-
Previous Article
Equivariant versal unfoldings for linear retarded functional differential equations
- DCDS Home
- This Issue
-
Next Article
Stability criteria for linear Hamiltonian systems with uncertain bounded periodic coefficients
Asymptotics toward strong rarefaction waves for $2\times 2$ systems of viscous conservation laws
| 1. | Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong |
| 2. | School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China |
$u_t+F(u)_x=(B(u)u_x)_x,\quad u\in R^2,\qquad $ ($*$)
$u(0,x)=u_0(x)\rightarrow u_\pm\quad$ as $x\rightarrow \pm\infty.$
Assume that the corresponding Riemann problem
$u_t+F(u)_x=0,$
$ u(0,x)=u^r_0(x)=u_-,\quad x<0, and u_+,\quad x>0$
can be solved by one rarefaction wave. If $u_0(x)$ in ($*$) is a small perturbation of an approximate rarefaction wave constructed in Section 2, then we show that the Cauchy problem ($*$) admits a unique global smooth solution $u(t,x)$ which tends to $ u^r(t,x)$ as the $t$ tends to infinity. Here, we do not require $|u_+ - u_-|$ to be small and thus show the convergence of the corresponding global smooth solutions to strong rarefaction waves for $2\times 2$ viscous conservation laws.
| [1] |
Evgeny Yu. Panov. On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws. Networks & Heterogeneous Media, 2016, 11 (2) : 349-367. doi: 10.3934/nhm.2016.11.349 |
| [2] |
Shuichi Kawashima, Shinya Nishibata, Masataka Nishikawa. Asymptotic stability of stationary waves for two-dimensional viscous conservation laws in half plane. Conference Publications, 2003, 2003 (Special) : 469-476. doi: 10.3934/proc.2003.2003.469 |
| [3] |
Laurent Lévi, Julien Jimenez. Coupling of scalar conservation laws in stratified porous media. Conference Publications, 2007, 2007 (Special) : 644-654. doi: 10.3934/proc.2007.2007.644 |
| [4] |
Tung Chang, Gui-Qiang Chen, Shuli Yang. On the 2-D Riemann problem for the compressible Euler equations I. Interaction of shocks and rarefaction waves. Discrete & Continuous Dynamical Systems, 1995, 1 (4) : 555-584. doi: 10.3934/dcds.1995.1.555 |
| [5] |
Alberto Bressan, Marta Lewicka. A uniqueness condition for hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems, 2000, 6 (3) : 673-682. doi: 10.3934/dcds.2000.6.673 |
| [6] |
Kenta Nakamura, Tohru Nakamura, Shuichi Kawashima. Asymptotic stability of rarefaction waves for a hyperbolic system of balance laws. Kinetic & Related Models, 2019, 12 (4) : 923-944. doi: 10.3934/krm.2019035 |
| [7] |
Roberto Triggiani. A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: A critical subspace of $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^*)^{\frac{1}{2}})$ and implications. Evolution Equations & Control Theory, 2016, 5 (1) : 185-199. doi: 10.3934/eect.2016.5.185 |
| [8] |
K. T. Joseph, Manas R. Sahoo. Vanishing viscosity approach to a system of conservation laws admitting $\delta''$ waves. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2091-2118. doi: 10.3934/cpaa.2013.12.2091 |
| [9] |
Wen Shen. Traveling waves for conservation laws with nonlocal flux for traffic flow on rough roads. Networks & Heterogeneous Media, 2019, 14 (4) : 709-732. doi: 10.3934/nhm.2019028 |
| [10] |
Thomas Ward, Yuki Yayama. Markov partitions reflecting the geometry of $\times2$, $\times3$. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 613-624. doi: 10.3934/dcds.2009.24.613 |
| [11] |
Feng Xie. Nonlinear stability of combination of viscous contact wave with rarefaction waves for a 1D radiation hydrodynamics model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1075-1100. doi: 10.3934/dcdsb.2012.17.1075 |
| [12] |
Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 |
| [13] |
Huijiang Zhao, Yinchuan Zhao. Convergence to strong nonlinear rarefaction waves for global smooth solutions of $p-$system with relaxation. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1243-1262. doi: 10.3934/dcds.2003.9.1243 |
| [14] |
Min Ding, Hairong Yuan. Stability of transonic jets with strong rarefaction waves for two-dimensional steady compressible Euler system. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2911-2943. doi: 10.3934/dcds.2018125 |
| [15] |
Weishi Liu. Multiple viscous wave fan profiles for Riemann solutions of hyperbolic systems of conservation laws. Discrete & Continuous Dynamical Systems, 2004, 10 (4) : 871-884. doi: 10.3934/dcds.2004.10.871 |
| [16] |
Gianluca Crippa, Elizaveta Semenova, Stefano Spirito. Strong continuity for the 2D Euler equations. Kinetic & Related Models, 2015, 8 (4) : 685-689. doi: 10.3934/krm.2015.8.685 |
| [17] |
Alicia Cordero, José Martínez Alfaro, Pura Vindel. Bott integrable Hamiltonian systems on $S^{2}\times S^{1}$. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 587-604. doi: 10.3934/dcds.2008.22.587 |
| [18] |
Avner Friedman. Conservation laws in mathematical biology. Discrete & Continuous Dynamical Systems, 2012, 32 (9) : 3081-3097. doi: 10.3934/dcds.2012.32.3081 |
| [19] |
Mauro Garavello. A review of conservation laws on networks. Networks & Heterogeneous Media, 2010, 5 (3) : 565-581. doi: 10.3934/nhm.2010.5.565 |
| [20] |
Len G. Margolin, Roy S. Baty. Conservation laws in discrete geometry. Journal of Geometric Mechanics, 2019, 11 (2) : 187-203. doi: 10.3934/jgm.2019010 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]