A transport model with adsorption hysteresis. (English) Zbl 1004.35033

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.


35F20 Nonlinear first-order PDEs
47H06 Nonlinear accretive operators, dissipative operators, etc.
47J35 Nonlinear evolution equations