The L1-Stability of Boundary Layers for Scalar Viscous Conservation Laws

Let v=v(x) be a non-trivial bounded steady solution of a viscous scalar conservation law u _t+f(u) _x =u _xx on a half-line R⁺, with a Dirichlet boundary condition. The semi-group of this IBVP is known to be contractive for the distance d(u, u)u–u₁ induced by L ¹(R⁺). We prove here that v is asymptotically stable with respect to d: if u ₀–vL ¹, then u(t)–v₁0 as t+. When v is a constant, we show that this property holds if and only if f(v)0. These results complement our study of the Cauchy problem 2].