Searcharxiv⌕ Search

arXiv subjects

Justin T. Webster

Publications and source records attributed to Justin T. Webster.

At least 19 recordsLinked to original sources
arXiv2026

Weak Solutions and Inertial Limits for Quasi-static Filtrations

A quasi-static filtration system, comprising a poroelastic solid coupled to an incompressible free-flow, is considered in 3D. Across a flat 2D interface, the Beavers-Joseph-Saffman coupling conditions are taken. The system constitutes a doubly elliptic-parabolic coupling and can be seen as a degenerate case of the inertial Biot-Stokes dynamics. These dynamics cannot be easily recovered by simply sending the inertial parameters to zero; however, inserting a viscoelastic regularization of the inertial Biot system allows us to construct weak solutions in the inertial limit. Subsequently, we can pass to the limit in the regularization parameter to obtain quasi-static weak solutions. This addresses an open singular/degenerate limiting problem in the class of filtration models, and permits future analyses of uniqueness and regularity of solutions. This linear construction also provides a foundation for the later incorporation of physically-motivated nonlinear poroelastic effects via fixed point methods.

math.AP↗
arXiv2026

From Polynomial Stability to Periodic Well-posedness in Partially Dissipative Systems

The study of resonances (and well-posedness) for complex systems under time-periodic loading is of broad interest in application. The work of Galdi et al.~(2014) connects asymptotic stability of solutions to an unforced Cauchy problem to solvability of the time-periodic forced problem. Uniform stability of the solution semigroup gives periodic well-posedness for all forces in the natural mild forcing class, whereas strong stability yields only existence of a dense set of forcings for which resonance can be excluded. We address an intermediate regime for polynomial (also: rational or semiuniform) stability. Working with a Fourier decomposition in Hilbert space, we demonstrate that polynomial stability of the semigroup yields an explicit characterization of the dense forcing set on which periodic well-posedness holds. More precisely, resolvent bounds translate directly into certain losses of time derivatives on the forcing required to ensure well-posedness. Our result is motivated by partially dissipative models -- including the famous heat-wave interaction problem idealizing fluid-structure interactions, as well as some thermoelastic, viscoelastic, and weakly damped hyperbolic systems -- for which polynomial decay is the natural regime.

math.AP↗
arXiv2026

Steady weak solutions to an inflow/outflow driven compressible fluid-structure interaction problem

We study a stationary 3D/2D fluid-structure interaction problem between an elastic structure described by the linear plate equation and a fluid described by the compressible Navier-Stokes equations with hard-sphere pressure and inflow/outflow boundary data. This problem is motivated by wind-tunnel configuration and by the need for physically relevant steady states about which compressible flow-plate dynamics can be linearized. The main difficulty in the analysis is the lack of uniform estimates, both for approximate and weak solutions. In particular, the fixed-point construction for approximate solution yields a density estimate depending on approximate parameter, while the pressure estimate for the weak solution is only finite and non-quantifiable. As a result, large pressure loads can drive outward volume growth, while low pressure regions may lead to contact and therefore domain degeneration. This necessitates a novel approach based on a Lipschitz \emph{domain-correction} (barrier) mechanism that provides a framework in which solutions can be constructed without volume blow-up or degeneration of the domain. Constrained by the possibly very large fluid pressure load, our main result is the existence of a weak solution for a sufficiently large plate stiffness. Keywords: fluid-structure interaction, compressible Navier-Stokes, stationary weak solutions, hard-sphere pressure, inflow/outflow, linear plate, mathematical aeroelasticity

math.AP↗
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

Large Deflections of A Flow-Driven Cantilever with Kutta-Joukowski Flow Conditions

We consider a canonical flow-structure system modeling airflow over a cantilevered beam. Flow-beam interactions arise in flight systems as well as alternative energy technologies, such as piezoelectric energy harvesters. A potential flow, given through a hyperbolic equation, captures the airflow interacting with a beam clamped on one end and free on the other. The dynamic coupling occurs through an impermeability condition across the beam; in the wake the Kutta-Joukowski flow condition is imposed. Several challenges arise in the analysis, including the unboundedness of the flow domain, lack of interface trace regularity, and flow conditions which give rise to a dynamic and mixed boundary value problem. Additionally, we consider a recent nonlinear model capturing the cantilever large deflections through the effects of inextensibility. We produce a viable underlying semigroup for the model's linearization, which includes a flow regularity theory. Then, exploiting higher regularity nonlinear estimates for the beam, we utilize a semigroup perturbation to obtain local-in-time strong solutions for the nonlinear dynamics.

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 (Lam\'e) 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

Time-Periodic Solutions for Hyperbolic-Parabolic Systems

Time-periodic weak solutions for a coupled hyperbolic-parabolic system are obtained. A linear heat and wave equation are considered on two respective $d$-dimensional spatial domains that share a common $(d-1)$-dimensional interface $\Gamma$. The system is only partially damped, leading to an indeterminate case for existing theory (Galdi et al., 2014). We construct periodic solutions by obtaining novel a priori estimates for the coupled system, reconstructing the total energy via the interface $\Gamma$. As a byproduct, geometric constraints manifest on the wave domain which are reminiscent of classical boundary control conditions for wave stabilizability. We note a ``loss" of regularity between the forcing and solution which is greater than that associated with the heat-wave Cauchy problem. However, we consider a broader class of spatial domains and mitigate this regularity loss by trading time and space differentiations, a feature unique to the periodic setting. This seems to be the first constructive result addressing existence and uniqueness of periodic solutions in the heat-wave context, where no dissipation is present in the wave interior. Our results speak to the open problem of the (non-)emergence of resonance in complex systems, and are readily generalizable to related systems and certain nonlinear cases.

math.AP↗
arXiv2024

Resonance and Periodic Solutions for Harmonic Oscillators with General Forcing

We discuss the notion of resonance, as well as the existence and uniqueness of periodic solutions for a forced simple harmonic oscillator. While this topic is elementary, and well-studied for sinusoidal forcing, this does not seem to be the case when the forcing function is general (perhaps discontinuous). Clear statements of theorems and proofs do not readily appear in standard textbooks or online. For that reason, we provide a characterization of resonant solutions, written in terms of the relationship between the forcing and natural frequencies, as well as a condition on a particular Fourier mode. While our discussions involve some notions from $L^2$-spaces, our proofs are elementary, using this the variation of parameters formula; the main theorem and its proof should be readable by students who have completed a differential equations course and have some experience with analysis. We provide several examples, and give various constructions of resonant solutions. Additionally, we connect our discussion to notions of resonance in systems of partial differential equations, including fluid-structure interactions and partially damped systems.

math.CA↗
arXiv2024

Analysis of a nonlinear fish-bone model for suspension bridges with rigid hangers in the presence of flow effects

We consider a dynamic system of nonlinear partial differential equations modeling the motions of a suspension bridge. This fish-bone model captures the flexural displacements of the bridge deck's mid-line, and each chordal filament's rotation angle from the centerline. These two dynamics are strongly coupled through the effect of cable-hanger, appearing through a sublinear function. Additionally, a structural nonlinearity of Woinowsky-Krieger type is included, allowing for large displacements. Well-posedness of weak solutions is shown and long-time dynamics are studied. In particular, to force the dynamics, we invoke a non-conservative potential flow approximation which, although greatly simplified from the full multi-physics fluid-structure interaction, provides a driver for non-trivial end behaviors. We describe the conditions under which the dynamics are uniformly stable, as well as demonstrate the existence of a compact global attractor under all nonlinear and non-conservative effects. To do so, we invoke the theory of quasi-stability, first explicitly constructing an absorbing ball via stability estimates and, subsequently, demonstrating a stabilizability estimate on trajectory differences applied to the aforesaid absorbing ball. Finally, numerical simulations are performed to examine the possible end behaviors of the dynamics.

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↗
arXiv2023

Mathematical Effects of Linear Visco-elasticity in Quasi-static Biot Models

We investigate and clarify the mathematical properties of linear poro-elastic systems in the presence of classical (linear, Kelvin-Voigt) visco-elasticity. In particular, we quantify the time-regularizing and dissipative effects of visco-elasticity in the context of the quasi-static Biot equations. The full, coupled pressure-displacement presentation of the system is utilized, as well as the framework of implicit, degenerate evolution equations, to demonstrate such effects and characterize linear poro-visco-elastic systems. We consider a simple presentation of the dynamics (with convenient boundary conditions, etc.) for clarity in exposition across several relevant parameter ranges. Clear well-posedness results are provided, with associated a priori estimates on the solutions. In addition, precise statements of admissible initial conditions in each scenario are given.

math.AP↗
arXiv2022

Weak Solutions in Nonlinear Poroelasticity with Incompressible Constituents

We consider quasi-static poroelastic systems with incompressible constituents. The nonlinear permeability is taken to be dependent on solid dilation, and physical types of boundary conditions (Dirichlet, Neumann, and mixed) for the fluid pressure are considered. Such dynamics are motivated by applications in biomechanics and, in particular, tissue perfusion. This system represents a nonlinear, implicit, degenerate evolution problem. We provide a direct fixed point strategy for proving the existence of weak solutions, which is made possible by a novel result on the uniqueness of weak solution to the associated linear system (the permeability a given function of space and time). The linear uniqueness proof is based on novel energy estimates for arbitrary weak solutions, rather than just for constructed solutions (as limits of approximants). The results of this work provide a foundation for addressing strong solutions, as well uniqueness of weak solutions for the nonlinear porous media system.

math.AP↗
arXiv2021

Strong Stabilization of a 3D Potential Flow via a Weakly Damped von Karman Plate

The elimination of aeroelastic instability (resulting in sustained oscillations of bridges, buildings, airfoils) is a central engineering and design issue. Mathematically, this translates to strong asymptotic stabilization of a 3D flow by a 2D elastic structure. The stabilization (convergence to the stationary set) of a aerodynamic wave-plate model is established here. A 3D potential flow on the half-space has a spatially-bounded von Karman plate embedded in the boundary. The physical model, then, is a Neumann wave equation with low regularity of coupling conditions. Motivated on empirical observations, we examine if intrinsic panel damping can stabilize the subsonic flow-plate system to a stationary point. Several partial results have been established through partial regularization of the model. Without doing so, classical approaches attempting to treat the given wave boundary data have fallen short, owing to the failure of the Lopatinski condition (in the sense of Kreiss, Sakamoto) and the associated regularity defect of the hyperbolic Neumann mapping. Here, we operate on the panel model as in the engineering literature with no regularization or modifications; we completely resolve the question of stability by demonstrating that weak plate damping strongly stabilizes system trajectories. This is accomplished by microlocalizing the wave data (given by the plate) and observing an "anisotropic" a microlocal compensation by the plate dynamics precisely where the regularity of the 3D wave is compromsed (in the characteristic sector). Several additional stability results for both wave and plate subsystems are established to "push" strong stability of the plate onto the flow.

math.AP↗
arXiv2021

Multilayered Poroelasticity Interacting with Stokes Flow

We consider the interaction between an incompressible, viscous fluid modeled by the dynamic Stokes equation and a multilayered poroelastic structure which consists of a thin, linear, poroelastic plate layer (in direct contact with the free Stokes flow) and a thick Biot layer. The fluid flow and the elastodynamics of the multilayered poroelastic structure are fully coupled across a fixed interface through physical coupling conditions (including the Beavers-Joseph-Saffman condition), which present mathematical challenges related to the regularity of associated velocity traces. We prove existence of weak solutions to this fluid-structure interaction problem with either (i) a linear, dynamic Biot model, or (ii) a nonlinear quasi-static Biot component, where the permeability is a nonlinear function of the fluid content (as motivated by biological applications). The proof is based on constructing approximate solutions through Rothe's method, and using energy methods and a version of Aubin-Lions compactness lemma (in the nonlinear case) to recover the weak solution as the limit of approximate subsequences. We also provide uniqueness criteria and show that constructed weak solutions are indeed strong solutions to the coupled problem if one assumes additional regularity.

math.AP↗
arXiv2021

Dynamic Equations of Motion for Inextensible Beams and Plates

The large deflections of cantilevered beams and plates are modeled and discussed. Traditional nonlinear elastic models (e.g., that of von Karman) employ elastic restoring forces based on the effect of stretching on bending, and these are less applicable to cantilevers. Recent experimental work indicates that elastic cantilevers are subject to nonlinear inertial and stiffness effects. We review a recently established (quasilinear and nonlocal) cantilevered beam model, and consider some natural extensions to two dimensions -- namely, inextensible plates. Our principal configuration is that of a thin, isotropic, homogeneous rectangular plate, clamped on one edge and free on the remaining three. We proceed through the geometric and elastic modeling to obtain equations of motion via Hamilton's principle for the appropriately specified energies. We enforce {\em effective} inextensibility constraints through Lagrange multipliers. Multiple plate analogs of the established 1D model are obtained, based on various assumptions. For each plate model, we present the modeling hypotheses and the resulting equations of motion. It total, we present three distinct nonlinear partial differential equation models, and, additionally, describe a class of ``higher order" models. Each model has particular advantages and drawbacks for both mathematical and engineering analyses. We conclude with an in depth discussion and comparison of the various systems and some analytical problems.

math.AP↗
arXiv2021

Weak Solutions for a Poro-elastic Plate System

We consider a recent plate model obtained as a scaled limit of the three dimensional Biot system of poro-elasticity. The result is a "2.5" dimensional linear system that couples traditional Euler-Bernoulli plate dynamics to a pressure equation in three dimensions, where diffusion acts only transversely. We alow the permeability function to be time-dependent, making the problem non-autonomous and disqualifying much of the standard abstract theory. Weak solutions are defined in the so called quasi-static case, and the problem is framed abstractly as an implicit, degenerate evolution problem. Utilizing the theory for weak solutions to implicit evolution equations, we obtain existence of solutions. Uniqueness is obtained under additional hypotheses on the regularity of the permeability. We address the inertial case in an appendix, by way of semigroup theory. The work here provides a baseline theory of weak solutions for the poro-elastic plate, and exposits a variety of interesting related models and associated analytical investigations.

math.AP↗
arXiv2021

Theory of Solutions for An Inextensible Cantilever

Recent equations of motion for the large deflections of a cantilevered elastic beam are analyzed. In the traditional theory of beam (and plate) large deflections, nonlinear restoring forces are due to the effect of stretching on bending; for an inextensible cantilever, the enforcement of arc-length preservation leads to quasilinear stiffness effects and inertial effects that are both nonlinear and nonlocal. For this model, smooth solutions are constructed via a spectral Galerkin approach. Additional compactness is needed to pass to the limit, and this is obtained through a complex procession of higher energy estimates. Uniqueness is obtained through a non-trivial decomposition of the nonlinearity. The confounding effects of nonlinear inertia are overcome via the addition of structural (Kelvin-Voigt) damping to the equations of motion. Local well-posedness of smooth solutions is shown first in the absence of nonlinear inertial effects, and then shown with these inertial effects present, taking into account structural damping. With damping in force, global-in-time, strong well-posedness result is obtained by achieving exponential decay for small data.

math.AP↗