Open research questions in Advanced Mathematical Modeling in Engineering

What the literature leaves open

• To study the problem with Dirichlet-Neumann conditions for the time variable and almost periodicity conditions with respect to the spatial coordinates in other contexts. To establish conditions for correct solvability of the problem in the scale of Sobolev spaces for inhomogeneous equations.

  Dirichlet–Neumann Problem for Partial Differential Equations Unsolved With Respect to the Highest Time Derivative · 2026 · DOI

• The lack of conditions for unique solvability in the scale of Sobolev spaces for inhomogeneous equations. The need for a series representation of the solution and the construction of Green's functions.

  Dirichlet–Neumann Problem for Partial Differential Equations Unsolved With Respect to the Highest Time Derivative · 2026 · DOI

• Complex, random microstructures and high-dimensional input spaces. The need to preserve physical symmetries in the prediction of homogenized properties. The challenge of modeling complex, multi-modal distributions inherent in stochastic systems.

  Symmetry-Preserving Neural Operator Flows for High-Dimensional Stochastic Homogenization in Disordered Media · 2026 · DOI

• The system of differential equations is nonlinear and has higher-order derivatives. The unknown functions v and ρ enter the system through higher-order derivatives and in a nonlinear manner.

  SOLVABILITY OF THE INITIAL BOUNDARY VALUE PROBLEM FOR A NONSTATIONARY DIFFUSION MODEL OF AN INHOMOGENEOUS FLUID · 2026 · DOI

• The paper suggests future research on the regularity theory for stochastic partial differential equations. The paper suggests future research on the numerical approximation of stochastic partial differential equations on evolving curves.

  A Surface Finite Element Scheme for a Stochastic PDE on an Evolving Curve · 2026 · DOI

• The paper identifies a gap in the literature for a continuous model framework for the stochastic partial differential equation on an evolving curve. The paper identifies a gap in the literature for a suitable variational solution concept and a fully discrete finite element method discretization.

  A Surface Finite Element Scheme for a Stochastic PDE on an Evolving Curve · 2026 · DOI

• The paper only considers the viscous case with σ > 0. The construction of counterexamples is limited to one dimension. The paper does not provide a general result for higher dimensions.

  Loss of Quasiconvexity in the Periodic Homogenization of Viscous Hamilton–Jacobi Equations · 2026 · DOI

• Future research should focus on the study of the viscous case in higher dimensions. The construction of counterexamples should be extended to higher dimensions. The paper's findings should be generalized to other nonlinear partial differential equations.

  Loss of Quasiconvexity in the Periodic Homogenization of Viscous Hamilton–Jacobi Equations · 2026 · DOI

• The relaxation of spatial anisotropy in non-linear field continua is not well understood. A linearized treatment commonly leads to exponential relaxation, but strongly deformed configurations can exhibit an initially non-linear relaxation regime.

  Universal_Topological_Relaxation_Constant__Γ.pdf · 2026 · DOI

• The uniqueness of finite-dimensional distributions is established via iteration methods from [12] and [21], but the paper does not explicitly address uniqueness and existence of solutions to the full martingale problem for time-dependent test functions in F ∈ C₀¹(ℝ₊;DMax) when the martingale property must be preserved under limits in H⁻¹₀ norm for singular SPDEs with unbounded domains.

  Energy solutions of singular SPDEs on Hilbert spaces with applications to domains with boundary conditions · 2026 · DOI

• The approximation via mollification (ιN ∗ h) ◦ pN in equation (71) with 'slowed' shifting near boundaries is introduced for the Burgers equation on semi-infinite domains, but convergence rates and bounds for equation (74) with parameter κ ∈ (0,1/2) are not fully explored for non-smooth initial data or when the shifted mollifier pN becomes non-monotone near domain boundaries.

  Energy solutions of singular SPDEs on Hilbert spaces with applications to domains with boundary conditions · 2026 · DOI

• The integration-by-parts estimate in the proof of Theorem 4.7 (equation preceding ∥∇̂ₙU∥_{L∞} ≤ C) assumes periodic boundary conditions on Tᵈˣⁿ, but the paper does not address how the convergence rates and constants C depend on domain geometry for non-periodic boundary conditions or for coefficients with finite support in the slow variable x.

  Optimal Convergence Rates in Multiscale Elliptic Homogenization · 2026 · DOI

• The Sobolev embedding application in Theorem 4.7 relies on setting ℓ₀ = [nd] + 1 derivative orders to guarantee L∞ bounds for the flux and solution, but the paper does not characterize how this bound ℓ₀ evolves with problem dimension d and scale number n, or whether weaker regularity assumptions on A could reduce the required derivative order.

  Optimal Convergence Rates in Multiscale Elliptic Homogenization · 2026 · DOI

• The exponential decay condition on γ (required for temporal regularity in Proposition 5.8) restricts the class of admissible weight functions, but the paper does not investigate whether polynomial decay γ(z) ~ z^(-α) for α > 0 allows convergence under modified temporal estimates for long-time integration of SPDEs on graphs.

  Asymptotic-preserving approximations for stochastic incompressible viscous fluids and SPDEs on graph · 2026 · DOI

• The weighted L̄²Aγ norm is used throughout the error analysis for the AP scheme, but comparative performance analysis with alternative weighted norms (e.g., L̄²Fγ or graph-based Sobolev norms) is absent, leaving open whether the chosen weight function A(z)γ(z) is optimal for capturing discretization errors in practical simulations.

  Asymptotic-preserving approximations for stochastic incompressible viscous fluids and SPDEs on graph · 2026 · DOI

• The method is limited to heterogeneous coefficients and domains with nontrivial topology. The paper does not consider other types of coefficients or domains.

  Multiscale model reduction and two-level Schwarz preconditioner for H(curl) elliptic problems · 2026 · DOI

• The efficient solution of linear systems arising from curl-conforming finite element discretizations of H(curl) elliptic problems is a major challenge. There is a need for a model reduction method that can be used to solve these problems.

  Multiscale model reduction and two-level Schwarz preconditioner for H(curl) elliptic problems · 2026 · DOI

• Future research can focus on applying the results to specific stochastic partial differential equations. Future research can investigate the application of the results to other types of noise.

  Discretizations of Stochastic Evolution Equations in Variational Approach Driven by Jump-Diffusion · 2026 · DOI

• The paper identifies a gap in the analysis of Discrete Exterior Calculus (DEC) approximations. The authors note that prior work has not provided a framework for interpreting DEC numerical schemes in terms of Finite Element Exterior Calculus (FEEC).

  A framework for analysis of DEC approximations to Hodge-Laplacian problems using generalized Whitney forms · 2026 · DOI

• The inability of traditional methods to efficiently and accurately predict homogenized properties in high-dimensional stochastic homogenization. The lack of physical consistency in conventional neural operators.

  Symmetry-Preserving Neural Operator Flows for High-Dimensional Stochastic Homogenization in Disordered Media · 2026 · DOI

• The gap is the lack of rigorous error analysis for unfitted finite element methods in evolving domains with topological changes. The gap is the need for a framework to understand the behavior of the method.

  Analysis of a finite element method for PDEs in evolving domains with topological changes · 2026 · DOI

• Further study of the Cahn-Hilliard equation with nonlinear diffusion in higher dimensions. Investigation of the behavior of solutions in unbounded domains.

  On the Cahn-Hilliard equation with nonlinear diffusion: the non-convex case · 2026 · DOI

• Previous results relied on strong convexity assumptions, excluding relevant cases. The paper addresses this gap by removing the convexity condition.

  On the Cahn-Hilliard equation with nonlinear diffusion: the non-convex case · 2026 · DOI

• Lack of a rigorous measure-theoretic framework for analyzing differential and integral operators in the presence of extremely non-smooth interfaces - Inability of classical approaches to capture the singular behaviors localized at these boundaries

  On the Measure-Theoretic Foundations of Extremely Non-Smooth Interfaces via Structural Calculus (MDPI) · 2026 · DOI

• Spectral Theory of Structural Operators: The spectral analysis of the structural Laplacian Dˢ² remains an open and deeply promising domain. 3 Future Horizons and Open Challenges The mathematical framework formalized in this work opens several fertile avenues for future theoretical investigation and computational deployment: Computational Discretization and Meshless Methods: Standard numerical techniques (such as Level-Set or Immersed Boundary methods) suffer from severe computational overhead and numerical diffusion when capturing extreme interface sharpness. Extension to Singular Millennium Geometries: The capacity of Structural Calculus to absorb localized singular energies without blowing up global energy bounds provides a powerful toolkit for addressing long-standing open problems in mathematical physics, including the global regularity of the 3D Navier–Stokes equations and the analytical formulation of non-perturbative quantum field theories across multi-scale topological defects.

  On the Measure-Theoretic Foundations of Extremely Non-Smooth Interfaces via Structural Calculus (MDPI) · 2026 · DOI

Most-cited papers in Advanced Mathematical Modeling in Engineering

• Upscaling and its application in numerical simulation of long‐term CO₂ storage · Greenhouse Gases Science and Technology · 2012 · 33 citations
• Role of Graphs in Blending Physical and Mathematical Meaning of Partial Derivatives in the Context of the Heat Equation · International Journal of Science and Mathematics Education · 2022 · 14 citations
• Error Estimation and Adaptivity for Differential Equations with Multiple Scales in Time · Computational Methods in Applied Mathematics · 2021 · 7 citations
• Macroscopic Limits of Microscopic Models · International Journal of Mechanical Engineering Education · 2014 · 6 citations
• Variational Approximation for a Non-Isothermal Coupled Phase-Field System: Structure-Preservation & Nonlinear Stability · Computational Methods in Applied Mathematics · 2024 · 5 citations
• A Gradient Discretisation Method for Anisotropic Reaction–Diffusion Models with Applications to the Dynamics of Brain Tumors · Computational Methods in Applied Mathematics · 2021 · 3 citations
• A Convergent Entropy-Dissipating BDF2 Finite-Volume Scheme for a Population Cross-Diffusion System · Computational Methods in Applied Mathematics · 2023 · 3 citations
• Porous media equation on locally finite graphs · Archivum Mathematicum · 2022 · 2 citations
• Averaged reaction for nonlinear boundary conditions on a grill-type Winkler foundation · Mathematical Modelling and Analysis · 2024 · 2 citations
• Anisotropic Adaptive Finite Elements for an Elliptic Problem with Strongly Varying Diffusion Coefficient · Computational Methods in Applied Mathematics · 2022 · 2 citations

Most recent work

• Multicontinuum Homogenization for Two-Phase Model with Linear Transport · Computational Methods in Applied Mathematics · 2026
• Homogenization and dimensional reduction for nonlinear multilayered plates including biological growth when the plate thickness and the size of the heterogeneities are not of the same order of magnitude · Mathematics and Mechanics of Solids · 2026
• A framework for analysis of DEC approximations to Hodge-Laplacian problems using generalized Whitney forms · Mathematics of Computation · 2026
• Resource-efficient numerical computing and elastic wave behavior analysis in complex poroelastic media · Computers and Geotechnics · 2026
• Energy solutions of singular SPDEs on Hilbert spaces with applications to domains with boundary conditions · Stochastics and Partial Differential Equations: Analysis and Computations · 2026
• Optimal Convergence Rates in Multiscale Elliptic Homogenization · Communications on Pure and Applied Mathematics · 2026
• Asymptotic-preserving approximations for stochastic incompressible viscous fluids and SPDEs on graph · IMA Journal of Numerical Analysis · 2026
• Modelling tumour micro-scale dynamics via virtual element methods on polygonized square and disk domains · International Journal of Computer Mathematics · 2026
• Multiscale model reduction and two-level Schwarz preconditioner for H(curl) elliptic problems · Mathematics of Computation · 2026
• Discretizations of Stochastic Evolution Equations in Variational Approach Driven by Jump-Diffusion · Applied Mathematics & Optimization · 2026

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