Well-posedness of a Nonlinear Acoustics -- Structure Interaction Model

We establish local-in-time and global in time well-posedness for small data, for a coupled system of nonlinear acoustic structure interactions. The model consists of the nonlinear Westervelt equation on a bounded domain with non homogeneous boundary conditions, coupled with a 4th order linear equation defined on a lower dimensional interface occupying part of the boundary of the domain, with transmission boundary conditions matching acoustic velocities and acoustic pressures. While the well-posedness of the Westervelt model has been well studied in the literature, there has been no works on the literature on the coupled structure acoustic interaction model involving the Westervelt equation. Another contribution of this work, is a novel variational weak formulation of the linearized system and a consideration of various boundary conditions

On the interaction between compressible inviscid flow and an elastic plate

We address a free boundary model for the compressible Euler equations where the free boundary, which is elastic, evolves according to a weakly damped fourth order hyperbolic equation forced by the fluid pressure. This system captures the interaction of an inviscid fluid with an elastic plate. We establish a priori estimates on local-in-time solutions in low regularity Sobolev spaces, namely with velocity and density initial data v0,R0 in H3. The main new device is a variable coefficients space tangential-time differential operator of order 1 with non-homogeneous boundary conditions, which captures the hyperbolic nature of the compressible Euler equations as well as the coupling with the structural dynamics

The Mathematics of Fluids and Solids

Fluid-structure interaction is a rich and active field of mathematics that studies the interaction between fluids and solid objects. In this short article, we give a glimpse into this exciting field, as well as a sample of the most significant questions that mathematicians try to answer

First volumetric records of airborne Cladosporium and Alternaria spores in the atmosphere of Al Khor (northern Qatar): a preliminary survey

Daily monitoring of airborne fungal spores was carried out for the first time in Al Khor city, Qatar, using a Hirst type 7-day recording volumetric spore trap, from May 2017 to May 2019. During the sampling period, the annual and monthly fluctuations, as well as intradiurnal variations of airborne fungal spore concentrations, were evaluated. Cladosporium, followed by Alternaria, were the spore types most abundant in the atmosphere of the city, with a strong interannual variability in the atmospheric concentrations being observed. The Annual Spore Integrals (ASIns) were 3334 and 1172 spore * day/m3 (2017–2018), and 6796 and 1538 spore * day/m3 (2018–2019) for Cladosporium and Alternaria, respectively. Total daily spore concentrations showed significantly positive correlations with mean, minimum, and maximum temperatures but significantly negative correlations with relative humidity. However, due to the scarce rainfalls’ days, we did not find a statistically significant correlations between Cladosporium and Alternaria spore concentrations and this parameter. Despite this, the spore peaks were strongly related to precipitations that occurred during the previous month. In general, no significant correlations were found with wind speed but, regarding wind direction, the higher percentage of spores were collected when wind blows from the 4th quadrant (NW). According to the intradiurnal pattern, Cladosporium fungal spores displayed their maximum daily concentration during 8:00–10:00 h in the morning, with a second peak in the afternoon, while for Alternaria, the maximum peaks were observed between 08:00 and 14:00 h. Because no consistent previous aerobiological studies exist from Qatar, the aim of this study is to define the seasonality and intradiurnal behaviour of these two airborne fungal spore and the role that, in such arid scene, the meteorological parameters play on the spore concentrations.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Funding for open access charge: Universidad de Málaga / CBUA. We declare that the research reported in this manuscript received supported grant funding from the Qatar National Research Fund (QNRF)-Qatar (Project NPRP 9–241-3–043)

Differential Riccati equations for the Bolza problem associated with point boundary control of singular estimate control systems

Recent developments in boundary and point control theory have provided strong motivation for studying Riccati equations with unbounded coefficients. By unbounded coefficients we mean the coefficients of the Riccati equations that are given in terms of unbounded, and even uncloseable, operators. These typically result from unbounded control actions or unbounded observations. The unboundedness of the coefficients is, of course, a source of mathematical difficulties. Standard methods for establishing wellposedness to these nonlinear equations are no longer applicable. The problem is particularly acute when the coefficients in the nonlinear term of the Riccati equation are unbounded. This latter case corresponds to the unbounded control action. In these cases, the well-posedness of Riccati equations may fail altogether, as evidenced in [21]. However, the situation is very different when the dynamics is generated by an analytic semigroup. This is to say that eAt is a generator of an analytic semigroup on a given Hilbert space H. For this class of dynamics the Riccati theory is by now well understood. The regularizing effect of analyticity compensates for the unboundedness of the coefficients, leading to a well-posed nonlinear problem

Riccati theory and singular estimates for a Bolza control problem arising in linearized fluid-structure interaction

We consider a Bolza boundary control problem involving a fluid-structure interaction model. The aim of this paper is to develop an optimal feedback control synthesis based on Riccati theory. The model considered consists of a linearized Navier-Stokes equation coupled on the interface with a dynamic wave equation. The model incorporates convective terms resulting from the linearization of the Navier-Stokes equation around equilibrium. The existence of the optimal control and its feedback characterization via a solution to a Riccati equation is established. The main mathematical difficulty of the problem is caused by unbounded action of control forces which, in turn, result in Riccati equations with unbounded coefficients and in singular behavior of the gain operator. This class of problems has been recently studied via the so-called Singular Estimate Control Systems (SECS) theory, which is based on the validity of the Singular Estimate (SE) [G. Avalos, Differential Riccati equations for the active control of a problem in structural acoustic, J. Optim. Theory, Appl. 91 (1996) 695-728; I. Lasiecka, Mathematical Control Theory of Coupled PDE\u27s, in: NSF- CMBS Lecture Notes, SIAM, 2002. with Unbounded Controls; I. Lasiecka, A. Tuffaha, Riccati Equations for the Bolza Problem arising in boundary/point control problems governed by c0 semigroups satisfying a singular estimate, J. Optim. Theory Appl. 136 (2008) 229-246]. It is shown that the fluid-structure interaction does satisfy the Singular Estimate (SE) condition. This is accomplished by showing that the maximal abstract parabolic regularity is transported via hidden hyperbolic regularity of the boundary traces on the interface. Thus, the established Singular Estimate allows for the application of recently developed general theory which, in turn, implies well-posedness of the feedback synthesis and of the associated Riccati Equation. Moreover, the singularities in the optimal control and in the feedback operator at the terminal time are quantitatively described. © 2009 Elsevier B.V. All rights reserved

Optimal feedback synthesis for bolza control problem arising in linearized fluid structure interaction

Bolza boundary control problem defined for linearized fluid structure interaction model is considered. The aim of this paper is to develop an optimal feedback control synthesis based on Riccati theory. The main mathematical challenge of the problem is caused by unbounded action of control forces which, in turn, give rise to Riccati equations with unbounded coefficients and singular behavior of the gain operator. This class of problems has been recently studied via the so-called Singular Estimate Control Systems. (SECS) theory, which is based on the validity of the so-called Singular Estimate (SE) [4, 27, 32]. It is shown that the fluid structure interaction does satisfy Singular Estimate (SE) condition. This is accomplished by showing that the maximal abstract parabolic regularity is transported, onto the wave dynamics, via hidden hyperbolic regularity of the boundary traces on the interface. The established Singular Estimate allows for the application of recently developed general theory which, in turn, implies well-posedness of feedback synthesis and of the associated Riccati Equation. Blow up rates of optimal control and of the feedback operator at the terminal time are provided

Boundary feedback control in fluid-structure interactions

We consider a boundary control system for a Fluid Structure Interaction Model. This system describes the motion of an elastic structure inside a viscous fluid with interaction taking place at the boundary of the structure, and with the possibility of controlling the dynamics from this boundary. Our aim is to construct a real time feedback control based on a solution to a Riccati Equation. The difficulty of the problem under study is due to the unboundedness of the control action, which is typical in boundary control problems. However, this class of unbounded control systems, due to its physical relevance, has attracted a lot of attention in recent literature (cf. [5], [18], [11]). It is known that Riccati feedback (unbounded) controls may develop strong singularities which destroy the well-posedness of Riccati equations. This makes computational implementations problematic, to say the least. However, as shown recently, this pathology does not happen for certain classes of unbounded control systems usually referred to as Singular Estimate Control Systems (SECS) (cf. [11], [21]). For such systems, there is a full and optimal Riccati theory in place, which leads to the well-posedness of feedback dynamics. Our objective is to show that the boundary control problem in question falls in the class of Singular Estimate Control Systems (SECS). Once this is accomplished, an application of the theory in [21] leads to the main result of this paper which is well-posedness of Riccati equations and of the Riccati feedback synthesis. © 2008 IEEE