Gilbert Peralta

Local well-posedness of a class of hyperbolic PDE–ODE systems on a bounded interval

The well-posedness theory for hyperbolic systems of first-order quasilinear PDEs with ODEs boundary conditions (on a bounded interval) is discussed. Such systems occur in multi-scale blood flow models, as well as valveless pumping and fluid mechanics. The theory is presented in the setting of Sobolev spaces Hm (m ≥ 3 being an integer), which is an appropriate set-up when it comes to proving existence of smooth solutions using energy estimates. A blow-up criterion is also derived, stating that if the maximal time of existence is finite, then the state leaves every compact subset of the hyperbolicity region, or its first-order derivatives blow-up. Finally, we discuss physical examples which fit in the general framework presented.

Stabilization of Viscoelastic Wave Equations with Distributed or Boundary Delay

The wave equation with viscoelastic boundary damping and internal or boundary delay is considered. The memory kernel is assumed to be integrable and completely monotonic. Under certain conditions on the damping factor, delay factor and the memory kernel it is shown that the energy of the solutions decay to zero either asymptotically or exponentially. In the case of internal delay, the result is obtained through spectral analysis and the Gearhart-Pruss Theorem, whereas in the case of boundary delay, it is obtained using the energy method.

Well-posedness and regularity of linear hyperbolic systems with dynamic boundary conditions

We consider first-order hyperbolic systems on an interval with dynamic boundary conditions. These systems occur when the ordinary differential equation dynamics on the boundary interact with the waves in the interior. The well-posedness for linear systems is established using an abstract Friedrichs theorem. Due to the limited regularity of the coefficients, we need to introduce the appropriate space of test functions for the weak formulation. It is shown that the weak solutions exhibit a hidden regularity at the boundary as well as at interior points. As a consequence, the dynamics of the boundary components satisfy an additional regularity. Neither result can be achieved from standard semigroup methods. Nevertheless, we show that our weak solutions and the semigroup solutions coincide. For illustration, we give three particular physical examples that fit into our framework.

Feedback Stabilization of a Linear Fluid–Membrane System with Time Delay

A coupled parabolic–hyperbolic system of partial differential equations modelling the interaction of a fluid and a membrane is considered. The model is reformulated as an abstract Cauchy problem and thereby constructing a semigroup for the evolution. This is done by eliminating the pressure. The system is stabilized through a feedback force applied to the membrane incorporating a time delay. The spectral properties and stability are considered under suitable conditions on the fluid viscosity, damping coefficient and delay coefficient.