2019/10/18 by George Avalos, Avalos, George, Pelin G. Geredeli +3 · 1 citation
Mathematics · Engineering · #Advanced Mathematical Physics Problems #Navier-Stokes equation solutions #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.1910.08596
n this work we consider a multilayered heat-wave system where a 3-D heat\nequation is coupled with a 3-D wave equation via a 2-D interface whose dynamics\nis described by a 2-D wave equation. This system can be viewed as a\nsimplification of a certain fluid-structure interaction (FSI) PDE model where\nthe structure is of composite-type; namely it consists of a textquotedblleft\nthin textquotedblright layer and a textquotedblleft thick textquotedblright \nlayer. We associate the wellposedness of the system with a strongly continuous\nsemigroup and establish its asymptotic decay.\n Our first result is semigroup well-posedness for the (FSI) PDE dynamics.\nUtilizing here a Lumer-Phillips approach, we show that the fluid-structure\nsystem generates a C0-semigroup on a chosen finite energy space of data. As\nour second result, we prove that the solution to the (FSI) dynamics generated\nby the C0-semigroup tends asymptotically to the zero state for all initial\ndata. That is, the semigroup of the (FSI) system is strongly stable. For this\nstability work, we analyze the spectrum of the generator mathbf A and show\nthat the spectrum of mathbf A does not intersect the imaginary axis.\n vskip.3cm noindent \Key terms: Fluid-structure interaction, heat-wave\nsystem, well-posedness, semigroup, strong stability.\n