Well-posedness of general boundary-value problems for scalar conservation laws

Abstract:

In this paper we investigate well-posedness for the problem $u_t+ \operatorname {div} \varphi (u)=f$ on $(0,T)\!\times \!\Omega$, $\Omega \subset \mathbb {R}^N$, with initial condition $u(0,\cdot )=u_0$ on $\Omega$ and with general dissipative boundary conditions $\varphi (u)\cdot \nu \in \beta _{(t,x)}(u)$ on $(0,T)\!\times \!\partial \Omega$. Here for a.e. $(t,x)\in (0,T)\!\times \!\partial \Omega$, $\beta _{(t,x)}(\cdot )$ is a maximal monotone graph on $\mathbb {R}$. This includes, as particular cases, Dirichlet, Neumann, Robin, obstacle boundary conditions and their piecewise combinations.

As for the well-studied case of the Dirichlet condition, one has to interpret the formal boundary condition given by $\beta$ by replacing it with the adequate effective boundary condition. Such effective condition can be obtained through a study of the boundary layer appearing in approximation processes such as the vanishing viscosity approximation. We claim that the formal boundary condition given by $\beta$ should be interpreted as the effective boundary condition given by another monotone graph $\tilde \beta$, which is defined from $\beta$ by the projection procedure we describe. We give several equivalent definitions of entropy solutions associated with $\tilde \beta$ (and thus also with $\beta$).

For the notion of solution defined in this way, we prove existence, uniqueness and $L^1$ contraction, monotone and continuous dependence on the graph $\beta$. Convergence of approximation procedures and stability of the notion of entropy solution are illustrated by several results.