L^1 singular limit for relaxation and viscosity approximations of extended traffic flow models
This paper considers the Cauchy problem for an extended traffic flow model with $L^1$-bounded initial data. A solution of the corresponding equilibrium equation with $L^1$-bounded initial data is given by the limit of solutions of viscous approximations of the original system as the dissipation para...
| Main Authors: | , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Texas State University
2003-03-01
|
| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2003/23/abstr.html |
| _version_ | 1818360511833571328 |
|---|---|
| author | Christian Klingenberg Yun-Guang Lu Hui-Jiang Zhao |
| author_facet | Christian Klingenberg Yun-Guang Lu Hui-Jiang Zhao |
| author_sort | Christian Klingenberg |
| collection | DOAJ |
| description | This paper considers the Cauchy problem for an extended traffic flow model with $L^1$-bounded initial data. A solution of the corresponding equilibrium equation with $L^1$-bounded initial data is given by the limit of solutions of viscous approximations of the original system as the dissipation parameter $epsilon$ tends to zero more slowly than the response time $au$. The proof of convergence is obtained by applying the Young measure to solutions introduced by DiPerna and, based on the estimate $$ | ho(t,x)| leq sqrt {| ho_0(x)|_1/(ct)} $$ derived from one of Lax's results and Diller's idea, the limit function $ ho(t,x)$ is shown to be a $L^1$-entropy week solution. A direct byproduct is that we can get the existence of $L^1$-entropy solutions for the Cauchy problem of the scalar conservation law with $L^1$-bounded initial data without any restriction on the growth exponent of the flux function provided that the flux function is strictly convex. Our result shows that, unlike the weak solutions of the incompressible fluid flow equations studied by DiPerna and Majda in [6], for convex scalar conservation laws with $L^1$-bounded initial data, the concentration phenomenon will never occur in its global entropy solutions. |
| first_indexed | 2024-12-13T21:01:58Z |
| format | Article |
| id | doaj.art-dc1bc303d0f74eadb2a3aee259902fcd |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-13T21:01:58Z |
| publishDate | 2003-03-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-dc1bc303d0f74eadb2a3aee259902fcd2022-12-21T23:31:35ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912003-03-01200323111L^1 singular limit for relaxation and viscosity approximations of extended traffic flow modelsChristian KlingenbergYun-Guang LuHui-Jiang ZhaoThis paper considers the Cauchy problem for an extended traffic flow model with $L^1$-bounded initial data. A solution of the corresponding equilibrium equation with $L^1$-bounded initial data is given by the limit of solutions of viscous approximations of the original system as the dissipation parameter $epsilon$ tends to zero more slowly than the response time $au$. The proof of convergence is obtained by applying the Young measure to solutions introduced by DiPerna and, based on the estimate $$ | ho(t,x)| leq sqrt {| ho_0(x)|_1/(ct)} $$ derived from one of Lax's results and Diller's idea, the limit function $ ho(t,x)$ is shown to be a $L^1$-entropy week solution. A direct byproduct is that we can get the existence of $L^1$-entropy solutions for the Cauchy problem of the scalar conservation law with $L^1$-bounded initial data without any restriction on the growth exponent of the flux function provided that the flux function is strictly convex. Our result shows that, unlike the weak solutions of the incompressible fluid flow equations studied by DiPerna and Majda in [6], for convex scalar conservation laws with $L^1$-bounded initial data, the concentration phenomenon will never occur in its global entropy solutions.http://ejde.math.txstate.edu/Volumes/2003/23/abstr.htmlSingular limittraffic flow modelrelaxation and viscosity approximation. |
| spellingShingle | Christian Klingenberg Yun-Guang Lu Hui-Jiang Zhao L^1 singular limit for relaxation and viscosity approximations of extended traffic flow models Electronic Journal of Differential Equations Singular limit traffic flow model relaxation and viscosity approximation. |
| title | L^1 singular limit for relaxation and viscosity approximations of extended traffic flow models |
| title_full | L^1 singular limit for relaxation and viscosity approximations of extended traffic flow models |
| title_fullStr | L^1 singular limit for relaxation and viscosity approximations of extended traffic flow models |
| title_full_unstemmed | L^1 singular limit for relaxation and viscosity approximations of extended traffic flow models |
| title_short | L^1 singular limit for relaxation and viscosity approximations of extended traffic flow models |
| title_sort | l 1 singular limit for relaxation and viscosity approximations of extended traffic flow models |
| topic | Singular limit traffic flow model relaxation and viscosity approximation. |
| url | http://ejde.math.txstate.edu/Volumes/2003/23/abstr.html |
| work_keys_str_mv | AT christianklingenberg l1singularlimitforrelaxationandviscosityapproximationsofextendedtrafficflowmodels AT yunguanglu l1singularlimitforrelaxationandviscosityapproximationsofextendedtrafficflowmodels AT huijiangzhao l1singularlimitforrelaxationandviscosityapproximationsofextendedtrafficflowmodels |