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Abstract
We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions.
Highlights
There are different applicative models that lead to consider hyperbolic conservation laws with flux function discontinuous in the state space
G(u(t, 0−)) for almost every t, which is the Rankine–Hugoniot condition at x = 0. It provides the conservation of the quantity u through the discontinuity points
We will show that there exists a class of Riemann Solvers, which determine a unique compatible solution to the Cauchy problem depending in a Lipschitz fashion from initial data
Summary
There are different applicative models that lead to consider hyperbolic conservation laws with flux function discontinuous in the state space. We will show that there exists a class of Riemann Solvers, which determine a unique compatible solution to the Cauchy problem depending in a Lipschitz fashion from initial data Observe that this corresponds to a particular choice of the Riemann Solver at the discontinuity point Using this approach, Gimse proved existence and uniqueness for small time assuming regularity and monotonicity hypotheses for the solutions along the boundary x = 0. Motivations for entropy solutions are given starting from a model of two-phase flows in a porous medium In this case, undercompressive waves are not allowed, shocks can not enter simultaneously from both sides of the discontinuity point x = 0. In this case we prove uniqueness of solutions, using the doubling method of Kruzkov
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