Abstract
In this article, we investigate the global existence and asymptotic behavior of classical solutions to Goursat problem for diagonalizable quasilinear hyperbolic system. Under the assumptions that system is strictly hyperbolic and linearly degenerate, we obtain the global existence and uniqueness of C^{1 }solutions with the bounded L^{1 }∩ L^{∞ }norm of the boundary data as well as their derivatives. Based on the existence result, we can prove that when t tends to in nity, the solutions approach a combination of piecewised C^{1 }traveling wave solutions. As the important example, we apply the results to the chaplygin gas system.
Mathematics Subject Classi cation (2000): 35B40; 35L50; 35Q72.
Keywords:
Goursat problem; global classical solutions; linearly degenerate; asymptotic behavior; traveling wave solutions.1 Introduction and main results
For the general first order quasilinear hyperbolic systems,
the global existence of classical solutions of Cauchy problem has been established for linearly degenerate characteristics or weakly linearly degenerate characteristics with various smallness assumptions on the initial data by Bressan [1], Li [2], Li and Zhou [3,4], Li and Peng [5,6], and Zhou [7]. The asymptotic behavior has been obtained by Kong and Yang [8], Dai and Kong [9,10]. For linearly degenerate diagonalizable quasilinear hyperbolic systems with "large" initial data, asymptotic behavior of the global classical solutions has been obtained by Liu and Zhou [11]. For the initialboundary value problem in the first quadrant Li and Wang [12] proved the global existence of classical solutions for weakly linearly degenerate positive eigenvalues with small and decay initial and boundary data. The asymptotic behavior of the global classical solutions is studied by Zhang [13]. The global existence and asymptotic behavior of classical solutions of the initialboundary value problem of diagonalizable quasilinear hyperbolic systems in the first quadrat was obtained in [14].
However, relatively little is known for the Goursat problem with characteristic boundaries. Global existence of the global classical solutions for the Goursat problem of reducible quasilinear hyperbolic system was obtained in [15]. Under the assumptions of boundary data is small and decaying, the global existence and asymptotic behavior to classical solutions can be obtained by Liu [16,17]. The asymptotic behavior of classical solutions of Goursat problem for reducible quasilinear hyperbolic system was shown in [18].
In this article, we consider the following diagonalizable quasilinear hyperbolic system:
where u = (u_{1}, ..., u_{n})^{T }is unknown vectorvalued function of (t, x). λ_{i}(u) is given by C^{2 }vectorvalued function of u and is linearly degenerate, i.e.,
The system (1) is strictly hyperbolic, i.e.,
Suppose that there exists a positive constant δ such that
Consider the Goursat problem for the strictly quasilinear hyperbolic system (1), in which the solutions to system (1) is asked to satisfy the following characteristic boundary conditions:
and
where x = x_{1}(t) and x = x_{n}(t) are the leftmost and the rightmost characteristics passing through the origin (t, x) = (0, 0), respectively, such that
and
moreover,
and
where ϕ (t) = (ϕ_{1}(t), ..., ϕ_{n}(t))^{T }and ψ(t) = (ψ_{1}(t), ..., ψ_{n}(t))^{T }are any given C^{1 }vector functions satisfying the conditions of C^{1 }compatibility at the origin (0, 0):
and
l_{i}(u) be a left eigenvector corresponding to λ_{i}(u) and A(u) = diag{λ_{1}(u), ..., λ_{n}(u)}.
Our goal in this article is to get the global existence and asymptotic behavior of the global classical solutions of the Goursat problem (1), (5), and (6) with "large" boundary data. With the assumptions that
we can prove the following results:
Theorem 1.1. The above assumptions and the conditions of C^{1 }compatibility at the point (0, 0) are satisfied, the Goursat problem (1), (5), and (6) admits a unique global C^{1 }solutions u = u(t, x) on the domain D = {(t, x) t ≥ 0, x_{1}(t) ≤ × ≤ x_{n}(t)}.
If the leftmost and rightmost characteristics are convex, we can get the following result:
Theorem 1.2. Under the assumptions of Theorem 1.1, there exists a unique piecewised C^{1 }vectorvalued function Φ(x) = (Φ_{1}(x), ..., Φ_{n}(x))^{T }such that
uniformly as t tends to infinity, where e_{i }= (0, ..., 0, 1^{i}, 0, ..., 0)^{T}.
Remark 1.1. If the system (1) is nonstrictly hyperbolic but each characteristic has constant multiplicity, then the result is similar as Theorems above.
2 Global existence of C^{1 }solutions
In this section, we will obtain some uniform a priori estimate which also play an important role in the proof of Theorem 1.1. In order to proving the global existence of classical solutions of the Goursat problem (1), (5), and (6), we will prove that is bounded. For any fixed T ≥ 0, we denote D_{T }= {(t, x)0 ≤ t ≤ T, x_{1}(t) ≤ x ≤ x_{n}(t)} and introduce
where stands for any given jth characteristic , L_{j }stands for any given radial that has the slope λ_{j}(0) on the domain D_{T }.
Lemma 2.1. Under the assumptions of Theorem 1.1, there exists a positive constant C such that, the following estimates hold
Remark 2.1. The positive constant C is only depend on δ, M_{0 }and independent of M, N_{1}, N_{2}, T. In the following, the meaning of C is similar but may change from line to line.
Proof. For any fixed point (t, x) ∈ D_{T }, we draw the ith characteristic through this point and intersecting x_{1}(t) or x_{n}(t) at a point (t_{*}, x_{1}(t_{*})) or (t_{*}, x_{n}(t_{*})). Noting system (1), u_{i}(t, x) is a constant along the ith characteristic, then we have u_{i}(t, x) = ϕ_{i}(t_{*}) or u_{i}(t, x) = ψ_{i}(t_{*}). Then
Then, we obtain the estimate (23).
Differentiating (1) with respect to x we can get
We rewrite (25) as
Multiplying the system above by sign(w_{i}), we have
There are only the following cases(as shown in Figure 1):
Figure 1. Cases in estimating .
Case 1. For any fixed t_{0 }∈ R^{+}, let stands for any given jth characteristic, passing through any point A(t_{0}, x_{1}(t_{0})) on the boundary x = x_{1}(t) and intersects t = T at point P. We draw an ith characteristic from P downward, intersecting x = x_{1}(t) at a point B(t_{1}, x_{1}(t_{1})). Without loss of generality, we assume t_{0 }< t_{1}, then j > i. Integrating (26) in the region APB to get
Along the 1th characteristic, , then
Noting (4), (13), and (23), we can get
Case 2. For any fixed t_{0 }∈ R^{+}, passing through the point A(t_{0}, x_{1}(t_{0})), we draw and intersecting t = T at point P. We draw the ith characteristic from P downward, intersecting x = x_{n}(t) at B(t_{1}, x_{n}(t_{1})). Then, we integrate (26) in the region PAOB to get
Noting i = 1, ..., m i < j, Equations (4), (13) and using the above procedure, we have
If the point A is on the characteristic boundary x = x_{n}(t), using the same method as above cases we can get the desired estimates. Then, we can obtain
In the similar way, substituting L_{j }for in the above cases we also can get
In the following, we will get the estimate W_{1}(T) ≤ CN_{2}. Integrating Equation (26) with respect to x from x_{1}(t) to x_{n}(t) for any t ∈ [0, T ] leads to
Using the same procedure as (28), (29), we can get
Then, we can get the desired estimate.
Rewriting the Equation (25), we get
Case 1. When i = 1, ..., m, we draw the ith characteristic intersecting x = x_{n}(t) with the point (t_{0}, x_{n}(t_{0})). Solving the ODE system we can get
Noting and the estimate (30), then
Case 2. When i = m + 1, ..., n and the ith characteristic intersecting x = x_{1}(t) with a point (t_{0}, x_{1}(t_{0})), then
Then, we can get
Then, we can obtain the estimate
In the following we estimate .
Noting system (1), we denote the multiplier H_{i }∈ C^{1}, (i = 1, ..., n) such that,
Noting (1), then
Let H_{i}(t, x_{1}(t)) = 1 and H_{i}(t, x_{n}(t)) = 1 we can get
We note the system (1) is linearly degenerate, then
Noting Equation (38), we can rewrite it as
There are only the following cases like we estimate :
Case 1. We can integrate (40) in the region APB which is same to the case 1 of proof of to get
Then
Case 2. Integrating the Equation (40) in the region PAOB which is same to the case 2 of proof of , we can get
Using the above procedures, we can get
Then
In the similar way, substituting L_{j }for we can get
Combining (24), (30), (31), (34), (37), (43), and (44) together we can obtain the conclusion of Lemma 2.1.
Proof of Theorem 1.1.
Noting the conclusion of Lemma 2.1, we can get
Therefore, we can obtain that the system (1), (5), and (6) have global classical solutions on the domain D = {(t, x) t ≥ 0, x_{1}(t) ≤ x ≤ x_{n}(t)}.
3 Asymptotic behavior of global classical solutions
In this section, under the assumption of the leftmost and rightmost characteristics, we will study the asymptotic behavior of the global classical solutions of system (1), (5), and (6) and give the proof of Theorem 1.2.
Let
Obviously,
where . Thus, noting system (1)
Using the Hadamard's Lemma, we can obtain
For any fixed point (t, x) ∈ D, Passing through (t, x), we draw down the characteristic x = x_{r}(t), which intersect with the characteristic boundary in the point . Then (where r = 1, when i = m + 1, ..., n or r = n, when i = 1, ..., m).
Then, it follows from Equation (51) that
Noting (46) and (47), we can get
This implies that the integral converges uniformly for α ∈ R. Then, there exists a unique function Φ_{i}(α) such that,
Moreover, noting (13) and Equations (52), (53), we have
In what follows, we will study the regularity of Φ(α). Noting Equation (52), we can get Φ(α) ∈ C^{0 }(R). From any fixed point , we draw a characteristic intersecting the boundary x = x_{1}(t) or x = x_{n}(t) at .
Then, integrating it along the ith characteristic, we obtain
Then, we can get the following Lemma
Lemma 3.1 Under the assumptions of Theorem 1.1, for the θ_{i}(t, α) defined above, there exists a unique ϑ_{i}(α), such that
Proof. Using the Hardarmad's formula, we can rewrite (55) as following
where
are C^{1 }functions. Therefore
where r is either 1 or n. Then, we know that when t tends to ∞, the right hand of (58) convergence absolutely. For any given α, the right hand of (57) convergence to some function with respect to α. That implies that there exists a unique function ϑ(α), such that
Lemma 3.2 Under the assumptions of Theorem 1.1, for any given point on the boundary, there exists a unique function Ψ_{i}(ϑ(α)) ∈ C^{0}, such that
uniformly for any α ∈ R.
Proof. Noting
In the following, we prove that there exists a unique Ψ_{i}(ϑ(α)) ∈ C^{0}, such that
Integrating (35) along the ith characteristic gives
where . Noting (45) and (47), we can get
Then, as t tends to +∞, the integrals in the righthand side of (63) convergence, i.e., there exists a unique function Ψ_{i}(ϑ_{i}(α)), such that
Lemma 3.3
Proof. For any fixed α ∈ R, we calculate
Thus, when i = 1, ..., s, i.e., r = 1, we can get
Using the similar procedure, when α > 0, i.e., i = s + 1, ..., n and r = n, we can get
Noting the above lemmas, we get the conclusion of Theorem 1.2.
4 Applications
Recent years observations of the luminosity of type Ia distant supernovae point towards an accelerated expansion of the universe, which implies that the pressure p and the energy density ρ of the universe should violate the strong energy condition, i.e., ρ + 3p < 0. Here, we consider a recently proposed class of simple cosmological models based on the use of peculiar perfect fluids [19]. In the simplest case, we study the model of a universe filled with the so called Chaplygin gas, which is a perfect fluid characterized by the following equation of state , where A is a positive constant.
In Lagrange coordinate, the 1D gas dynamics equations in isentropic case can be written as
Noting (69), in isentropic case we can get the system of one dimensional Chaplygin gas model
Nothing systems (70) is linear systems, it is easy to get the eigenvalues
and left eigenvectors
Introduce the Riemann invariants
we can rewrite the system as following
Consider the Goursat problem for system (69) with following characteristic boundary conditions:
Then, the above system satisfies the assumptions of Theorems 1.1 and 1.2. More precisely, we can get the following theorems:
Theorem 4.1. The Goursat problem (69) and (75) admits a unique global C^{1 }solution (τ, u) (t, x) on the domain .
Theorem 4.2. There exists a unique piecewised C^{1 }vectorvalue function Φ(x) = (Φ_{1 }(x), ...,Φ_{n}(x))^{T }such that
uniformly as t tends to infinity, where e_{i }= (0, ..., 0, 1^{i}, 0, ..., 0)^{T}.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
The first author was supported by NSFCTianyuan Special Foundation (No. 11126058), Excellent Young Teachers Program of Shanghai and the Shanghai Leading Academic Discipline Project (No. J50101). The second author was supported by China Postdoctoral Science Foundation (No. 2011M501295) and the Fundamental Research Funds for the Central Universities (No. 2011QNZT102). The authors would like to thank Professor Zhou Yi for his guidance and encouragements.
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