arXiv subjects

Alain Miranville

Publications and source records attributed to Alain Miranville.

At least 19 recordsLinked to original sources

arXiv2026

Existence of pullback $\mathcal{D}$-attractors for Kirchhoff wave equations with strong damping and delay

This paper studies the existence of pullback $\mathcal{D}$-attractors for non-autonomous Kirchhoff wave equations with strong damping and delay effects in the phase space $\mathcal{E} = C_{H^1_0(Ω)} \times C_{L^2(Ω)}$. The model contains a strong damping term $-Δ\partial_t u$, a nonlocal Kirchhoff term $Ψ(\|\nabla u\|^2)$, a state-dependent delay term $ϕ(t, u_t)$, and a time-dependent external force $h(x, t)$. There appear to be no results on the pullback attractors of Kirchhoff wave equations involving both strong damping and delay effects in the literature. To this end, we establish delicate uniform estimates and employ the contraction function method to prove the pullback asymptotic compactness of the associated process, which yields the existence of pullback $\mathcal{D}$-attractors.

math.AP↗

arXiv2026

Pullback Attractors for a Non-Autonomous Plate System with Strong Damping and Delay

This paper addresses the existence of pullback attractors for a class of nonlinear non-autonomous strongly damped plate equations with delay. The model contains the biharmonic operator, the strong damping term, a nonlinear source term and a delay operator acting on the history of the solution. The interaction between the strong damping mechanism and the hereditary delay effect gives rise to several analytical difficulties in deriving uniform estimates and proving pullback asymptotic compactness. To overcome these difficulties, we construct a modified energy functional adapted to the strong damping structure and employ the contractive function method to handle the delay term in the history phase space. By establishing the existence and uniqueness of weak solutions, uniform pullback estimates and the pullback asymptotic compactness of the associated non-autonomous process, we prove the existence of pullback attractors for the strongly damped plate equation with delay.

math.AP↗

arXiv2026

Quaternionic Response Geometry for Proteins: Toward a Noncommutative Theory of Ordered Deformations

Protein function may depend not only on endpoint conformations but also on the ordered deformation histories through which they are reached. This distinction is relevant to allostery, conformational switching, mutation-induced rearrangements, and epistatic effects, where different perturbation sequences may produce similar visible structures while retaining distinct internal transport histories. Current state-centered or endpoint-centered representations do not always preserve this order-sensitive information. The practical motivation is therefore to provide a foundation for future descriptors of protein deformation trajectories that can distinguish ordered histories even when endpoint conformations are similar. We propose a deformation-first geometric framework based on quaternionic frame transport along the protein backbone. Local backbone frames are lifted to quaternionic variables, with infinitesimal rotation encoded by $Ω(\ell)=2\,q(\ell)^{-1}\partial_\ell q(\ell).$ Ordered concatenation of admissible deformation paths generates a noncommutative transport algebra, recording that deformation A followed by B need not be equivalent to B followed by A. From this ordered transport layer, we construct a spectral-response layer comprising a global Dirac-type operator, local spectral germs, a renormalized spectral density, and a mixed response form. A minimal realization on an idealized $α$-helix shows how localized pitch and bending perturbations can yield similar endpoint descriptors while producing a nonzero endpoint-derived ordered-transport discrepancy. At the formal level, the framework separates an order-sensitive transport-memory sector, lost under a commutative shadow, from a spectral-response sector that remains visible.

q-bio.BM↗

arXiv2026

On generalised d'Alembert-type integral representations for damped wave equations on the quarter-plane

We rigorously construct and verify a posteriori new closed-form solutions for the forced Maxwell-Cattaneo-Vernotte equation (also broadly known as the damped wave equation, hyperbolic heat, and telegrapher's equation on lossy transmission lines) posed on the spatiotemporal quarter-plane with general initial and boundary data in classical function spaces. For this purpose, the modern complex-analytic unified transform method of Fokas (originally developed for elliptic PDE and evolution equations with polynomial dispersion relations) is here, for the first time, extended for analysis of hyperbolic-parabolic problems on the semi-infinite interval. Importantly, we then establish theorems which pertain to regularity, boundary and asymptotic properties of the new analytical formulae as well as to well-posedness of the addressed boundary-value problems. Notably, the nature and generality of problems considered, combined with the semi-unboundedness of the domain, induce substantial analytic challenges which demand delicate treatment, both in appropriately interpreting oscillatory integral terms of the solution formulae and in proving the proposed results. In this process, crucially, certain compatibility conditions, between initial, boundary and forcing data at the origin, are revealed, which guarantee the existence of a smooth solution across the whole domain of interest. Our explicit integral representations are of direct utility for numercal benchmarking purposes, for exploring connections with modelling in continuum mechanics, mathematical physics, biology and the natural sciences, and for the investigation of well-posedness for nonlinear counterparts too.

math.AP↗

arXiv2026

Global random attractor for 3D stochastic Navier-Stokes equation with nonlinear colored noise

In this paper, we first introduce the definitions of random evolutionary system that associate with random evolutionary semigroup and the corresponding global weak or strong random attractor. Then we establish the existence result about global weak or strong random attractor and it's properties like invariance and weak or strong tracking features. Finally, we use our established results to the 3D stochastic Navier-Stokes equation with nonlinear colored noise.

math.DS↗

arXiv2026

The linear Cahn-Hilliard equation with an interface

We obtain new integral representations, expressed as contour integrals in the complex Fourier plane, for the solution of fully nonhomogeneous interface problems for the linearized Cahn-Hilliard equation with arbitrary initial data on the line and general interface conditions prescribed at the origin. Cahn-Hilliard-type models emerge in applied mathematics in connection to a spectacular variety of phenomena of mathematical physics, continuum mechanics, chemistry and biology. A novel implementation of Fokas' unified transform method is in force herein for a fourth-order operator for the first time, with particular challenges arising due to the nature and the generality of the problems under consideration. Our explicit formulae directly lend themselves to exploration of the solution's qualitative properties such as regularity and asymptotic behavior. This work is also useful in the investigation of well-posedness for nonlinear counterparts as well as in the study of free-boundary and diffuse-interface problems.

math.AP↗

arXiv2025

An Allen-Cahn tumor growth model with temperature

In this paper, we propose a new non-isothermal Allen-Cahn (Ginzburg-Landau) model for tumor growth. After deriving it using a microforces approach, we study its well-posedness. In particular, we are able to prove the existence and uniqueness of a local and global-in-time solution to our PDE system.

math.AP↗

arXiv2025

WAN3DNS: Weak Adversarial Networks for Solving 3D Incompressible Navier-Stokes Equations

The 3D incompressible Navier-Stokes equations model essential fluid phenomena, including turbulence and aerodynamics, but are challenging to solve due to nonlinearity and limited solution regularity. Despite extensive research, the full mathematical understanding of the 3D incompressible Navier-Stokes equations continues to elude scientists, highlighting the depth and difficulty of the problem. Classical solvers are costly, and neural network-based methods typically assume strong solutions, limiting their use in underresolved regimes. We introduce WAN3DNS, a weak-form neural solver that recasts the equations as a minimax optimization problem, allowing learning directly from weak solutions. Using the weak formulation, WAN3DNS circumvents the stringent differentiability requirements of classical physics-informed neural networks (PINNs) and accommodates scenarios where weak solutions exist, but strong solutions may not. We evaluated WAN3DNS's accuracy and effectiveness in three benchmark cases: the 2D Kovasznay, 3D Beltrami, and 3D lid-driven cavity flows. Furthermore, using Galerkin's theory, we conduct a rigorous error analysis and show that the $L^{2}$ training error is controllably bounded by the architectural parameters of the network and the norm of residues. This implies that for neural networks with small loss, the corresponding $L^{2}$ error will also be small. This work bridges the gap between weak solution theory and deep learning, offering a robust alternative for complex fluid flow simulations with reduced regularity constraints. Code: https://github.com/Wenran-Li/WAN3DNS

physics.flu-dyn↗

arXiv2025

Optimal control on a brain tumor growth model with lactate metabolism, viscoelastic effects, and tissue damage

In this paper, we study an optimal control problem for a brain tumor growth model that incorporates lactate metabolism, viscoelastic effects, and tissue damage. The PDE system, introduced in [G. Cavalleri, P. Colli, A. Miranville, E. Rocca, On a Brain Tumor Growth Model with Lactate Metabolism, Viscoelastic Effects, and Tissue Damage (2025)], couples a Fisher-Kolmogorov type equation for tumor cell density with a reaction-diffusion equation for the lactate, a quasi-static force balance governing the displacement, and a nonlinear differential inclusion for tissue damage. The control variables, representing chemotherapy and a lactate-targeting drug, influence tumor progression and treatment response. Starting from well-posedness, regularity, and continuous dependence results already established, we define a suitable cost functional and prove the existence of optimal controls. Then, we analyze the differentiability of the control-to-state operator and establish a necessary first-order condition for treatment optimality.

math.AP↗

arXiv2025

On a Brain Tumor Growth Model with Lactate Metabolism, Viscoelastic Effects, and Tissue Damage

In this paper, we study a nonlinearly coupled initial-boundary value problem describing the evolution of brain tumor growth including lactate metabolism. In our modeling approach, we also take into account the viscoelastic properties of the tissues as well as the reversible damage effects that could occur, possibly caused by surgery. After introducing the PDE system, coupling a Fischer-Kolmogorov type equation for the tumor phase with a reaction-diffusion equation for the lactate, a quasi-static momentum balance with nonlinear elasticity and viscosity matrices, and a nonlinear differential inclusion for the damage, we prove the existence of global in time weak solutions under reasonable assumptions on the involved functions and data. Strengthening these assumptions, we subsequently prove further regularity properties of the solutions as well as their continuous dependence with respect to the data, entailing the well-posedness of the Cauchy problem associated with the nonlinear PDE system.

math.AP↗

arXiv2025

From thermodynamics to protein design: Diffusion models for biomolecule generation towards autonomous protein engineering

Protein design with desirable properties has been a significant challenge for many decades. Generative artificial intelligence is a promising approach and has achieved great success in various protein generation tasks. Notably, diffusion models stand out for their robust mathematical foundations and impressive generative capabilities, offering unique advantages in certain applications such as protein design. In this review, we first give the definition and characteristics of diffusion models and then focus on two strategies: Denoising Diffusion Probabilistic Models and Score-based Generative Models, where DDPM is the discrete form of SGM. Furthermore, we discuss their applications in protein design, peptide generation, drug discovery, and protein-ligand interaction. Finally, we outline the future perspectives of diffusion models to advance autonomous protein design and engineering. The E(3) group consists of all rotations, reflections, and translations in three-dimensions. The equivariance on the E(3) group can keep the physical stability of the frame of each amino acid as much as possible, and we reflect on how to keep the diffusion model E(3) equivariant for protein generation.

q-bio.QM↗

arXiv2024

Pullback attractors for nonclassical diffusion equations with a delay operator

In this paper, we consider the asymptotic behavior of weak solutions for nonclassical non-autonomous diffusion equations with a delay operator in time-dependent spaces when the nonlinear function $g$ satisfies subcritical exponent growth conditions, the delay operator $φ(t, u_t)$ contains some hereditary characteristics and the external force $k \in L_{l o c}^{2}\left(\mathbb{R} ; L^{2}(Ω)\right)$. First, we prove the well-posedness of solutions by using the Faedo-Galerkin approximation method. Then after a series of elaborate energy estimates and calculations, we establish the existence and regularity of pullback attractors in time-dependent spaces $C_{\mathcal{H}_{t}(Ω)}$ and $C_{\mathcal{H}^{1}_{t}(Ω)}$, respectively.

math.AP↗

arXiv2024

Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system

The motion of two contiguous incompressible and viscous fluids is described within the diffuse interface theory by the so-called Model H. The system consists of the Navier-Stokes equations, which are coupled with the Cahn-Hilliard equation associated to the Ginzburg-Landau free energy with physically relevant logarithmic potential. This model is studied in bounded smooth domain in R^d, d=2 and d=3, and is supplemented with a no-slip condition for the velocity, homogeneous Neumann boundary conditions for the order parameter and the chemical potential, and suitable initial conditions. We study uniqueness and regularity of weak and strong solutions. In a two-dimensional domain, we show the uniqueness of weak solutions and the existence and uniqueness of global strong solutions originating from an initial velocity u_0 in V, namely u_0 in H_0^1 such that div u_0=0. In addition, we prove further regularity properties and the validity of the instantaneous separation property. In a three-dimensional domain, we show the existence and uniqueness of local strong solutions with initial velocity u_0 in V.

math.AP↗

arXiv2024

Existence and upper semicontinuity of pullback attractors for Kirchhoff wave equations in time-dependent spaces

In this paper, we shall investigate the existence and upper semicontinuity of pullback attractors for non-autonomous Kirchhoff wave equations with a strong damping in the time-dependent space $X_t$. After deriving the existence and uniqueness of solutions by the Faedo-Galerkin approximation method, we establish the existence of pullback attractors. Later on, we prove the upper semicontinuity of pullback attractors between the Kirchhoff-type wave equations with $δ\geq 0$ and the conventional wave equations with $δ=0$ by a series of complex energy estimates.

math.AP↗

arXiv2023

Existence and dimensions of global attractors for a delayed reaction-diffusion equation on an unbounded domain

The purpose of this paper is to investigate the existence and Hausdorff dimension as well as fractal dimension of global attractors for a delayed reaction-diffusion equation on an unbounded domain. The noncompactness of the domain causes the Laplace operator has a continuous spectrum, the semigroup generated by the linear part and the Sobolev embeddings are no longer compact, making the problem more difficult compared with the equations on bounded domains. We first obtain the existence of an absorbing set for the infinite dimensional dynamical system generated by the equation by a priori estimate of the solutions. Then, we show the asymptotic compactness of the solution semiflow by an uniform a priori estimates for far-field values of solutions together with the Arzelà-Ascoli theorem, which facilitates us to show the existence of global attractors. By decomposing the solution into three parts and establishing a squeezing property of each part, we obtain the explicit upper estimation of both Hausdorff and fractal dimension of the global attractors, which only depend on the inner characteristic of the equation, while not related to the entropy number compared with the existing literature.

math.AP↗

arXiv2023

Existence and regularity of pullback attractors for nonclassical non-autonomous diffusion equations with delay

In this paper, we consider the asymptotic behavior of weak solutions for non-autonomous diffusion equations with delay in time-dependent spaces when the nonlinear function $f$ is critical growth, the delay term $g(t, u_t)$ contains some hereditary characteristics and the external force $h \in L_{l o c}^{2}\left(\mathbb{R} ; L^{2}(Ω)\right)$. Firstly, we prove the well-posedness of solutions by using the Faedo-Galerkin approximation method. Then after a series of elaborate energy estimates and calculations, we establish the existence and regularity of pullback attractors in time-dependent spaces $C_{\mathcal{H}_{t}(Ω)}$ and $C_{\mathcal{H}^{1}_{t}(Ω)}$ respectively.

math.AP↗

arXiv2023

Evaluating The Robustness of Self-Supervised Representations to Background/Foreground Removal

Despite impressive empirical advances of SSL in solving various tasks, the problem of understanding and characterizing SSL representations learned from input data remains relatively under-explored. We provide a comparative analysis of how the representations produced by SSL models differ when masking parts of the input. Specifically, we considered state-of-the-art SSL pretrained models, such as DINOv2, MAE, and SwaV, and analyzed changes at the representation levels across 4 Image Classification datasets. First, we generate variations of the datasets by applying foreground and background segmentation. Then, we conduct statistical analysis using Canonical Correlation Analysis (CCA) and Centered Kernel Alignment (CKA) to evaluate the robustness of the representations learned in SSL models. Empirically, we show that not all models lead to representations that separate foreground, background, and complete images. Furthermore, we test different masking strategies by occluding the center regions of the images to address cases where foreground and background are difficult. For example, the DTD dataset that focuses on texture rather specific objects.

cs.CV↗

arXiv2023

Strong global attractors for a three dimensional nonclassical diffusion equation with memory

In this paper, we study the strong global attractors for a three dimensional nonclassical diffusion equation with memory. First, we prove the existence and uniqueness of strong solutions for the equations by the Galerkin method. Then we prove the existence of global attractors for the equations in $H^2(Ω)\cap H^1_0(Ω)\times L^2_μ(\mathbb{R}^+;H^2(Ω)\cap H^1_0(Ω))$ by the condition (C).

math.AP↗