ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO CONSERVATION LAWS WITH PERIODIC FORCING

2006 ◽  
Vol 03 (02) ◽  
pp. 387-401 ◽  
Author(s):  
DEBORA AMADORI ◽  
DENIS SERRE
Keyword(s):  
Asymptotic Behavior ◽  
Steady State ◽  
Initial Data ◽  
Conservation Law ◽  
Inflection Point ◽  
Global Minimum ◽  
Asymptotic Behavior Of Solutions ◽  
Periodic Forcing ◽  
Scalar Conservation Law ◽  
Scalar Conservation

We consider the asymptotic behavior of the solution of a forced scalar conservation law, where both the forcing and the initial data are periodic. We prove that there exists a steady state toward which the solution converges in the L1 norm, as t → ∞. We do not assume any smallness or smoothness of the initial data. The limit steady state can be discontinuous, as effect of resonance, and it can be identified when the potential of the forcing term has a unique global minimum, thanks to the conservation of mass. The flux is assumed to be strictly convex; the relevance of this assumption is justified by the construction of solutions with a lattice of periods for a flux with an inflection point.

ASYMPTOTIC SHAPES OF SHOCK CURVES FOR SOME CRITICAL CASES

2013 ◽  
Vol 10 (03) ◽  
pp. 415-430
Author(s):  
TIANHONG LI ◽  
XING LI
Keyword(s):  
Initial Data ◽  
Conservation Law ◽  
Critical Case ◽  
Asymptotic Curves ◽  
Scalar Conservation Law ◽  
Scalar Conservation

We consider the scalar conservation law with convex flux. When the antiderivative Φ of the initial data satisfies Φ(∞) > inf y∈ℝΦ(y), Lax obtained that asymptotic shapes of shock curves at the far ends are [Formula: see text] with some constant C. In this paper, we consider the critical case when Φ(∞) = inf y∈ℝΦ(y), and show that asymptotic shapes of shock curves are [Formula: see text] instead of [Formula: see text]. We give the formula for asymptotic curves, and provide examples showing rich phenomena of the critical case.

scholarly journals Smooth Approximations of Global in Time Solutions to Scalar Conservation Laws

2009 ◽  
Vol 2009 ◽  
pp. 1-26 ◽  
Author(s):  
V. G. Danilov ◽  
D. Mitrovic
Keyword(s):  
Approximate Solution ◽  
Initial Data ◽  
Conservation Laws ◽  
Asymptotic Method ◽  
Nonlinear Waves ◽  
Conservation Law ◽  
Scalar Conservation Laws ◽  
Scalar Conservation Law ◽  
Scalar Conservation ◽  
Smooth Approximations

We construct global smooth approximate solution to a scalar conservation law with arbitrary smooth monotonic initial data. Different kinds of singularities interactions which arise during the evolution of the initial data are described as well. In order to solve the problem, we use and further develop the weak asymptotic method, recently introduced technique for investigating nonlinear waves interactions.

A Hölder continuous ODE related to traffic flow

2003 ◽  
Vol 133 (4) ◽  
pp. 759-772 ◽  
Author(s):  
Rinaldo M. Colombo ◽  
Andrea Marson
Keyword(s):  
Vector Field ◽  
Initial Data ◽  
Differential Equations ◽  
Traffic Flow ◽  
Conservation Law ◽  
Continuous Dependence ◽  
Well Posedness ◽  
Hölder Continuous ◽  
Scalar Conservation Law ◽  
Scalar Conservation

This paper is devoted to the proof of the well posedness of a class of ordinary differential equations (ODEs). The vector field depends on the solution to a scalar conservation law. Forward uniqueness of Filippov solutions is obtained, as well as their Hölder continuous dependence on the initial data of the ODE. Furthermore, we prove the continuous dependence in C0 of the solution to the ODE from the initial data of the conservation law in L1.This problem is motivated by a model of traffic flow.

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