠ 0, 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 A(u) = Df(u) the Jacobian matrix of partial derivatives of f.

We consider the general system of Keyfitz-Kranzer type to get some general observations about this type of system. Let F(u, v) = (uϕ(u, v), vϕ(u, v)) in (19), and we have that the eigenvalues and eigenvector of the Jacobian matrix Df are given by From (20) and (21), we have that ∇ϕ · r₁ = 0 and ∇Z(u, v) · r₂ = 0, where Z(u, v) = u/v, and then the Riemann invariants are given by

We consider the parabolic regularization of the system (1). Namely, with initial data where j_ϵ is a mollifier. In this case, ϕ(u, v) = ϕ(r), with $$. By (20), the eigenvectors and eigenvalues are given by The following conditions will be necessary in our next discussion:

•  

  C₁ rϕ(r) → 0, as r → 0, rϕ^′(r) ≠ 0;

•  

  C₂ a > b.

The condition (C₁) guarantees the strict hyperbolicity to the system (27) according to Lemma 7, while condition (C₂) ensures the existence of a positive invariant region. Now, we consider the following subset of $$: We affirm that Σ is an invariant region. Let h(u, v) = (au, bv), if $$, where γ₁ is the level curve of W = ϕ(r). We have that and if $$, where γ₂ is the level curve of Z = u/v, we have that By Theorem 14.7 of [5], Σ is an invariant region for the system (27). It is easy to verify that (au, bv) satisfies the condition H₁ ⋯ H₅ in [3]; thus, we have the following Lemma.

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 $$. Then, we have that Multiplying (1) by ∇η, we have that Now, we choose $$ as a function, such as |h^′(x)| ≤ 1, |h^′′(x)| ≤ 1, and h(x) = |x| for |x| ≥ 1 and set k(x) = e^−h(x). Then, k^′(x) ≤ k(x). Multiplying by k(x) in (37) and integrating over x, we have By the inequality (36), we have If $$, we have By Gronwall’s inequality, we have Thus, we have Passing to limit m → ∞, in (42), we have the inequality (10).