arXiv subjects

Boris Muha

Publications and source records attributed to Boris Muha.

At least 19 recordsLinked to original sources

arXiv2026

Analyticity of the data-to-solution map for a stationary Navier-Stokes fluid-structure interaction problem

We consider a stationary fluid--structure interaction problem in which the steady Navier--Stokes equations are coupled, through a free elastic interface, with a clamped Euler-Bernoulli beam equation. Using a complexification of the fixed-domain formulation and the holomorphic implicit function theorem, we prove that, in a neighbourhood of the trivial solution, the mapping from the right-hand side to the weak solution is real analytic. As a byproduct we obtain a small-data existence and local uniqueness result for the coupled system. Our motivation comes from data-driven reduced-order modelling for parametric PDEs, where approximation properties are closely related to the regularity of the solution map. Numerically, a manufactured-solution test exhibits approximately second-order convergence in the reported relative $L^2$ errors, while a proper orthogonal decomposition study for a parametric force family shows rapid decay of the empirical reconstruction error until a numerical floor is reached.

math.AP↗

arXiv2026

Exponential Decay of Solutions to a Fluid-Plate Model with Small Initial Data

We consider a three-dimensional fluid-structure interaction problem coupling the incompressible Navier-Stokes equations in a time-dependent domain with a square-root damped plate equation, which is posed on the moving upper boundary of the fluid. We prove, a priori, the exponential decay of strong solutions for initial data that are sufficiently small in a suitable Sobolev space. The proof combines higher-order energy estimates, Stokes-type regularity bounds, and a nonlinear bootstrap scheme that closes under the smallness assumption.

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

A Temperature-Coupled Cahn-Hilliard-Stokes-Heat Model for Thermally Driven Phase Separation

We study a diffuse-interface model for thermally driven phase separation in viscous incompressible mixtures. The system couples a convective Cahn-Hilliard equation for the order parameter with a Stokes subsystem for the velocity-pressure field and a heat equation for the temperature. Temperature enters the bulk free energy through a Landau-type coefficient, while the phase field affects the flow through concentration-dependent density and viscosity. The model serves as a proxy for temperature-triggered condensation-like phase separation; humidity, latent heat, vapor pressure, and capillary forcing are absorbed into the choice of the threshold temperature $\Theta_S$. We motivate the chemical potential through a temperature-dependent Landau free energy and use a regularized auxiliary formulation to prove local-in-time existence of weak solutions. For the numerical analysis, we employ a first-order sequential finite-element discretization of a simplified quasi-static formulation. The heat equation is advanced by implicit diffusion, the variable-coefficient Stokes problem is treated by a Taylor-Hood discretization, and the Cahn-Hilliard bulk derivative is evaluated at the previous time level, so each algebraic subproblem is linear. An isothermal diffusive test confirms mass conservation to roundoff and exhibits monotone discrete-energy decay for the tested parameters. Time-step and mesh-refinement studies show first-order temporal and approximately second-order spatial behavior. The remaining computations provide qualitative, parameter-specific illustrations; no global discrete energy law is claimed for the non-isothermal sequential scheme.

math.AP↗

arXiv2026

AE-ViT: Stable Long-Horizon Parametric Partial Differential Equations Modeling

Deep Learning Reduced Order Models (ROMs) are becoming increasingly popular as surrogate models for parametric partial differential equations (PDEs) due to their ability to handle high-dimensional data, approximate highly nonlinear mappings, and utilize GPUs. Existing approaches typically learn evolution either on the full solution field, which requires capturing long-range spatial interactions at high computational cost, or on compressed latent representations obtained from autoencoders, which reduces the cost but often yields latent vectors that are difficult to evolve, since they primarily encode spatial information. Moreover, in parametric PDEs, the initial condition alone is not sufficient to determine the trajectory, and most current approaches are not evaluated on jointly predicting multiple solution components with differing magnitudes and parameter sensitivities. To address these challenges, we propose a joint model consisting of a convolutional encoder, a transformer operating on latent representations, and a decoder for reconstruction. The main novelties are joint training with multi-stage parameter injection and coordinate channel injection. Parameters are injected at multiple stages to improve conditioning. Physical coordinates are encoded to provide spatial information. This allows the model to dynamically adapt its computations to the specific PDE parameters governing each system, rather than learning a single fixed response. Experiments on the Advection-Diffusion-Reaction equation and Navier-Stokes flow around the cylinder wake demonstrate that our approach combines the efficiency of latent evolution with the fidelity of full-field models, outperforming DL-ROMs, latent transformers, and plain ViTs in multi-field prediction, reducing the relative rollout error by approximately $5$ times.

cs.LG↗

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

The diffuse interface approximation to fluid-structure interaction

We consider a fluid-structure interaction problem in the Eulerian, phase-field formulation. The problem is described using the Navier--Stokes equations for a viscous, incompressible fluid, coupled with the incompressible hyperelasticity system, both written in the Eulerian coordinates. This allows the problem to be written in a unified formulation, using a single field for the fluid and structure velocities. To track the position of the domain, we use a phase-field approach, resulting in a coupled Cahn--Hilliard--Navier--Stokes-type of problem for the diffuse interface fluid-structure interaction. Under certain assumptions, we prove the convergence of the diffuse interface model to the sharp interface fluid-structure interaction problem. To solve the problem numerically, we propose a novel, strongly coupled, second-order partitioned computational method where the system is decoupled into the Cahn--Hilliard problem, the transport problem for the left Cauchy--Green deformation tensor, and the Navier--Stokes problem. The problems are solved iteratively until convergence at each time step. The performance of the method is illustrated on two computational examples.

math.NA↗

arXiv2025

A global existence result on weak solutions for the 3D Navier-Stokes-plate system with no contact

We consider the three-dimensional fluid-structure interaction system modeling a system consisting of a viscous incompressible fluid and an elastic plate forming its moving upper boundary. The fluid is described by the incompressible Navier-Stokes equations with a free upper boundary that evolves according to the motion of the structure, coupled via the velocity- and stress-matching conditions. We show that under a rather general condition on the initial data, there exists a global-in-time weak solution of the system. In particular, there is no contact between the plate and the bottom boundary.

math.AP↗

arXiv2025

A no-contact result for a plate-fluid interaction system in dimension three

We address the fluid-structure interaction between a viscous incompressible fluid and an elastic plate forming its moving upper boundary in three dimensions. The fluid is described by the incompressible Navier-Stokes equations with a free upper boundary that evolves according to the motion of the structure, coupled via the velocity- and stress-matching conditions. Under the natural energy bounds and additional regularity assumptions on the weak solutions, we prove a non-contact property with a uniform separation of the plate from the rigid boundary. The result does not require damping in the plate equation.

math.AP↗

arXiv2025

A Regularized Interface Method for Fluid-Poroelastic Structure Interaction Problems with Nonlinear Geometric Coupling

We introduce a new regularized interface method for proving existence of weak solutions to nonlinear moving boundary problems with low-regularity interfaces. We study a fluid-poroelastic structure interaction (FPSI) problem coupling the Navier-Stokes equations for an incompressible viscous fluid with the Biot system for a bulk poroelastic medium. The two phases occupy domains of the same spatial dimension, separated by a moving interface defined by the trace of the poroelastic displacement, which exhibits low regularity and strong geometric nonlinearities. Despite its importance in applications, no existence theory has been available for this nonlinear moving-domain setting, primarily because the lack of interface regularity precludes even the formulation of a weak solution framework. To address this gap, we (1) introduce a regularization of the Biot displacement via spatial convolution at scale $\delta > 0$, which defines regularized moving domains and interface, and (2) modify the weak formulation in a way that preserves energy consistency with the original problem. For each fixed $\delta > 0$, we prove existence of a weak solution to the resulting regularized interface problem. The proof strategy involves inserting a thin plate of thickness $h > 0$ at the interface, applying a time-discretization via a Lie operator splitting scheme, establishing uniform a priori bounds, and employing Aubin-Lions compactness on moving domains. The analysis is particularly involved, partly because the thin plate allows displacements in all spatial directions. Passing to the limit $h \to 0$ with uniform-in-h estimates and compactness arguments yields a regularized interface weak solution. The regularization introduced in this manuscript is essential to maintain uniform geometric control of the moving interface and to accommodate vector-valued structural displacements.

math.AP↗

arXiv2025

Three-dimensional Navier-Stokes-Biot coupling via a moving reticular plate interface: existence of weak solutions

We prove the existence of finite-energy weak solutions to a regularized three-dimensional fluid-structure interaction (FSI) problem involving an incompressible, viscous, Newtonian fluid and a multilayered poro(visco)elastic structure. The structure consists of a thick layer modeled by the Biot equations and a thin reticular plate with inertia and elastic energy, transparent to fluid flow. The coupling is nonlinear in the sense that it takes place on a moving interface that is not known a priori but is defined by the solution itself, making the problem a moving-boundary problem. This nonlinear free-boundary coupling, combined with the limited regularity of the Biot displacement, renders the classical weak formulation ill-defined at finite energy. To address this, we introduce a minimally invasive regularization based on a suitable extension and convolution of the Biot displacement, chosen so that the regularized problem remains consistent with the original model. We then construct approximate solutions to the regularized problem via a Lie operator-splitting scheme and derive uniform energy bounds. While these bounds ensure weak and weak* convergence, passing to the limit in the nonlinear terms requires refined compactness arguments, including variants of the Aubin-Lions lemma and tools adapted to moving non-Lipschitz interfaces. The result applies in particular to the purely elastic case (without structural damping) as well as the poroviscoelastic case. This work extends the two-dimensional analysis of Kuan-\v{C}ani\'c-Muha 2024 to the fully three-dimensional setting and, to our knowledge, provides the first existence result for a nonlinearly coupled, multilayer 3D Navier-Stokes-Biot FSI system with a permeable interface.

math.AP↗

arXiv2025

Existence, uniqueness, and long-time asymptotic behavior of regular solutions in multidimensional thermoelasticity

We study a simplified nonlinear thermoelasticity model on two- and three-dimensional tori. A novel functional involving the Fisher information associated with temperature is introduced, extending the previous one-dimensional approach from the first two authors (SIAM J.\ Math.\ Anal.\ \textbf{55} (2023), 7024--7038)) to higher dimensions. Using this functional, we prove global/local existence of unique regular solutions for small/large initial data. Furthermore, we analyze the asymptotic behavior as time approaches infinity and show that the temperature stabilizes to a constant state, while the displacement naturally decomposes into two distinct components: a divergence-free part oscillating indefinitely according to a homogeneous wave equation and a curl-free part converging to zero. Analogous results for the Lam\'e operator are also stated.

math.AP↗

arXiv2025

On Self-Propulsion by Oscillations in a Viscous Liquid

Suppose that a body $\mathscr B$ can move by translatory motion with velocity $\boldsymbol{\gamma}$ in an otherwise quiescent Navier-Stokes liquid, $\mathscr L$, filling the entire space outside $\mathscr B$. Denote by $\Omega = \Omega(t)$, $t\in\mathbb{R}$, the one-parameter family of bounded, sufficiently smooth domains of $\mathbb{R}^3$, each one representing the configuration of $\mathscr B$ at time $t$ with respect to a frame with the origin at the center of mass $G$ and axes parallel to those of an inertial frame. We assume that there are no external forces acting on the coupled system $\mathscr S := \mathscr B +\mathscr L$ and that the only driving mechanism is a prescribed change in shape of $\Omega$ with time. The self-propulsion problem that we would like to address can be thus qualitatively formulated as follows. Suppose that $\mathscr B$ changes its shape in a given time-periodic fashion, namely, $\Omega(t+T) = \Omega(t)$, for some $T > 0$ and all $t \in \mathbb{R}$. Then, find necessary and sufficient conditions on the map $t\mapsto \Omega(t)$ securing that $\mathscr B$ self-propels, that is, $G$ covers any given finite distance in a finite time. We show that this problem is solvable, in a suitable function class, provided the amplitude of the oscillations is below a given constant. Moreover, we provide examples where the propelling velocity of $\mathscr B$ is explicitly evaluated in terms of the physical parameters and the frequency of oscillations.

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

Analysis of an Inelastic Contact Problem for the Damped Wave Equation

In this paper, we examine the dynamic behavior of a viscoelastic string oscillating above a rigid obstacle in a one-dimensional setting, accounting for inelastic contact between the string and the obstacle. We construct a global-in-time weak solution to this problem by using an approximation method that incorporates a penalizing repulsive force of the form $\frac1\varepsilon\chi_{\{\eta<0\}} (\partial_t\eta)^-$. The weak solution exhibits well-controlled energy dissipation, occurring only during contact on a set of zero measure and exclusively when the string moves downward. Furthermore, the velocity is shown to vanish after contact in a specific weak sense. This model serves as a simplified framework for studying contact problems in fluid-structure interaction contexts.

math.AP↗

arXiv2024

Inviscid fluid interacting with a nonlinear two-dimensional plate

We address a moving boundary problem that consists of a system of equations modeling an inviscid fluid interacting with a two-dimensional nonlinear Koiter plate at the boundary. We derive a priori estimates needed to prove the local-in-time existence of solutions. We use the Arbitrary Lagrange Euler (ALE) coordinates to fix the domain and obtain careful estimates for the nonlinear Koiter plate, ALE velocity, and pressure {without any viscoelastic smoothing}. For the nonlinear Koiter plate, higher order energy estimates are obtained, whereas estimates for the ALE pressure are obtained by setting up an elliptic problem. For the ALE velocity, the bounds are obtained through div-curl estimates by estimating the ALE vorticity. We then extend our results in two directions: (1) to include fractional Sobolev spaces and (2) to incorporate the normalized second fundamental form.

math.AP↗

arXiv2024

Existence and Regularity Results for a Nonlinear Fluid-Structure Interaction Problem with Three-Dimensional Structural Displacement

In this paper we investigate a nonlinear fluid-structure interaction (FSI) problem involving the Navier-Stokes equations, which describe the flow of an incompressible, viscous fluid in a 3D domain interacting with a thin viscoelastic lateral wall. The wall's elastodynamics is modeled by a two-dimensional plate equation with fractional damping, accounting for displacement in all three directions. The system is nonlinearly coupled through kinematic and dynamic conditions imposed at the time-varying fluid-structure interface, whose location is not known a priori. We establish three key results, particularly significant for FSI problems that account for vector displacements of thin structures. Specifically, we first establish a hidden spatial regularity for the structure displacement, which forms the basis for proving that self-contact of the structure will not occur within a finite time interval. Secondly, we demonstrate temporal regularity for both the structure and fluid velocities, which enables a new compactness result for three-dimensional structural displacements. Finally, building on these regularity results, we prove the existence of a local-in-time weak solution to the FSI problem. This is done through a constructive proof using time discretization via the Lie operator splitting method. These results are significant because they address the well-known issues associated with the analysis of nonlinearly coupled FSI problems capturing vector displacements of elastic/viscoelastic structures in 3D, such as spatial and temporal regularity of weak solutions and their well-posedness.

math.AP↗

arXiv2024

Reduced Order Modeling of Partial Differential Equations on Parameter-Dependent Domains Using Deep Neural Networks

Partial differential equations (PDEs) are widely used for modeling various physical phenomena. These equations often depend on certain parameters, necessitating either the identification of optimal parameters or the solution of the equations over multiple parameters. Performing an exhaustive search over the parameter space requires solving the PDE multiple times, which is generally impractical. To address this challenge, reduced order models (ROMs) are built using a set of precomputed solutions (snapshots) corresponding to different parameter values. Recently, Deep Learning ROMs (DL-ROMs) have been introduced as a new method to obtain ROM, offering improved flexibility and performance. In many cases, the domain on which the PDE is defined also varies. Capturing this variation is important for building accurate ROMs but is often difficult, especially when the domain has a complex structure or changes topology. In this paper, we propose a Deep-ROM framework that can automatically extract useful domain parametrization and incorporate it into the model. Unlike traditional domain parameterization methods, our approach does not require user-defined control points and can effectively handle domains with varying numbers of components. It can also learn from domain data even when no mesh is available. Using deep autoencoders, our approach reduces the dimensionality of both the PDE solution and the domain representation, making it possible to approximate solutions efficiently across a wide range of domain shapes and parameter values. We demonstrate that our approach produces parametrizations that yield solution accuracy comparable to models using exact parameters. Importantly, our model remains stable under moderate geometric variations in the domain, such as boundary deformations and noise - scenarios where traditional ROMs often require remeshing or manual adjustment.

math.NA↗