Discontinuous traveling waves for scalar hyperbolic-parabolic balance law
- Tianyuan Xu^{1},
- Chunhua Jin^{1} and
- Shanming Ji^{1}Email author
Received: 16 November 2015
Accepted: 24 January 2016
Published: 5 February 2016
Abstract
This paper is concerned with the existence of traveling waves for the scalar hyperbolic-parabolic balance law. Using a phase-plane analysis method, we first prove the existence of an increasing traveling wave solution in \(C^{1}(\mathbb{R})\). Then we construct a family of discontinuous periodic traveling wave entropy solutions.
Keywords
1 Introduction
This paper is organized as follows. In Section 2, after introducing some definitions and notations, we present some auxiliary lemmas and state the main results. Subsequently, in Section 3, we prove the existence of discontinuous traveling waves.
2 Preliminaries and main results
- (A1)
\(f\in C^{2}(\mathbb{R})\), \(f'\) is strictly increasing;
- (A2)
\(g\in C^{1}(\mathbb{R})\), \(g'\le k_{g}\) for some constant \(k_{g}\);
- (A3)
\(u_{0}\in \mathit{BV}_{L}(\mathbb{R})\), where \(\mathit{BV}_{L}\) is the space of functions which are locally of bounded variation and are L-periodic for some given constant \(L>0\);
- (A4)
The zeros of g are simple, i.e., \(g'(v)\ne0\) for any v such that \(g(v)=0\).
To any zero \(v_{2k}\) of g there can be associated a continuous traveling wave solution and a family of discontinuous periodic traveling waves of (1.2) with speed \(f'(v_{2k})\).
It is well known that the problem (1.2)-(2.1) does not have global classical solutions even if \(u_{0}\) is smooth [4]. On the other hand, discontinuous solutions in the distributional sense may not be unique. This leads to the definition of entropy solutions as follows.
Definition 2.1
([4])
A function u is a solution of (1.2) in the sense of distributions with the entropy condition (2.2) if and only if u satisfies Definition 2.2, which is derived by a vanishing viscosity method [15].
Definition 2.2
With this notion of entropy solutions and under the assumptions (A1)-(A3), the problem (1.2)-(2.1) is well posed, as is shown by [15] and references therein.
Definition 2.3
It was Vol’pert and Hudjaev [16] who first treated the solvability of the initial value problem for the hyperbolic-parabolic balance law (1.1). The uniqueness of entropy solution of equation (1.1) subject to given initial value was proved by Wu and Yin [17] in an equivalent form of discontinuity condition.
Now we show that for piecewise continuous functions, the definition of entropy solutions can be valid by satisfying the assumptions in the following lemma.
Lemma 2.1
- (i)
in any piecewise continuous domain, u satisfies (1.1) in the classical sense;
- (ii)along any discontinuous line \(x=x(t)\) the following Rankine-Hugoniot condition holds:$$ \bigl(u\bigl(x^{+},t\bigr)-u\bigl(x^{-},t\bigr)\bigr) \frac{dx}{dt}=f\bigl(u\bigl(x^{+},t\bigr)\bigr)-f\bigl(u\bigl(x^{-},t\bigr) \bigr); $$(2.5)
- (iii)
along any discontinuous line \(x=x(t)\) the entropy condition (2.2) is valid and there exists a \(1\le k\le2K\) such that \(u(x^{+},t), u(x^{-},t)\in (b_{k-1},a_{k})\).
Proof
Since \(a(s)\equiv0\) for \(s\in(b_{k-1},a_{k})\) and \(u(x^{+},t), u(x^{-},t)\in (b_{k-1},a_{k})\), the second derivative term \(A(u)_{xx}=(a(u)u_{x})_{x}\) is strongly degenerate. The rest of the proof is similar to the proof of the equivalence between the inequality (2.3) in Definition 2.2 and the definition of entropy solutions for the first order equation (1.2) in Definition 2.1. □
We state our main result here and leave its proof to the next section.
Theorem 2.1
3 Proof of the main results
Lemma 3.1
Proof
- (i)
along the curve \(\Gamma_{0}\), \(\Phi>0\), \(\Psi=0\);
- (ii)
in the domain \(G_{1}=\{(\phi,\psi);\psi>\frac{g(\phi)a(\phi)}{f'(\phi)-f'(v_{2k})},\phi\in(a_{2k},b_{2k})\}\), \(\Phi>0\), \(\Psi>0\);
- (iii)
while in the domain \(G_{2}=G_{0}\backslash\overline{G}_{1}\), \(\Phi>0\), \(\Psi<0\).
- (i)
the curve \(\Gamma_{1}\) lies in the domain \(G_{1}\);
- (ii)
any trajectory intersecting with the curve \(\Gamma_{1}\) all runs through \(\Gamma_{1}\) from \(G_{1}''\) into \(G_{1}'\).
Let \(\Gamma_{\varepsilon}\) and \(\Gamma_{\delta}\) be the trajectories that arrive at the point \((b_{2k},\varepsilon)\) and \((b_{2k}-\delta,0)\), respectively, where \(\varepsilon>0\), \(0<\delta<b_{2k}-a_{2k}\). Since \(\Phi>0\), \(\Psi<0\) in the domain \(G_{2}\), \(\Gamma_{\varepsilon}\) cannot intersect with the segment \(L=\{(\phi ,0);\phi\in(a_{2k},b_{2k})\}\) and \(\Gamma_{\delta}\) cannot intersect with L except the point \((b_{2k}-\delta,0)\). On the other hand, according to the properties of the curve \(\Gamma_{1}\), all the trajectories \(\Gamma_{\varepsilon}\) and \(\Gamma_{\delta}\) do not intersect with the curve \(\Gamma_{1}\). Thus \(\Gamma_{\varepsilon}\) and \(\Gamma_{\delta}\) are all starting from the singular point \((a_{2k},0)\). By the continuity of system (3.3), there exists a trajectory Γ that connects \((a_{2k},0)\) and \((b_{2k},0)\). It follows that equation (3.1) admits a solution \(\phi(\xi)\) such that ϕ is defined on the interval \((\xi^{+},\xi_{1})\), ϕ is strictly increasing and \(\phi(\xi_{1})=b_{2k}\). Combining with (3.1), we see that \(\phi'(\xi_{1})=\frac{g(b_{2k})}{f'(b_{2k})-f'(v_{2k})}>0\), which implies that \(\xi_{1}\) is a finite real number.
The left-hand extension of ϕ to the interval \((-\infty,\xi^{-})\) is similar.
Lemma 3.2
Proof
The proof of this lemma is similar to that of Lemma 3.1. □
Utilizing the above two lemmas, we can prove that the degenerate parabolic equation (1.1) admits a strictly increasing traveling wave solution whose range is \((v_{2k-1},v_{2k+1})\).
Lemma 3.3
Assume that the conditions (3.2) and (3.5) hold, then equation (1.1) admits a continuous traveling wave \(\phi(\xi)\in C^{1}(\mathbb{R})\) such that ϕ is piecewise \(C^{2}\) continuous and strictly increasing, \(\phi(0)=v_{2k}\), and the range of ϕ is \((v_{2k-1},v_{2k+1})\).
Concerned with the Rankine-Hugoniot condition (2.5), we give the following property of convex functions.
Lemma 3.4
Proof
According to the strict convexity of f, we conclude the above assertions. □
Using the continuous traveling wave of equation (1.1), we can construct a family of discontinuous periodic traveling waves. Suppose that the conditions (3.2) and (3.5) are fulfilled. We note that either the condition (3.6) or the condition (3.7) is true. Without loss of generality, suppose that (3.6) holds. Lemma 3.3 implies that equation (1.1) admits a continuous traveling wave, denoted by \(\phi_{2k}(\xi)\) with \(\xi=x-f'(v_{2k})t\) and the speed of this traveling wave is \(f'(v_{2k})\).
Lemma 3.5
Assume that the conditions (3.2) and (3.5) are fulfilled. The periodic function \(u(x,t)=\psi_{2k}(\xi)\) with \(\xi=x-f'(v_{2k})t\) defined by (3.9) is a discontinuous traveling wave entropy solution with speed \(f'(v_{2k})\) for equation (1.1).
Proof
By the construction of \(\psi_{2k}(\xi)\), we see that the two single-side limits of \(\psi_{2k}(\xi)\) at the discontinuous point \(\xi_{0}+jT+ \sum_{i=1}^{m}T_{i}+\xi_{m}^{-}\) are \(\phi_{2k}(\xi_{m}^{+})=\eta_{m}\) and \(\phi_{2k}(\xi_{m}^{-})=\mu_{m}\), for \(j\in \mathbb{Z}\), \(1\le m\le N\). Since \(a(s)\equiv0\) for \(s\in(b_{2k-1},a_{2k})\) and the equality (3.8), we prove that \(\psi_{2k}\) satisfies the Rankine-Hugoniot condition (2.5) at any discontinuous points. The entropy condition (2.2) is fulfilled as \(\mu_{m}<\eta_{m}\). Clearly, \(\mu_{m}, \eta_{m}\in(b_{2k-1},a_{2k})\). Lemma 2.1 implies that \(u(x,t)\) is an entropy solution of (1.1). □
Declarations
Acknowledgements
The second author was supported by the Program for New Century Excellent Talents in University of the Ministry of Education (NCET-13-0804), NSFC (11471127,11371153), Guangdong Natural Science Funds for Distinguished Young Scholar (2015A030306029), The Excellent Young Teachers Program of Guangdong Province (HS2015007), and Special support program of Guangdong Province.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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