arXiv subjects

George Avalos

Publications and source records attributed to George Avalos.

At least 19 recordsLinked to original sources

arXiv2026

Semigroup Solutions for A Multilayered Filtration System

We investigate solutions to a coupled system of partial differential equations that describe a multilayered filtration system. Namely, we study the interaction of a viscous incompressible flow with bulk poroelasticity, via a poroelastic interface. The configuration consists of two 3D toroidal subdomains connected via a plate interface, which permits elastic deformation and perfusive fluid dynamics. The governing dynamics comprise Stokes equations in the bulk fluid region, Biot's equations in the bulk poroelastic region, and the recent poroplate of Mikeli\'c at the interface. Coupling occurs on the top and lower surfaces of the plate, and involves conservation of mass, stress balance, and a certain slip condition for the fluid free-flow. We seek strong (and mild) solutions in the Hilbert space framework via the Lumer-Phillips theorem. The resolvent analysis employs a nonstandard mixed variational formulation which captures the complex, multi-physics coupling at the interface. We explicitly characterize the infinitesimal generator associated to the linear Cauchy problem and establish the generation of a $C_0$-semigroup on a suitably chosen finite-energy space. With the semigroup in hand, we may treat elastic nonlinearities for plate displacements through perturbation theory. These result parallel those for Biot-Stokes filtration systems, and complement the recently established weak solution theory for multilayer filtrations. The agency of the semigroup straightforwardly admits structural (plate) nonlinearity into the dynamics. Future stability and regularity analyses for multilayer filtrations are also made possible by these results, as well as a comparison of spectral and regularity properties between filtration configurations, and the elucidation of the mitigating poroplate dynamics as possibly regularizing and stabilizing.

math.AP↗

arXiv2025

A Revisiting of the Pressure Elimination for a Fluid-Structure PDE Interaction and Its Implications

In this paper we construct a novel technique for eliminating and recovering the pressure for a fluid-structure interaction model. This pressure elimination methodology is valid for general bounded Lipschitz domains. The specific fluid-structure interaction (FSI) that we consider is a well-known model of Stokes flow coupled to a system of linear elasticity, which constitutes a coupled parabolic-hyperbolic system. The coupling between the two distinct PDE dynamics occurs across a boundary interface, with each of the components evolving on its own distinct geometry, with the domains of each being Lipschitz. Our new pressure elimination technique admits of an explicit $C_{0}$-semigroup generator representation $\mathcal{A}: D(\mathcal{A}) \subset \mathbf{H} \to \mathbf{H}$, where $\mathbf{H}$ is the associated finite energy space of fluid-structure initial data. This leads to a novel proof of well-posedness in the explicit semigroup sense of the continuous PDE, now valid in general geometries. Subsequently, we illustrate an immediate consequence of our semigroup well-posedness result; namely a finite element method (FEM) with associated rates of convergence for a static version of the FSI, posed on polygonal domains.

math.AP↗

arXiv2025

A novel approach to study the wellposedness of the 3D fluid-2D plate interaction PDE System

We consider a certain fluid-structure interaction (FSI) system with a view of obtaining an alternative methodology for establishing its strongly continuous semigroup wellposedness. (Semigroup generation for this FSI was originally considered in Avalos-Clark (2014).) The FSI model under consideration describes the vibrations of an incompressible fluid within a 3D cavity as it interacts with the elastic membrane on the ``free" upper boundary of the cavity. Such coupled PDE systems appear in variety of natural settings such as biomedicine, aeroelasticity, and fluid dynamics. Our proof of $C_0$-semigroup wellposedness is based on a proper application of Lumer Phillips Theorem. In this regard, our main challenge is to show the maximality of the corresponding semigroup generator. To this end, we develop a ``nonstandard" inf-sup approach which avoids the use of technical nonlocal maps in the associated bilinear forms--unlike the earlier paper Avalos-Clark (2014)--and allows for the solution of the fluid and plate solution variables simultanously. Our new inf-sup strategy will lead to a more efficient mixed finite element method (FEM) for approximating solutions to the FSI problem, inasmuch our novel variational formulation avoids bilinear forms which are free from the computationally-intensive nonlocal solution operators invoked in Avalos-Clark (2014). We also perform numerical tests based on this formulation using a benchmark problem and present numerical results to demonstrate the effectiveness of our approach.

math.AP↗

arXiv2025

Uniqueness of Weak Solutions for Biot-Stokes Interactions

We resolve the issue of uniqueness of weak solutions for linear, inertial fluid-poroelastic-structure coupled dynamics. The model comprises a 3D Biot poroelastic system coupled to a 3D incompressible Stokes flow via a 2D interface, where kinematic, stress-matching, and tangential-slip conditions are prescribed. Our previous work provided a construction of weak solutions, these satisfying an associated finite energy inequality. However, several well-established issues related to the dynamic coupling, hinder a direct approach to obtaining uniqueness and continuous dependence. In particular, low regularity of the hyperbolic (Lamé) component of the model precludes the use of the solution as a test function, which would yield the necessary a priori estimate. In considering degenerate and non-degenerate cases separately, we utilize two different approaches. In the former, energy estimates are obtained for arbitrary weak solutions through a systematic decoupling of the constituent dynamics, and well-posedness of weak solutions is inferred. In the latter case, an abstract semigroup approach is utilized to obtain uniqueness via a precise characterization of the adjoint of the dynamics operator. The results here can be adapted to other systems of poroelasticity, as well as to the general theory of weak solutions for hyperbolic-parabolic coupled systems.

math.AP↗

arXiv2024

Gevrey Regularity for a Fluid-Structure Interaction Model

A result of Gevrey regularity is ascertained for a semigroup which models a fluid-structure interaction problem. In this model, the fluid evolves in a piecewise smooth or convex geometry $\mathcal{O}$. On a portion of the boundary, a fourth order plate equation is coupled with the fluid through pressure and matching velocities. The key to obtaining the conclusion of Gevrey regularity is an appropriate estimation of the resolvent of the associated $C_0$-semigroup operator. Moreover, a numerical scheme and example is provided which empirically demonstrates smoothing of the fluid-structure semigroup.

math.AP↗

arXiv2024

Weak and Strong Solutions for A Fluid-Poroelastic-Structure Interaction via A Semigroup Approach

A filtration system, comprising a Biot poroelastic solid coupled to an incompressible Stokes free-flow, is considered in 3D. Across the flat 2D interface, the Beavers-Joseph-Saffman coupling conditions are taken. In the inertial, linear, and non-degenerate case, the hyperbolic-parabolic coupled problem is posed through a dynamics operator on an appropriate energy space, adapted from Stokes-Lamé coupled dynamics. A semigroup approach is utilized to circumvent issues associated to mismatched trace regularities at the interface. $C_0$-semigroup generation for the dynamics operator is obtained with a non-standard maximality argument. The latter employs a mixed-variational formulation in order to invoke the Babuška-Brezzi theorem. The Lumer-Philips theorem yields semigroup generation, and thereby, strong and generalized solutions are obtained. As the dynamics are linear, a standard argument by density obtains weak solutions; we extend this argument to the case where the Biot compressibility of constituents degenerates. Thus, for the inertial Biot-Stokes filtration, we provide a clear elucidation of strong and weak solutions, as well as their regularity through associated estimates.

math.AP↗

arXiv2022

An inf-sup approach to semigroup wellposedness for a compressible flow - incompressible fluid interactive PDE system

This work presents qualitative and numerical results on a system of partial differential equations (PDEs) which models certain fluid-fluid interaction dynamics. This system models a compressible fluid in a domain $Ω^+ \subset \mathbb{R}^2$, coupled to an incompressible fluid modeled by Stokes flow in domain $Ω^- \subset \mathbb{R}^2$, with the strong coupling implemented through certain boundary conditions on the shared interface, $Γ$. The wellposedness of this system is established by means of constructing for it a semigroup generator representation. This representation is accomplished by eliminating one of the pressure variables via identifying it as the solution of a certain boundary value problem, while the wellposedness is established via a nonstandard usage of the Babuska-Brezzi Theorem. In later sections, we demonstrate how the earlier constructive proof of wellposedness naturally lends itself to a certain finite element method (FEM), by which to numerically approximate solutions of the given coupled PDE system. This FEM is provided with error estimates and associated rates of convergence.

math.AP↗

arXiv2022

Rational Decay of A Multilayered Structure-Fluid PDE System

In this work, we consider a certain multilayered (thick layer) wave--(thin layer) wave--heat (fluid) interactive PDE system. Such coupled PDE systems have been used in the literature to describe the blood transport process in mammalian vascular systems. In particular, the deformations of the boundary interface (thin layer) are described via the two dimensional elastic equation. The present work constitutes an investigation of the extent of the stabilizing effects of the underlying fluid dissipation -- across the boundary interface -- upon both the thick and thin structural components. (All three PDE components evolve on their respective geometries.) In this regard, our main result is the derivation of uniform decay rates for classical solutions of this multilayered PDE model. To obtain these estimates, necessary a priori inequalities for certain static multilayered PDE models are generated here to ultimately allow an application of a wellknown resolvent criterion for rational decay.

math.AP↗

arXiv2021

A Resolvent Criterion Approach to Strong Decay of a Multilayered Lamé-Heat System

We consider a multilayer hyperbolic-parabolic PDE system which constitutes a coupling of 3D thermal - 2D elastic - 3D elastic dynamics, in which the boundary interface coupling between 3D fluid and 3D structure is realized via a 2D elastic equation. Our main result here is one of strong decay for the given multilayered - heat system. That is, the solution to this composite PDE system is stabilized asymptotically to the zero state. Our proof of strong stability takes place in the "frequency domain" and ultimately appeals to the pointwise resolvent condition introduced by Tomilov [45]. This very useful result, however, requires that the semigroup associated with our multilayered FSI system be completely non-unitary (c.n.u). Accordingly, we firstly establish that the semigroup $\{e^{\mathcal {A}t}\}_{t\geq 0}$ is indeed c.n.u., in part by invoking relatively recent results of global uniqueness for overdetermined Lamé systems on nonsmooth domains. Although the entire proof also requires higher regularity results for some trace terms, this \textit{"resolvent criterion approach"} allows us to establish a "classially soft" proof of strong decay. In particular, it avoids the sort of technical PDE multipliers invoked in [9].

math.AP↗

arXiv2019

Wellposedness, Spectral Analysis and Asymptotic Stability of a Multilayered Heat-Wave-Wave System

n this work we consider a multilayered heat-wave system where a 3-D heat equation is coupled with a 3-D wave equation via a 2-D interface whose dynamics is described by a 2-D wave equation. This system can be viewed as a simplification of a certain fluid-structure interaction (FSI) PDE model where the structure is of composite-type; namely it consists of a \textquotedblleft thin\textquotedblright\ layer and a \textquotedblleft thick\textquotedblright\ layer. We associate the wellposedness of the system with a strongly continuous semigroup and establish its asymptotic decay. Our first result is semigroup well-posedness for the (FSI) PDE dynamics. Utilizing here a Lumer-Phillips approach, we show that the fluid-structure system generates a $C_0$-semigroup on a chosen finite energy space of data. As our second result, we prove that the solution to the (FSI) dynamics generated by the $C_0$-semigroup tends asymptotically to the zero state for all initial data. That is, the semigroup of the (FSI) system is strongly stable. For this stability work, we analyze the spectrum of the generator $\mathbf A$ and show that the spectrum of $\mathbf A$ does not intersect the imaginary axis. \vskip.3cm \noindent \textbf{Key terms:} Fluid-structure interaction, heat-wave system, well-posedness, semigroup, strong stability.

math.AP↗

arXiv2018

A Linearized Viscous, Compressible Flow-Plate Interaction with Non-dissipative Coupling

We address semigroup well-posedness for a linear, compressible viscous fluid interacting at its boundary with an elastic plate. We derive the model by linearizing the compressible Navier-Stokes equations about an arbitrary flow state, so the fluid PDE includes an ambient flow profile $\mathbf{U}$. In contrast to model in [Avalos, Geredeli, Webster, 2017], we track the effect of this term at the flow-structure interface, yielding a velocity matching condition involving the material derivative of the structure; this destroys the dissipative nature of the coupling of the dynamics. We adopt here a Lumer-Phillips approach, with a view of associating fluid-structure solutions with a $C_{0}$-semigroup $\left\{e^{\mathcal{A}t}\right\} _{t\geq 0}$ on a chosen finite energy space of data. Given this approach, the challenge becomes establishing the maximal dissipativity of an operator $\mathcal{A}$, yielding the flow-structure dynamics.

math.AP↗

arXiv2017

Stability Analysis of Coupled Structural Acoustics PDE Models under Thermal Effects and with no Additional Dissipation

In this study we consider a coupled system of partial differential equations (PDE's) which describes a certain structural acoustics interaction. One component of this PDE system is a wave equation, which serves to model the interior acoustic wave medium within a given three dimensional chamber $% Ω$. This acoustic wave equation is coupled on a boundary interface ($% Γ_{0}$) to a two dimensional system of thermoelasticity: this thermoelastic PDE comprises a structural beam or plate equation, which governs the vibrations of flexible wall portion $Γ_{0}$ of the chamber $Ω$; the elastic dynamics is coupled to a heat equation which also evolves on $Γ_{0}$, and which imparts a thermal damping onto the entire structural acoustic system. As we said, the interaction between the wave and thermoelastic PDE components takes place on the boundary interface $% Γ_{0}$, and involves coupling boundary terms which are above the level of finite energy. We analyze the stability properties of this coupled structural acoustics PDE model, in the absence of \ any additive feedback dissipation on the hard walls $Γ_{1}$ of the boundary $\partial Ω$. Under a certain geometric assumption on $Γ_{1}$, an assumption which has appeared in the literature in conection with structural acoustic flow, and which allows for the invocation of a recently derived microlocal boundary trace estimate, we show that classical solutions of this thermally damped structural acoustics PDE decay uniformly to zero, with a rational rate of decay.

math.AP↗

arXiv2017

Exponential Stability and Supporting Spectral Analysis of a Linearized Compressible Flow-Structure PDE Model

In this work, a result of exponential stability is obtained for solutions of a compressible flow-structure partial differential equation (PDE) model which has recently appeared in the literature. In particular, a compressible flow PDE and its associated state equation for the associated pressure variable, each evolving within a three dimensional domain $\mathcal{O}$, are coupled to a fourth order plate equation which holds on a flat portion $% Ω$ of the boundary $\partial \mathcal{O}$. Moreover, since this coupled PDE model is the result of a linearization of the compressible Navier-Stokes equations about an arbitrary state, the flow PDE component contains a generally nonzero ambient flow profile $\mathbf{U}$. By way of obtaining the aforesaid exponential stability, a \textquotedblleft frequency domain\textquotedblright\ approach is adopted here, an approach which is predicated on obtaining a uniform estimate on the resolvent of the associated flow-structure semigroup generator.

math.AP↗

arXiv2017

Semigroup Well-posedness of A Linearized, Compressible Fluid with An Elastic Boundary

We address semigroup well-posedness of the fluid-structure interaction of a linearized compressible, viscous fluid and an elastic plate (in the absence of rotational inertia). Unlike existing work in the literature, we linearize the compressible Navier-Stokes equations about an arbitrary state (assuming the fluid is barotropic), and so the fluid PDE component of the interaction will generally include a nontrivial ambient flow profile $ \mathbf{U}$. The appearance of this term introduces new challenges at the level of the stationary problem. In addition, the boundary of the fluid domain is unavoidably Lipschitz, and so the well-posedness argument takes into account the technical issues associated with obtaining necessary boundary trace and elliptic regularity estimates. Much of the previous work on flow-plate models was done via Galerkin-type constructions after obtaining good a priori estimates on solutions (specifically \cite {Chu2013-comp}---the work most pertinent to ours here); in contrast, we adopt here a Lumer-Phillips approach, with a view of associating solutions of the fluid-structure dynamics with a $C_{0}$-semigroup $\left\{ e^{ \mathcal{A}t}\right\} _{t\geq 0}$ on the natural finite energy space of initial data. So, given this approach, the major challenge in our work becomes establishing of the maximality of the operator $\mathcal{A}$ which models the fluid-structure dynamics. In sum: our main result is semigroup well-posedness for the fully coupled fluid-structure dynamics, under the assumption that the ambient flow field $ \mathbf{U}\in \mathbf{H}^{3}(\mathcal{O})$ has zero normal component trace on the boundary (a standard assumption with respect to the literature). In the final sections we address well-posedness of the system in the presence of the von Karman plate nonlinearity, as well as the stationary problem associated with the dynamics.

math.AP↗

arXiv2015

Stability analysis of degenerately-damped oscillations

Presented here is a study of well-posedness and asymptotic stability of a "degenerately damped" PDE modeling a vibrating elastic string. The coefficient of the damping may vanish at small amplitudes thus weakening the effect of the dissipation. It is shown that the resulting dynamical system has strictly monotonically decreasing energy and uniformly decaying lower-order norms, however, is not uniformly stable on the associated finite-energy space. These theoretical findings were motivated by numerical simulations of this model using a finite element scheme and successive approximations. A description of the numerical approach and sample plots of energy decay are supplied. In addition, for certain initial data the solution can be determined in closed form up to a dissipative nonlinear ordinary differential equation. Such solutions can be used to assess the accuracy of the numerical examples.

math.AP↗

arXiv2014

Backward Uniqueness for a PDE Fluid-Structure Interaction

In this work, we establish the so-called backward uniqiueness property for a coupled system of partial differential equations (PDEs) which governs a certain fluid-structure interaction. In particular, a three-dimensional Stokes flow interacts across a boundary interface with a two-dimensional mechanical plate equation. By way of attaining this result, a certain estimate is obtained for the associated semigroup generator resolvent.

math.AP↗

arXiv2014

Exponential decay properties of a mathematical model for a certain fluid-structure interaction

In this work, we derive a result of exponential stability for a coupled system of partial differential equations (PDEs) which governs a certain fluid-structure interaction. In particular, a three-dimensional Stokes flow interacts across a boundary interface with a two-dimensional mechanical plate equation. In the case that the PDE plate component is rotational inertia-free, one will have that solutions of this fluid-structure PDE system exhibit an exponential rate of decay. By way of proving this decay, an estimate is obtained for the resolvent of the associated semigroup generator, an estimate which is uniform for frequency domain values along the imaginary axis. Subsequently, we proceed to discuss relevant point control and boundary control scenarios for this fluid-structure PDE model, with an ultimate view to optimal control studies on both finite and infinite horizon. (Because of said exponential stability result, optimal control of the PDE on time interval $(0,\infty)$ becomes a reasonable problem for contemplation.)

math.AP↗

arXiv2014

A Mixed Variational Formulation for the Wellposedness and Numerical Approximation of a PDE Model Arising in a 3-D Fluid-Structure Interation

We will present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. The wellposedness of this PDE model is established by means of constructing for it a nonstandard semigroup generator representation; this representation is essentially accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain $\mathcal{O}$ being coupled to a fourth order plate equation, possibly with rotational inertia parameter $ρ>0$, which evolves on a flat portion $Ω$ of the boundary of $\mathcal{O}$. The coupling on $Ω$ is implemented via the Dirichlet trace of the Stokes system fluid variable - and so the no-slip condition is necessarily not in play - and via the Dirichlet boundary trace of the pressure, which essentially acts as a forcing term on this elastic portion of the boundary. We note here that inasmuch as the Stokes fluid velocity does not vanish on $Ω$, the pressure variable cannot be eliminated by the classic Leray projector; instead, the pressure is identified as the solution of a certain elliptic boundary value problem. Eventually, wellposedness of this fluid-structure dynamics is attained through a certain nonstandard variational (``inf-sup") formulation. Subsequently we show how our constructive proof of wellposedness naturally gives rise to a certain mixed finite element method for numerically approximating solutions of this fluid-structure dynamics.

math.AP↗