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- W2056593185 abstract "In this paper we investigate well-posedness for the problem $u_t+ div ph(u)=f$ on $(0,T)!times!Om$, $Om subset R^N$, with initial condition $u(0,cdot)=u_0$ on $Om$ and with general dissipative boundary conditions $varphi(u)cdot nu in beta_{(t,x)}(u)$ on $(0,T)!times!ptlOm$. Here for a.e. $(t,x)in(0,T)!times!ptlOm$, $beta_{(t,x)}(cdot)$ is a maximal monotone graph on $R$. 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 interprete the {it formal boundary condition} given by $beta$ by replacing it with the adequate {it 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." @default.
- W2056593185 created "2016-06-24" @default.
- W2056593185 creator A5039672326 @default.
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- W2056593185 date "2015-02-12" @default.
- W2056593185 modified "2023-10-18" @default.
- W2056593185 title "Well-posedness of general boundary-value problems for scalar conservation laws" @default.
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- W2056593185 doi "https://doi.org/10.1090/s0002-9947-2015-05988-1" @default.
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