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Programma anno 2021
Seminario Matematico e Fisico di Milano
Irena Lasiecka
University of Memphis
JMGT [Jordan-Moore Gibson-Thompson] dynamics arising in non- linear acoustics - a view from the boundary

Lunedì 01 Febbraio 2021, ore 15:00
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A third-order (in time) JMGT equation is a nonlinear (quasi-linear) Partial Differential Equation (PDE) model introduced to describe a non-linear propagation of high frequency acoustic waves. The interest in studying this type of problems is motivated by a large array of applications arising in engineering and medical sciences-including high intensity focused ultrasound [HIFU] technologies, lithotripsy, welding and others. The important feature is that the model avoids the infinite speed of propagation paradox associated with a classical second order in time equation referred to as Westervelt equation. Replacing a classical heat transfer by heat waves gives rise to the third order in time derivative scaled by a small parameter $\tau > 0$, the latter represents the thermal relaxation time parameter and is intrinsic to the properties of the medium where the dynamics occurs. The aim of the present lecture is to provide a brief overview of recent results in the area which are pertinent to both linear and non-linear dynamics. From the mathematical point of view JMGT, can be seen as a nonlinear perturbation of a third order strictly hyperbolic system, which however has a characteristic boundary. This feature has, of course, strong implications on boundary behavior [both regularity and controllability] which can not be patterned after classical hyperbolic systems theory [as it is the case for the wave equation]. As a consequence, the analysis of regularity [both forward and inverse estimates] is particularly challenging-even in the linear case. Several recent results pertaining to boundary stabilization, optimal control and asymptotic analysis of the solutions with vanishing time relaxation parameter will be presented and discussed. In all these case, peculiar features associated with the third order dynamics leads to novel phenomenological behaviors.
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