A transport model with adsorption hysteresis
The paper deals with an initial-boundary value problem for the transport equation \((u+v)_t + u_x = 0\), \(v = {\mathcal M}u\), \(u(0,t) = \varphi (t)\), \(u(x,0) = v(x,0) = 0\), \(t \in [0,T]\), \(x \in [0,L]\), where \(\mathcal M\) is a hysteresis operator, more specifically the play operator or its combinations (the so-called Prandtl-Ishlinskij operator). Considering alternatively \(x\) and \(t\) as the `time' variable and rewriting the problem as an abstract evolution equation of the form \(U' + AU \ni f\) in \(L^2 \times L^2\) with an \(m\)-accretive operator \(A\), the authors obtain two different existence, uniqueness and regularity results. In particular, as it is often the case in hyperbolic PDEs with convex hysteresis, shocks do not occur.
scientific article; zbMATH DE number 1453635