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Open research questions in Advanced Mathematical Modeling in Engineering

112 gap statements mined from Advanced Mathematical Modeling in Engineering papers in our 4.5M-paper local library, which holds 870 papers on the topic — drawn mostly from each paper's own stated research gap, future-work, challenge and limitation notes, and its abstract. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one.

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What the literature leaves open

  • The paper suggests that future research should focus on the numerical analysis of the proposed approach. The paper suggests that future research should explore the application of the proposed approach to other classes of SPDEs. The paper suggests that future research should investigate the use of Gelfand triples and infinite-dimensional analysis in Hilbert spaces for other problems.

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

  • The limited convergence rate obtained by the method of reiterated homogenization. The lack of a optimal convergence rate for multiscale elliptic homogenization.

    Optimal Convergence Rates in Multiscale Elliptic Homogenization · 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 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

  • Future research can build on the authors' methods to analyze other problems in stochastic homogenization. The study's results can be applied to other areas, such as elliptic operators.

    Sharp error estimates in stochastic homogenization of parabolic systems with time-dependent coefficients · 2026

  • The lack of understanding on how complex background structure alters dynamical behavior of active particles. The need for a general active diffusion-structure relation. The exploration of its active counterpart is still lacking.

    A universal scaling law for active diffusion in complex media · 2026 · DOI

  • The lack of regularity results for the minimizers of the multiphase optimal design problem. The need for a suitable infiltration lemma to establish regularity for the boundary of a single chamber. The absence of sufficiently strong decay estimates for the Dirichlet bulk energy.

    Ahlfors-regularity for minimizers of a multiphase optimal design problem · 2026 · DOI

  • The spectral analysis of the structural Laplacian Dˢ² remains an open and deeply promising domain.

    On the Measure-Theoretic Foundations of Extremely Non-Smooth Interfaces via Structural Calculus (MDPI) · 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 gap in the literature is the lack of study on the solvability of the initial-boundary value problem for a nonstationary diffusion model of an inhomogeneous fluid.

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

  • The numerical scheme is limited to a simple numerical scheme to approximate the L2-gradient flow. The computation is limited to a square calculation box Q with periodic boundary conditions. The error appears to be larger than in the two-dimensional case.

    A Cahn–Hilliard–Willmore phase field model for non-oriented interfaces · 2026 · DOI

  • Future research can focus on improving the numerical scheme to reduce the error. Future research can focus on applying the proposed model to various applications, such as materials science and biology.

    A Cahn–Hilliard–Willmore phase field model for non-oriented interfaces · 2026 · DOI

  • There is a gap in the existing literature regarding the regularity of free boundary surfaces minimizing degenerate area functionals. The weighted case is less understood than the unweighted case. The paper fills this gap by establishing an epsilon-regularity theorem at points in the free boundary.

    Regularity for free boundary surfaces minimizing degenerate area functionals · 2026 · DOI

  • Extending these methods to the transient regime has remained an open challenge: parabolic equations couple space and time through an anisotropic scaling, requiring joint sampling of spatial displacements and backward time steps whose distribution was not previously available in a unified, exact form.

    Walking on Heat Stars for Parabolic Heat Equations with Neumann Boundary Conditions · 2026

  • The paper suggests future research directions in the study of functions with differential constraints. The paper suggests applications of the results in material design and variational frameworks in solid mechanics.

    Rigidity and Functional Properties of $$\textrm{BD}_{\textrm{dev}}(\Omega )$$ · 2026 · DOI

  • The paper identifies a need for a rigidity theorem for BDdev-maps with constant polar vector.

    Rigidity and Functional Properties of $$\textrm{BD}_{\textrm{dev}}(\Omega )$$ · 2026 · DOI

  • The need to extend classical semi-Lagrangian methods beyond the Di Perna-Lions setting. The challenge of modeling socio-economic and collective behavior dynamics.

    Approximation of Viscous Transport and Conservative Equations with One Sided Lipschitz Velocity Fields · 2026 · DOI

  • There is a need to solve Eshelby's problem with a Steigmann-Ogden interface to account for these effects.

    A spherical Eshelby inclusion with a Steigmann-Ogden interface in a finite domain · 2026 · DOI

  • The paper does not provide a comprehensive treatment of all possible classes of SPDEs. The paper focuses on equations driven by highly irregular noise with bilinear nonlinearities. The paper does not address the numerical analysis of the proposed approach.

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

  • 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 study must obtain optimal homogenization error in strong and weak norms on C1 cylinders.

    Sharp error estimates in stochastic homogenization of parabolic systems with time-dependent coefficients · 2026

  • To extend the result to other types of coefficients. To apply the result to other problems in homogenization theory. To improve the design of composite materials using the result.

    Optimal Convergence Rates in Multiscale Elliptic Homogenization · 2026 · DOI

  • 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

  • Further testing of the universality of the scaling relation through simulations of active colloidal tracers with diverse reorientation dynamics, - Exploring the effect of inertia on the long-time diffusion of active tracers in complex media, - Investigating the application of the universal scaling law to other complex…

    A universal scaling law for active diffusion in complex media · 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

Most-cited papers in Advanced Mathematical Modeling in Engineering

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Open questions raised by papers listed on this page include: (1) The paper suggests that future research should focus on the numerical analysis of the proposed approach (“Energy solutions of singular SPDEs on Hilbert spaces with applications to…”, 2026); (2) The lack of a optimal convergence rate for multiscale elliptic homogenization (“Optimal Convergence Rates in Multiscale Elliptic Homogenization”, 2026); (3) Future research can focus on applying the results to specific stochastic partial differential equations (“Discretizations of Stochastic Evolution Equations in Variational Approach…”, 2026); (4) The lack of conditions for unique solvability in the scale of Sobolev spaces for inhomogeneous equations (“Dirichlet–Neumann Problem for Partial Differential Equations Unsolved With…”, 2026); (5) The lack of understanding on how complex background structure alters dynamical behavior of active particles (“A universal scaling law for active diffusion in complex media”, 2026). 112 gap statements have been mined from Advanced Mathematical Modeling in Engineering papers in our 4.5M-paper local library, which holds 870 papers on the topic; the gaps come from whichever of those papers state one. They are mostly the research gaps the authors state and the papers' abstracts, plus future-work, limitations and challenges passages. The ones listed below are a selection still marked open; each names the study that raised it, with a DOI link where the paper has one, so you can read the original claim in context.

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