Analysis of a Scalar Conservation Law with Space Discontinuous Advection Function in a Bounded Domain

Julien Jimenez (1) , Laurent Levi (1) , Monique Madaune-Tort (1)

1 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We deal with the scalar conservation law in a one dimensional bounded domain : $\Omega: \partial_t u + \partial_x(k(x)g(u)) = 0$, associated with a bounded initial value $u_0$. The function $k$ is supposed to be bounded, discontinuous at ${x_0 = 0}$, and with bounded variation. A weak entropy formulation for the Cauchy problem has been introduced by J.D Towers in [11]. In [10] the existence and the uniqueness is proved by N. Seguin and J. Vovelle through a regularization of the function $k$. We generalize the definition of J.D Towers and we adapt the method developed in [10] to establish an existence and uniqueness property in the case of the homogeneous Dirichlet boundary conditions.

Domaines: Equations aux dérivées partielles [math.AP]

Julien Jimenez, Laurent Levi, Monique Madaune-Tort. Analysis of a Scalar Conservation Law with Space Discontinuous Advection Function in a Bounded Domain. 2006. ⟨hal-00220437⟩