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Open research questions in Navier-Stokes equation solutions

162 gap statements mined from Navier-Stokes equation solutions papers in our 4.5M-paper local library, which holds 1,000 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.

What the literature leaves open

  • The lack of shear viscosity poses significant mathematical challenges. The control of nonlinear terms is a major challenge. The development of novel analytical approaches is required.

    Stabilizing effect of magnetic field on the compressible magnetohydrodynamic flow with vanishing shear viscosity · 2026 · DOI

  • The study of the degenerate viscosities and heat conductivity is challenging and has been less explored. The paper identifies a gap in the understanding of the behavior of solutions to the compressible Navier-Stokes equation.

    Local‐in‐time well‐posedness for the regular solution to the 2D full compressible Navier–Stokes equations with degenerate viscosities and heat conductivity · 2026 · DOI

  • The lack of global well-posedness for the compressible viscous MHD equations with no magnetic diffusion. The need for a fundamental framework for describing the dynamics of electrically conducting fluids.

    Global solutions to 3D compressible MHD equations with partial magnetic diffusion · 2026 · DOI

  • Further study of the behavior of the solution near the blow-up point is needed. Investigation of the applicability of the results to other areas of nonlinear physics.

    A sufficient condition for the existence of smooth solutions of the relativistic cold plasma equations on any given time interval · 2026 · DOI

  • The global regularity problem for the three-dimensional incompressible Navier-Stokes equations with NHL boundary conditions is not well understood. The presence of swirl introduces vortex stretching that may potentially lead to finite-time singularity formation.

    Global Regularity of Axisymmetric Navier-Stokes Equations with NHL Boundary Conditions under a Critical Smallness Condition · 2026

  • The lack of understanding of how magnetic forcing interacts with electrostatic screening to control ionic distributions. The limitation of classical MHD models in describing electrolyte flows.

    Electrolyte flows under magnetic fields: Manning-like counterion condensation in one dimension · 2026

  • The lack of a rigorous foundation for K41 theory. The need for a mathematical framework for understanding turbulence.

    Stability of Anomalous Dissipation for the Forced 3D Navier--Stokes Equations under Geometric Perturbations · 2026

  • The paper suggests that future research should focus on the construction of self-similar blow-up solutions in other cases, such as the non-axisymmetric case. The paper also suggests that future research should investigate the stability of the constructed solutions.

    Self-similar blow-up solutions of incompressible Euler equations in $\mathbb R^d, d\geq 3$ with $C^{1,1-2/d-}$ velocity · 2026

  • The gap is related to the lack of understanding of the behavior of the equations in the axisymmetric case without swirl.

    Self-similar blow-up solutions of incompressible Euler equations in $\mathbb R^d, d\geq 3$ with $C^{1,1-2/d-}$ velocity · 2026

  • The study is limited to a semi-infinite layer in SQG+. The conventional approach of finding a recurrence relation and solving with an eigenvalue solver cannot be directly applied to study SQG+ nonlinear stability.

    Stability of SQG Kolmogorov flow · 2026 · DOI

  • Future research should investigate the effects of ageostrophic dynamics on the stability of SQG systems. Future research should study the nonlinear stability of SQG systems using the correction terms derived by and.

    Stability of SQG Kolmogorov flow · 2026 · DOI

  • Future research should focus on extending the result to more general cases, such as viscoelastic plates with ν > 0. Future research should focus on understanding the physical implications of the result. Future research should focus on applying the result to various applications, such as microfluidic systems and lab-on-chip technologies.

    A Global Existence Result on Weak Solutions for the 3D Navier–Stokes-Plate System with no Contact · 2026 · DOI

  • The paper identifies a need for a global-in-time weak solution for the 3D Navier-Stokes-Plate system. The paper identifies a need for a better understanding of fluid-structure interaction phenomena.

    A Global Existence Result on Weak Solutions for the 3D Navier–Stokes-Plate System with no Contact · 2026 · DOI

  • The lack of a rigorous justification of the Baer–Nunziato system as a macroscopic description for a compressible liquid-vapor flow. The need for a global-in-time well-posedness result for the Cauchy problem of the non-local Navier–Stokes–Korteweg equations.

    Mathematical Justification of a Baer–Nunziato Model for a Compressible Viscous Fluid with Phase Transition · 2026 · DOI

  • The existing methods for simulating variable density incompressible flows have limitations. There is a need for a robust and efficient method that provides exact conservation of volume and pure advection of the density variable.

    A HHO formulation for variable density incompressible flows where the density is purely advected · 2026 · DOI

  • Further analysis of the front tracking scheme is needed. The scheme can be extended to more general fluxes. The results can be applied to other areas of mathematics and physics.

    Front tracking for scalar conservation laws with spatially heterogeneous flux · 2026 · DOI

  • The paper suggests further study of the behavior of the magnetic field in more general settings. The paper suggests further study of the applications of the results to astrophysical contexts.

    Effective Dynamics and Blow Up in a Model of Magnetic Relaxation · 2026 · DOI

  • The paper identifies a need to characterize the limit function in terms of the PDE it satisfies.

    Effective Dynamics and Blow Up in a Model of Magnetic Relaxation · 2026 · DOI

  • The manuscript addresses the lack of a non-PDE-native mathematical route to the official periodic problem.

    Carrier Universality and Global Smoothness for the Periodic 3D Navier–Stokes Equations · 2026 · DOI

  • Future research could focus on extending the results to more general classes of initial data. Future research could focus on applying the results to specific problems in geophysics and astrophysics.

    Global Well-Posedness of the 3D Inhomogeneous MHD Equations with Two-scale Anisotropic Initial Data · 2026 · DOI

  • The paper identifies a need for a new method for analyzing the well-posedness of the equations.

    Global Well-Posedness of the 3D Inhomogeneous MHD Equations with Two-scale Anisotropic Initial Data · 2026 · DOI

  • The study suggests the development of new numerical methods for the simulation of compressible fluids. The paper suggests the exploration of the global well-posedness of the solution. The study suggests the investigation of the behavior of solutions to the compressible Navier-Stokes equation in different regimes.

    Local‐in‐time well‐posedness for the regular solution to the 2D full compressible Navier–Stokes equations with degenerate viscosities and heat conductivity · 2026 · DOI

  • There are no dissipation estimates for the second-order derivatives of and.

    Optimal Decay Rate to the Contact Discontinuity for 1-D Compressible Radiation Hydrodynamics Model · 2026 · DOI

  • The study of relativistic effects associated with the dynamics of light particles is important.

    A sufficient condition for the existence of smooth solutions of the relativistic cold plasma equations on any given time interval · 2026 · DOI

  • Further study of the compressible MHD system with zero shear viscosity. Investigation of the effects of non-zero shear viscosity. Application of the results to astrophysical phenomena.

    Stabilizing effect of magnetic field on the compressible magnetohydrodynamic flow with vanishing shear viscosity · 2026 · DOI

Most-cited papers in Navier-Stokes equation solutions

Most recent work

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Open questions raised by papers listed on this page include: (1) The lack of shear viscosity poses significant mathematical challenges (“Stabilizing effect of magnetic field on the compressible magnetohydrodynamic…”, 2026); (2) The lack of global well-posedness for the compressible viscous MHD equations with no magnetic diffusion (“Global solutions to 3D compressible MHD equations with partial magnetic…”, 2026); (3) Further study of the behavior of the solution near the blow-up point is needed (“A sufficient condition for the existence of smooth solutions of the…”, 2026); (4) Future research should investigate the effects of ageostrophic dynamics on the stability of SQG systems (“Stability of SQG Kolmogorov flow”, 2026); (5) Future research should focus on extending the result to more general cases, such as viscoelastic plates with ν > 0 (“A Global Existence Result on Weak Solutions for the 3D Navier–Stokes-Plate…”, 2026). 162 gap statements have been mined from Navier-Stokes equation solutions papers in our 4.5M-paper local library, which holds 1,000 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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