Mathematical Theory of Navier-Stokes Equations

The three-dimensional Navier–Stokes equations form a system of nonlinear partial differential equations describing the evolution of velocity and pressure fields in viscous incompressible fluids. Since the pioneering work of Leray and Hopf, the existence of global weak solutions is established, yet the question of uniqueness and regularity in three dimensions remains one of the great open problems in mathematical physics. In two dimensions, global existence and smoothness follow from energy estimates and vorticity conservation, while in three dimensions conditional regularity can be proved under smallness or symmetry assumptions. The theory employs functional analytic frameworks such as Sobolev and Lebesgue spaces, alongside blow-up criteria that relate singularity formation to the growth of certain norms. Special classes of solutions—steady states, axially symmetric flows and weak-to-strong uniqueness scenarios—offer valuable insights into stability and structure of possible singularities. Liouville-type theorems impose rigidity in stationary and self-similar regimes, and various approximation schemes, including Faedo–Galerkin and penalisation methods, are used to construct and study weak solutions. Understanding these equations is central to modelling turbulence, predicting global behaviour, and tackling one of the Millennium Prize Problems that asks whether smooth solutions in three dimensions persist for all time.

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Mathematical Theory of Navier-Stokes Equations publication trend

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Technical terms

Incompressible flow: A fluid motion in which the divergence of the velocity field is zero, ensuring constant density.

Weak solution: A solution defined in an integral sense that may lack classical differentiability but satisfies the equations in a distributional framework.

Leray–Hopf weak solution: A global weak solution that also fulfils an energy inequality, named after the mathematicians who first established its existence.

Liouville-type theorem: A rigidity result stating that under certain decay or integrability constraints, a solution must be trivial (for example, identically zero).

Steady-state solution: A time-independent solution to the Navier–Stokes equations representing an equilibrium flow.

References

1. Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic equations. Discrete and Continuous Dynamical Systems (2016).
2. Some Remarks on regularity criteria of Axially symmetric Navier-Stokes equations. Communications on Pure and Applied Analysis (2019).
3. On the Stability of Steady-State Solutions to the Navier–Stokes Equations in the Whole Space. Journal of Mathematical Fluid Mechanics (2022).
4. On the Existence of Leray-Hopf Weak Solutions to the Navier-Stokes Equations. Fluids (2021).
5. On Liouville-type theorem for the stationary compressible Navier–Stokes equations in $ \mathbb{R}^{3} $. Electronic Research Archive (2023).
6. Navier-Stokes Equations—Millennium Prize Problems. Natural Science (2015).