Journals and Conference ProceedingsThe Journals and Conference Proceedings Collection includes publications reviewed and published by members of the university community and proceedings from conferences hosted by, or sponsored by, Texas State University.https://digital.library.txstate.edu/handle/10877/1362020-10-01T02:23:17Z2020-10-01T02:23:17ZL^1 Singular Limit for Relaxation and Viscosity Approximations of Extended Traffic Flow ModelsKlingenberg, ChristianLu, Yun-guangZhao, Hui-jianghttps://digital.library.txstate.edu/handle/10877/126402020-09-21T20:49:54Z2003-03-07T00:00:00ZL^1 Singular Limit for Relaxation and Viscosity Approximations of Extended Traffic Flow Models
Klingenberg, Christian; Lu, Yun-guang; Zhao, Hui-jiang
This paper considers the Cauchy problem for an extended traffic flow model with L1-bounded initial data. A solution of the corresponding equilibrium equation with L1-bounded initial data is given by the limit of solutions of viscous approximations of the original system as the dissipation parameter ∈ tends to zero more slowly than the response time τ. The proof of convergence is obtained by applying the Young measure to solutions introduced by DiPerna and, based on the estimate |p(t, x)| ≤ √|p0(x)|1/(ct) derived from one of Lax's results and Diller's idea, the limit function p(t, x) is shown to be a L1-entropy week solution. A direct byproduct is that we can get the existence of L1-entropy solutions for the Cauchy problem of the scalar conservation law with L1-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 L1-bounded initial data, the concentration phenomenon will never occur in its global entropy solutions.
2003-03-07T00:00:00ZNon-classical phase transitions at a sonic pointSable-Tougeron, Moniquehttps://digital.library.txstate.edu/handle/10877/126132020-09-15T18:55:53Z2003-03-07T00:00:00ZNon-classical phase transitions at a sonic point
Sable-Tougeron, Monique
The relevant mathematical features of phase transition for a general hyperbolic nonlinear system near a sonic discontinuity are clarified. A well-posed Riemann's problem is obtained, including non-classical undercompressive shocks, defined by a geometrical kinetic relation. A counterpart is the geometrical rejection of some compressive shocks. The result is consistent with the structure profiles of the elasticity model of Shearer-Yang and the combustion model of Majda.
2003-03-07T00:00:00ZMinimal and maximal solutions for two-point boundary-value problemsGrammatikopoulos, Myron K.Kelevedjiev, Petio S.https://digital.library.txstate.edu/handle/10877/126122020-09-15T18:49:05Z2003-02-28T00:00:00ZMinimal and maximal solutions for two-point boundary-value problems
Grammatikopoulos, Myron K.; Kelevedjiev, Petio S.
In this article we consider a boundary-value problem for the equation f (t, x, x', x'') = 0 with mixed boundary conditions. Assuming the existence of suitable barrier strips, and using the monotone iterative method, we obtain the minimal and maximal solutions.
2003-02-28T00:00:00ZBlow-up for p-Laplacian parabolic equationsLi, YuxiangXie, Chunhonghttps://digital.library.txstate.edu/handle/10877/126112020-09-14T21:17:39Z2003-02-28T00:00:00ZBlow-up for p-Laplacian parabolic equations
Li, Yuxiang; Xie, Chunhong
In this article we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem ut = ∇(|∇u| p - 2∇u) + λ|u| q - 2u, in ΩT, where p > 1. In particular, for p > 2, q = p is the blow-up critical exponent and we show that the sharp blow-up condition involves the first eigenvalue of the problem -∇(|∇ψ| p - 2 ∇ψ) = λ|ψ| p - 2ψ, in Ω; ψ|∂Ω = 0.
2003-02-28T00:00:00Z