From Euler to AI:
269 years of fluid equations

Córdoba, Martínez-Zoroa, and Zheng [8] extended this approach to hy-podissipative Navier–Stokes with small positive dissipation orders and forcing in a local well-posedness class.Navier–Stokes · p. 2

Recent preprints of Chen and Shkoller [5, 29] prove blowup on R3 for finite-energy axisymmetric velocities without swirl in C1,α, for every 0 < α < 1/3.Euler · p. 2

Recent preprints of Chen and Shkoller [5, 29] prove blowup on R3 for finite-energy axisymmetric velocities without swirl in C1,α, for every 0 < α < 1/3.Euler · p. 2

Córdoba, Martínez-Zoroa, and Zheng [13] proved blowup from finite-energy C1,α initial velocities that are smooth away from the origin.Euler · p. 1

Jeong, Martínez-Zoroa, and Ożański [23] constructed three-dimensional Euler solutions with a continuously decreasing Sobolev regularity threshold, starting from vorticity in Hs, 0 < s < 3/2.Euler · p. 2

For smooth initial data, Chen and Hou [6, 7] proved blowup in an axially periodic cylinder with an impermeable wall, combining analysis with rigorous numerical estimates.Euler · p. 2

For the incompressible porous media equation, Córdoba and Martínez-Zoroa [7] constructed singularities from smooth initial data by suc-cessive amplification of oscillatory layers, using approximations of increasing order to keep every spatial derivative of the source uniformly bounded.Navier–Stokes · p. 2

For the incompressible porous media equation, Córdoba and Martínez-Zoroa [11] constructed singularities from smooth initial data by successive amplification of oscillatory layers, using approximations of increasing order to keep every spatial derivative of the source uniformly bounded.Euler · p. 2

For two-dimensional Euler, Córdoba, Martínez-Zoroa, and Ożański [12] constructed global solutions with initial velocity in Hr, 1 < r < 2, that lose a positive amount of Sobolev regularity at every positive time.Euler · p. 2

Córdoba and Martínez-Zoroa [6] constructed finite-time singularities for the forced three-dimensional Euler equations through successive amplification of increasingly concentrated vortex layers.Navier–Stokes · p. 2

For each fixed 0 < ε < 1/2, Córdoba and Martínez-Zoroa [10, Theorem 1.1] constructed finite-energy solutions to the forced Euler equations on R3, with velocity in C3,1/2 before blowup and an external force uniformly bounded in C1,1/2−ε ∩L2.Euler · p. 2

Córdoba and Martínez-Zoroa also use a vorticity layer to amplify a more localized layer in their forced Euler construction [10, §1.2].Euler · p. 4

Elgindi and Pasqualotto [19] proved blowup from finite-energy C1,α initial velocities that are smooth away from a circle.Euler · p. 2

Albritton, Brué, and Colombo [1] subsequently constructed distinct suitable Leray–Hopf solutions with zero initial velocity and the same force, using an unstable vortex in similarity variables.Navier–Stokes · p. 2

Córdoba and Martínez-Zoroa [9] used this approach to prove ill-posedness and loss of regularity for the inviscid surface quasi-geostrophic equation.Euler · p. 2

For smooth initial data, Chen and Hou [6, 7] proved blowup in an axially periodic cylinder with an impermeable wall, combining analysis with rigorous numerical estimates.Euler · p. 2

Elgindi [15] proved finite-time blowup on R3 for axisymmetric velocities without swirl in C1,α, for sufficiently small α > 0.Euler · p. 1

Elgindi, Ghoul, and Masmoudi [16] proved stability of this blowup and obtained finite-energy examples with the same regularity.Euler · p. 1

Elgindi and Masmoudi [18] proved strong ill-posedness in the velocity spaces Ck ∩L2 for integers k ≥1, by estimating singular integral terms along the particle flow.Euler · p. 2

For the original equations, Buckmaster and Vicol [4] used convex integration to establish nonuniqueness among finite-energy weak solutions.Navier–Stokes · p. 2

Buckmaster, De Lellis, Székelyhidi, and Vicol [4] constructed solutions in this regularity range whose kinetic energy equals any prescribed smooth positive function of time.Euler · p. 2

On the same periodic domain, Isett [22] proved that, for every 0 < β < 1/3, there are solutions with spatial Cβ regularity whose kinetic energy is not constant.Euler · p. 2

Further precedents for the wave dynamics include the centrifugal-instability criteria of Leibovich and Stewartson [15] and of Billant and Gallaire [2, 3] and the exact viscous shearing waves of Singh and Sridhar [19].Navier–Stokes · p. 2

The use of oscillations to realize a prescribed stress is central to the Euler constructions of Daneri and Székelyhidi [10].Navier–Stokes · p. 2

Elgindi and Jeong [17] gave a construction in which localized vorticity generates a hyperbolic flow that stretches the vorticity gradient.Euler · p. 2

They later constructed continuous solutions on the three-dimensional torus with decreasing kinetic energy [27].Euler · p. 2

Further precedents for the wave dynamics include the centrifugal-instability criteria of Leibovich and Stewartson [15] and of Billant and Gallaire [2, 3] and the exact viscous shearing waves of Singh and Sridhar [19].Navier–Stokes · p. 2

De Lellis and Székelyhidi [26] constructed nonunique bounded weak solutions.Euler · p. 2

The use of an upper bound on the pressure Hessian to control this quadratic action has a precedent in Brenier’s short-time action-minimization criterion for Euler flows [3, Section 2].Euler · p. 16

At the nonlinear level, Cheverry [8] constructed oscillatory approximate solutions of incompressible Euler and analyzed the additional phase corrections required by their nonlinear evolution.Euler · p. 3

Le Dizès and Leblanc [25] showed that their construction generally satisfies the equations only along a single trajectory, preventing its use as the background for the next iteration.Euler · p. 2

Further precedents for the wave dynamics include the centrifugal-instability criteria of Leibovich and Stewartson [15] and of Billant and Gallaire [2, 3] and the exact viscous shearing waves of Singh and Sridhar [19].Navier–Stokes · p. 2

Fabijonas and Holm [20] proposed iterated superpositions of such waves.Euler · p. 2

Escauriaza, Seregin, and Šverák [11] proved regularity for the unforced Cauchy problem under a bounded scale-invariant L∞ t L3 x norm.Navier–Stokes · p. 2

The methods of Lifschitz and Hameiri and of Friedlander and Vishik [17, 14] describe the evolution of wavevectors and velocity polarizations along a background flow.Navier–Stokes · p. 2

For perturbations with a rapidly varying phase, Lifschitz and Hameiri and Friedlander and Vishik [28, 21] studied the leading equations for the phase gradient and the velocityEuler · p. 2

The methods of Lifschitz and Hameiri and of Friedlander and Vishik [17, 14] describe the evolution of wavevectors and velocity polarizations along a background flow.Navier–Stokes · p. 2

For perturbations with a rapidly varying phase, Lifschitz and Hameiri and Friedlander and Vishik [28, 21] studied the leading equations for the phase gradient and the velocityEuler · p. 2

Craik and Criminale [9] constructed exact finite-amplitude waves on affine background flows, exploiting the cancellation of the wave’s quadratic self-interaction.Navier–Stokes · p. 2

Craik and Criminale [14] constructed exact finite-amplitude waves on affine background flows, exploiting the cancellation of the wave’s quadratic self-interaction.Euler · p. 2

The Beale–Kato–Majda criterion [1] states that a smooth three-dimensional Euler solution continues past time T if ∫︁T 0 ∥curl u(t)∥L∞dt < ∞.Euler · p. 1

Further precedents for the wave dynamics include the centrifugal-instability criteria of Leibovich and Stewartson [15] and of Billant and Gallaire [2, 3] and the exact viscous shearing waves of Singh and Sridhar [19].Navier–Stokes · p. 2