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Analysis of small oscillations of a pendulum partially filled with a viscoelastic fluid
Essaouini H., Capodanno P.
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<abstract><p>This paper focuses on the spectral analysis of equations that describe the oscillations of a heavy pendulum partially filled with a homogeneous incompressible viscoelastic fluid. The constitutive equation of the fluid follows the simpler Oldroyd model. By examining the eigenvalues of the linear operator that describes the dynamics of the coupled system, it was demonstrated that under appropriate assumptions the equilibrium configuration remains stable in the linear approximation. Moreover, when the viscosity coefficient is sufficiently large the spectrum comprises three branches of eigenvalues with potential cluster points at $ 0 $, $ \beta $ and $ \infty $ where $ \beta $ represents the viscoelastic parameter of the fluid. These three branches of eigenvalues correspond to frequencies associated with various types of waves.</p></abstract>
Stokes-Dirac structures for distributed parameter port-Hamiltonian systems: An analytical viewpoint
Brugnoli A., Haine G., Matignon D.
<abstract><p>In this paper, we prove that a large class of linear evolution partial differential equations defines a Stokes-Dirac structure over Hilbert spaces. To do so, the theory of boundary control system is employed. This definition encompasses problems from mechanics that cannot be handled by the geometric setting given in the seminal paper by van der Schaft and Maschke in 2002. Many worked-out examples stemming from continuum mechanics and physics are presented in detail, and a particular focus is given to the functional spaces in duality at the boundary of the geometrical domain. For each example, the connection between the differential operators and the associated Hilbert complexes is illustrated.</p></abstract>
Conformal-type energy estimates on hyperboloids and the wave-Klein-Gordon model of self-gravitating massive fields
Duan S., MA Y., Zhang W.
<abstract><p>In this article we revisit the global existence result of the wave-Klein-Gordon model of the system of the self-gravitating massive field. Our new observation is that, by applying the conformal energy estimates on hyperboloids, we obtain mildly increasing energy estimate up to the top order for the Klein-Gordon component, which clarify the question on the hierarchy of the energy bounds of the Klein-Gordon component in our previous work. Furthermore, a uniform-in-time energy estimate is established for the wave component up to the top order, as well as a scattering result. These improvements indicate that the partial conformal symmetry of the Einstein-massive scalar system will play an important role in the global analysis.</p></abstract>
