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Open research questions in Navier-Stokes equation solutions

89 unresolved questions extracted from the limitations and future-work sections of 285 Navier-Stokes equation solutions papers in our library. Each links back to the study that raised it.

What the literature leaves open

  • The non-scaling invariant Navier-slip boundary condition is a challenge. The infinite two-dimensional wedge-shaped domain is a challenge. The need to prove higher regularity up to the tip of the wedge is a challenge.

    Well-posedness of the Stokes equations on a wedge with Navier-slip boundary conditions · 2026 · DOI

  • Constructing explicit solutions with mixed sign to the 2D Euler equations. Identifying the optimal regularity for mixed sign vortex sheets. Resolving the DiPerna-Majda 2D gap problem.

    A sparse resolution of the DiPerna–Majda gap problem for 2D Euler equations · 2026 · DOI

  • Further study of the properties of the measure constructed in the paper. Further study of the implications of the results for the study of turbulence and anomalous dissipation.

    A sparse resolution of the DiPerna–Majda gap problem for 2D Euler equations · 2026 · DOI

  • Further study of the properties of critical points in the axially symmetric cavity flow problem. Application of the results to the analysis of cavity flow problems and jet flow problems.

    Non-minimizing Axially Symmetric Cavity Flow · 2026 · DOI

  • Further study of the dynamics of the new equation. Extension of the numerical algorithm to 3D. Exploration of the rotational form of the nonlinearity in other equations.

    Burgers equation with a twist: a study on rotational-form equations · 2026 · DOI

  • The global well-posedness of the Navier-Stokes equations remains an open problem. The rotational form of the nonlinearity has not been fully explored.

    Burgers equation with a twist: a study on rotational-form equations · 2026 · DOI

  • The paper identifies a gap in the prior work on the 3D generalized Tropical Climate Model with damping. The paper's result fills this gap by establishing the global well-posedness of a strong solution to the model.

    Global Well-Posedness for the 3D Generalized Tropical Climate Model with Damping · 2026 · DOI

  • The lack of a dissipation mechanism in the inviscid compressible systems makes global solvability a challenging problem. The global well-posedness and large time behavior of solutions for the isentropic setting are not well understood.

    Global well-posedness of the inviscid resistive isentropic compressible MHD system · 2026 · DOI

  • The paper assumes certain conditions on the initial data, including smallness and regularity conditions. The paper does not address the question of uniqueness of solutions.

    Analyticity and asymptotic behavior of solutions to the compressible Navier-Stokes-Korteweg equations with the zero sound speed in scaling critical spaces · 2026 · DOI

  • To investigate the uniqueness of solutions. To study the behavior of solutions in other regimes, such as the high-frequency regime. To apply the results of the paper to the study of other systems modeled by the compressible Navier-Stokes-Korteweg equations.

    Analyticity and asymptotic behavior of solutions to the compressible Navier-Stokes-Korteweg equations with the zero sound speed in scaling critical spaces · 2026 · DOI

  • The nonlinearity of the equations poses a challenge. The roughness of the Kraichnan noise is a technical challenge. The lack of viscosity in the equations is a challenge.

    Strong uniqueness by Kraichnan transport noise for the 2D Boussinesq equations with zero viscosity · 2026 · DOI

  • Future research could explore the extension of the results to more general equations. The authors suggest that the strong uniqueness by Kraichnan transport noise could be used to study other fluid dynamics problems.

    Strong uniqueness by Kraichnan transport noise for the 2D Boussinesq equations with zero viscosity · 2026 · DOI

  • The paper assumes suitable assumptions on the dissipation and damping terms. The paper's result is limited to a > 1/2.

    Global Well-Posedness for the 3D Generalized Tropical Climate Model with Damping · 2026 · DOI

  • Surprisingly, even though the result in the multidimensional case is by now classical, the one-dimensional case has not been elucidated yet as far as we know.

    The One-Dimensional Compressible Navier–Stokes Equations in Critical Regularity Spaces · 2026 · DOI

  • The study of critical points in the axially symmetric cavity flow problem is not well understood. The rectifiability of the free boundary is not established.

    Non-minimizing Axially Symmetric Cavity Flow · 2026 · DOI

  • We also examine more exotic spatially modulated solutions that satisfy a third-order dynamical system, whose complete classification remains an open challenge.

    Couette-Taylor instabilities in the small gap regime: the very counter-rotating case · 2026

  • The stability analysis assumes periodic boundary conditions on the 2D torus T^2; the applicability of the stabilizing effect of the magnetic field to compressible MHD flows on other domains (e.g., bounded domains with non-slip boundaries, exterior domains) or with different boundary condition types remains unexplored.

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

  • The bounds on nonlinear terms like H · ∇H and H∇H in Sobolev spaces H^{[N/2]+3} are established using Banach algebra properties, but the paper does not provide explicit constants C or investigate their dependence on the magnetic field topology, domain geometry (T^2), or the ratio of magnetic pressure to kinetic energy.

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

  • The absence of heat conduction effects poses substantial difficulties in establishing energy estimates and decay estimates of the solution. 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 condition for smoothness is only sufficient and may be far from necessary. The numerical experiment is limited to a specific set of initial data.

    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 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 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 [5] and [6].

    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 gap in the mathematical theory of fluid-structure interaction systems. 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

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89 open questions have been extracted from the limitations and future-work passages of 285 Navier-Stokes equation solutions papers in our library. Each one below links back to the study that raised it, so you can read the original claim in context.

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