COMMENT NATURE REVIEWS | PHYSICS In many cases, the partial differential equations that describe a physical system embody a tension between linear effects, which are generally well understood and often act to stabilize or regularize the solution, and non- linear effects, which are more poorly understood and can lead to pathological behaviour such as development of singularities in finite time. A model case of this ten- sion arises in the incompressible Navier–Stokes equa- tions. The notorious global regularity problem for the Navier–Stokes equations 1 asks whether the finite time blow-up scenario is mathematically prohibited from occurring, which would imply that solutions to these equations will remain smooth for all time. The prob- lem remains unresolved despite many decades of inten- sive effort and countless failed proofs or disproofs of global regularity. The Navier–Stokes equations read ν ⋅∇ ∆ ∇ u u u u p ∂ + = - (1) t ∇⋅ u =0 (2) where R R → u : [0, +∞) × 3 3 is the velocity field of an incompressible viscous fluid, and R R → p : [0, +∞) × 3 3 is an unspecified pressure field, which can be deter- mined from the velocity field u and the incompressibility condition ∇⋅ u =0. In the mathematical literature one often selects units so that the kinematic viscosity ν is equal to 1; in this discussion I also neglect the effect of external forces such as gravity for the sake of simplic- ity. The main terms to focus on here are the convec- tion term ⋅∇ u u, which describes momentum flux in spatially non-uniform flow, and the viscosity term Δu. If the fluid has a typical speed v and wave number k (so that the velocity field oscillates at an amplitude ~v and wavelength ~2π/k), then the nonlinear convection term ⋅∇ u u would be expected to have magnitude kv ≈ 2 , whereas the linear viscosity term would have magnitude kv ≈ 2 . This leads to the following heuristic arguments: when the wave number greatly exceeds the typical speed, then viscous effects dominate and the fluid should dissi- pate its energy; however, when instead the speed greatly exceeds the wave number, convection (and incompress- ibility) effects dominate, and the fluid should behave in a highly nonlinear fashion. In principle, it might even be possible in this case that the speed and the wave num- ber both go to infinity in finite time, a scenario known as finite-time blow-up. Of course, such blow-up does not mean that a physical fluid such as water can exhibit this behaviour, but it does mean that the Navier–Stokes equations cease to be an accurate model for such a fluid in these cases. The above discussion can be made fully rigorous by using function space norms of the velocity field as precisely defined proxies for various combina- tions of informally defined concepts such as speed and wave number. Numerical studies show that although solutions to the Navier–Stokes equations may initially exhibit a cer- tain amount of turbulence (in which the wave number increases significantly), the speed does not keep pace with this increase in wave number, and eventually the solution reaches the viscosity-dominated regime in which the fluid begins to exhibit linear behaviour and settles down to equilibrium. This phenomenon can be partially explained by the energy identity R R ν ∇ ∫ ∫ utx dx utx dx ∂ 1 2 (, ) =- (, ) (3) t 2 2 3 3 where the right-hand side can be interpreted as the energy dissipation arising from the effect of viscosity (in units in which the fluid has unit density, for simplicity). This implies that the kinetic energy R ∣ ∣ ∫ ut x dx (, ) 1 2 2 3 is decreasing in time. Because the kinetic energy is propor- tional to v 2 multiplied by the volume of space in which the velocity field is concentrated, this decrease of energy suggests that the fluid speed is unlikely to increase fast enough to avoid the viscosity-dominated regime. However, viscosity effects can be overcome if the fluid exhibits a high amount of spatial intermittency, in the sense that u is highly concentrated in space (that is to say, the speed is negligible outside of a small region of space). The heart of the global regularity problem, there- fore, is to understand precisely how much intermittency Searching for singularities in the Navier–Stokes equations Terence Tao Despite much effort, the question of whether the Navier–Stokes equations allow solutions that develop singularities in finite time remains unresolved. Terence Tao discusses the problem, and possible routes to a solution. Department of Mathematics, University of California, Los Angeles, Los Angeles, CA, USA. e-mail: tao@math.ucla.edu https://doi.org/10.1038/ s42254-019-0068-9