Applied Mathematics Colloquium: Justin Webster (UMBC)
Location
Mathematics/Psychology : 103
Date & Time
September 20, 2024, 11:00 am – 12:00 pm
Description
Title: Stability and Periodicity in PDE Systems
Abstract: Time-periodic behavior can arise in multi-physics systems as (i) a response to periodic forcing (blood flow in human tissues) or (ii) as emergent ``self-excitations" (aeroelastic flutter). Taking fluid-structure interactions as a motivation, we consider an idealized heat-wave interaction with physical coupling conditions across a lower-dimensional interface, with time-periodic forcing. For undamped hyperbolic systems, one must be careful to avoid resonance under such forcing. For parabolic systems, the theory of existence and uniqueness of periodic solutions is well-understood. Yet the heat-wave system is only partially damped, yielding an indeterminate case. Indeed, owing to the work of Galdi et al., there is a deep connection between the stability of the solution semigroup and the existence and uniqueness of periodic solutions in the presence of time-periodic forcing.
For the heat-wave system, the underlying semigroup demonstrates polynomial ("almost uniform") stability, but not exponential stability. Thus the periodic behaviors of the system are not characterized by the existing abstract theory of dissipative systems. In this talk, we provide an explicit construction of a unique periodic solution for the partially damped heat-wave system. Without dissipation in the wave component, wave a priori estimates must be reconstructed through the interface from the dissipative heat equation using boundary control estimates. This introduces geometric constraints, and we demonstrate that for certain classes of domains, unique periodic solutions can be obtained from temporally smooth forcing; we thereby eliminate the possibility of resonance. It remains an open question whether the geometric control constraints are technical (arising from the proof method), or if the restrictions are truly necessary to circumvent resonance. This is joint work with B. Muha and S. Schwarzacher.
We will have the Departmental Coffee and Tea from 10 to 10:45 in M&P 422.
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