Linear Advection with Ill-Posed Boundary Conditions via $L^1$-Minimization

Abstract

It is proven that in dimension one the piecewise linear best $L^1$-approximation to the linear transport equation equipped with a set of ill-posed boundary conditions converges in $W_{loc}^{1,1}$ to the viscosity solution of the equation and the boundary layer associated with the ill-posed boundary condition is always localized in one mesh cell, i.e., the "last" one.