A linear equation cannot explode in finite time. A nonlinear one can, from perfectly smooth data, and the mathematician most associated with understanding how it happens in the equations of waves and fluids has just received the Breakthrough Prize. This interlude records what the sources say about him, and uses the smallest possible example to show what blow-up is.
The object. The equation u′ = up with u(0) = u0 > 0. For p = 1 the solution grows but is finite at every time. For p = 2 and p = 3 it is finite only up to a time T and then leaves every bound. Choose p and u0 and read off T and the values just before it. This is a toy: it is not Merle’s equation.
Frank Merle works on nonlinear partial differential equations, in particular on solutions that become singular in finite time. The facts below were read on 4 November 2026 from the pages listed at the end; they were not checked by the script. Where a source gave a citation, it is quoted exactly and briefly.
| Fact | Source |
|---|---|
| Frank Merle was born 22 November 1962 in Marseille. | Wikipedia; his CV gives the same date |
| He took his PhD in 1987 under Henri Berestycki, and has been a professor at Cergy-Pontoise from 1991. | Wikipedia; his CV |
| He holds the Chair of Analysis CY Cergy Paris Université – IHES. | IHES |
| He received the Bôcher Memorial Prize in 2005, for “his fundamental work in the analysis of nonlinear dispersive equations”. | Wikipedia, Bôcher Prize list |
| He received the Bôcher Memorial Prize in 2023, shared with Pierre Raphaël, Igor Rodnianski and Jérémie Szeftel, for work establishing the existence of blow-up solutions to certain defocusing nonlinear Schrödinger equations and to the compressible Euler and Navier–Stokes equations. | AMS |
| He received the Clay Research Award in 2023, with the same three colleagues, “in recognition of their profound contributions to the theory of nonlinear partial differential equations”, announced on 5 May 2023. | Clay Mathematics Institute |
| He was elected to the Académie des sciences on 12 December 2023 (the source does not say in which category). | IHES |
| His CV lists Academia Europaea in 2020, and he was a plenary speaker at the International Congress of Mathematicians in Seoul in 2014. | his CV |
| He received the Breakthrough Prize in Mathematics in 2026, announced on 18 April 2026, with the citation “For breakthroughs in nonlinear evolution equations”. The prize page lists $3 million for each Breakthrough Prize and no co-laureate in mathematics. | Breakthrough Prize |
Blow-up means a solution that is perfectly well behaved up to some time, and then, in finite time, grows past every bound. It needs a feedback that gets stronger as the quantity gets larger: the bigger u is, the faster it grows, and the faster it grows the bigger it gets. In u′ = u2 the rate is the square of the size, and the size doubles faster each time it doubles.
The derivative of 1/(c − t) is 1/(c − t)2, which is the square of 1/(c − t), so the formula solves u′ = u2; at t = 0 it equals u0. As t increases to T = 1/u0 the denominator tends to 0 from above. The product u(t)(T − t) is 1 by the formula. For u0 = 1 the blow-up time is T = 1, and u(0.5) = 2, u(0.9) = 10, u(0.99) = 100, u(0.999) = 1000: at each step the distance to T is divided by 10 and the solution is multiplied by 10. A solution that starts larger blows up sooner: T = 1/2 for u0 = 2 and T = 4 for u0 = 1/4.
The identity u(t)(T − t) = 1 is the simplest picture of a profile: multiply the solution by the rate at which it blows up, and what is left does not change. This is only an analogy with the word “profile” as it is used for partial differential equations; it is not a statement about Merle’s results. The script checks, exactly, that the formula satisfies u′ = u2 and u(0) = u0 for u0 in {1/4, 1/2, 1, 2} at ten times each, the identity u(t)(T − t) = 1, and the values 2, 10, 100, 1000 above. Lean: toy_quadratic_deriv, toy_quadratic_profile and toy_quadratic_unbounded, compiled.
For u′ = u (the case p = 1) the feedback is only proportional: doubling the size doubles the rate, and the solution u0et grows quickly but is finite at every time. Blow-up needs growth faster than proportional. And once the rate is not proportional to the size, the superposition of chapter 2 is gone: the sum of two solutions is no longer a solution.
(a) If u′ = u3, then (u−2)′ = −2u−3u′ = −2, so u−2 is a straight line of slope −2 and reaches 0 at t = 1/(2u02). For u0 = 1 that is T = 1/2; for u0 = 1/2, T = 2; for u0 = 3, T = 1/18. (b) u0et is finite for every t. (c) (u1 + u2)′ = u12 + u22, while (u1 + u2)2 = u12 + u22 + 2u1u2; the two differ by 2u1u2. At t = 1/3, for the solutions with u0 = 1/2 and u0 = 1/4, that difference is positive.
The general rule, which this page states without proof, is that for u′ = f(u) with f positive and increasing, the solution blows up in finite time exactly when the integral of du/f(u) out to infinity is finite. For f(u) = up that integral is infinite when p = 1 (it is a logarithm) and finite when p > 1, where it equals the blow-up time: 1/u0 for p = 2 and 1/(2u02) for p = 3.
The script checks (a) on u2 exactly for u0 in {1/2, 1, 3} at ten times each, and the three blow-up times; that et is finite at t = 10, 100 and 700; the partial sums behind the integral criterion for p = 1 and p = 2 (a finite check that illustrates it, not a proof of it); and the failure of superposition at t = 1/3, exactly.
The equation x′ = μ + x2 is the standard picture of a fold (a saddle-node bifurcation, the “fold” of Interlude 1). For μ < 0 it has two equilibria, one steady and one not. As μ rises to 0 they merge, and at μ = 0 the equation is exactly u′ = u2, the toy of Part 2. For μ > 0 there are no equilibria and every solution runs off to infinity in finite time. So, for this one ordinary differential equation, the blow-up and the fold are the same thing seen from two sides.
For μ < 0, μ + x2 = 0 exactly at x = ±√(−μ); for μ = −1/4, −4, −9 these are ±1/2, ±2, ±3. For μ > 0, the derivative of √μ tan(√μ t) is μ sec2(√μ t) = μ(1 + tan2(√μ t)) = μ + x2, and tan runs from −∞ to +∞ over an interval of length π, here π/√μ in t. The linearisation at 0 of u′ = u2 is v′ = 0 (the derivative 2u vanishes at 0): it predicts a constant solution, while the real solution starting at 1 reaches 100 at t = 0.99 and keeps going. This is the situation of Interlude 1, where linearisation said nothing.
This identification is exact for the ordinary differential equation and for nothing else here. Both words are polysemous, which is the hazard: “fold” here means only the saddle-node normal form x′ = μ + x2 (it also names the Whitney fold of a map of the plane, and a crease in many other settings), and “blow-up” means a solution leaving every bound in finite time (it also means other things in algebraic geometry and in numerical analysis). Two meanings that share a word are not thereby one meaning. Whether the singularities of partial differential equations that Merle and his collaborators study are folds in any sense is a different and much harder question, which this page does not address and the sources read do not claim; the words “blow-up” and “fold” sharing a picture in one dimension is not evidence about them. If a reader wants to claim a connection with another system, such as the DM3 system of this series, the claim needs its own derivation, not a resemblance. The script checks the equilibria for the three values of μ, the merger at μ = 0, the failed linearisation, the tangent solution (numerically, at twelve points, an identity that is proved above), and the passage times π/√μ for μ = 1/4, 1 and 4.
A wave packet of the linear Schrödinger equation spreads, because its short waves run faster than its long ones: this is dispersion. The nonlinear term in the critical equation pulls the packet together. Which one wins is decided by a single number, the energy, through an identity for the size of the packet, y(t) = ∫ x²|u|² dx. The identity says that the acceleration of y is a constant, 16 times the energy. A positive acceleration means spreading; a negative one means the size would have to turn negative, which is impossible, so the solution must collapse. This is the implosion, in the same sense as the one-line toy of Part 2, and dispersion is its opposite.
The first derivative. From ut = i(uxx + |u|4u), ûut + conj = 2 Re(i û uxx) + 2 Re(i|u|6) = −2 Im(û uxx) = −2 ∂x Im(û ux), because |u|6 is real. Integrating by parts, y′ = 4∫ x Im(û ux) dx. (Here û means the complex conjugate of u.)
The second derivative, organised by scaling. Take the dilation uλ(x) = λ1/2u(λx), which keeps the mass. Then ∫|∂xuλ|² = λ²∫|ux|² and ∫|uλ|6 = λ²∫|u|6: both terms of the energy scale with the same power, and that is what “critical” means. So E(uλ) = λ²E(u). The second derivative of y is 8 times the derivative of the energy along dilations at λ = 1, namely 8 × 2E = 16E; term by term, the kinetic part gives 8∫|ux|² = 16 × ½∫|ux|² and the nonlinear part gives −(8/3)∫|u|6 = 16 × (−1/6)∫|u|6. For another power of the nonlinearity the two terms scale differently, and y″ is no longer a multiple of E. The direct expansion by integration by parts takes a page and is not written out here; the identity is checked numerically below.
Solving for y. E is conserved, so y″ is the constant 16E, and integrating twice gives the parabola. If E < 0 its leading coefficient 8E is negative, so y < 0 for large t; since y ≥ 0 always, the solution cannot be defined for all t > 0. □
What was checked. The script integrates the equation numerically (split-step Fourier, floating point, not exact) for the initial data u0 = A e−x²/2e0.7ix with A = 0.5, 1.5, 2.5, and compares a finite-difference second derivative of y at t = 0 with 16E: the two agree to five or six digits in each case (energies +0.217, +0.031 and −36.2). It checks that Q solves its equation (residual about 3 × 10−11), that its mass is √3π/2, that E(Q) = 0, and that the inequality E ≥ ½(1 − (M/MQ)²)∫|ux|² holds for the three test functions, which illustrates the sharp inequality and does not prove it. The three test functions have masses 0.44, 3.99 and 11.1, so two of them lie above MQ; this is a check of an identity, not a study of blow-up.
What the virial argument cannot do. It needs y(0) finite, and it gives blow-up for E < 0 with no information on the rate. Removing the finiteness condition, and bounding the rate from above, near the threshold mass, is what the 2005 paper of Merle and Raphael does (Annals 161); this page has not read its proofs. The dispersive side below the threshold mass (global solutions) rests on the inequality above and is not proved here.
Analogy, not derivation. The competition between a term that spreads (here the Laplacian, in CATGT the coupling J) and a term that localises (here the nonlinearity, in CATGT the on-site binding λ) has the same shape as the self-trapping radius r*(λ) = a√(J/λ) of the CATGT paper. That is a shared functional form, not a shared invariant. r* is a crossover length on a lattice; the threshold here is a mass in a scale-invariant continuum equation. No derivation connects the two, and none is claimed. The compressible-fluid implosion of Merle, Raphaël, Rodnianski and Szeftel (Annals 196, 2022) is a different equation, and this page has not read those papers either.
Two plots from the numerics behind sections 5 and 6, drawn by interlude2-figs.py from its own data (stored in interlude2-figs.json). They are floating-point illustrations. The first shows the virial parabola against a computed solution, on one side of the threshold and the other. The second shows the lattice, where nothing can blow up, and what the nonlinearity does there instead.
The symbol r with an eye subscript is the limiting blow-up speed in Theorem 1.1 of Merle, Raphaël, Rodnianski and Szeftel, On the implosion of a three dimensional compressible fluid (arXiv:1912.11009), defined in their (1.4). With ℓ = 2/(γ−1) and dimension d:
reye(d,ℓ) = (d+ℓ)/(ℓ+√d) for ℓ < d, and reye(d,ℓ) = 1 + (d−1)/(1+√ℓ)² for ℓ > d.
The theorem (d = 3, ℓ ≠ 3, ℓ outside a countable exceptional set) gives a discrete sequence of blow-up speeds 1 < rk < reye(3,ℓ) with rk → reye(3,ℓ). The curve below is the formula only: no equation is solved here.
What is and is not checked. The formulas, the range ℓ > √3 for Navier–Stokes and the lower bound ℓ0(3) = √3 were read from the typeset pages 4 and 5 of arXiv:1912.11009v2 (PDF, 17 June 2020) and agree with the LaTeX source of the arXiv HTML version. A plain-text extraction of the paper had lost every square-root sign and so read ℓ > 3 and γ < (2+√3)/3; that was an extraction loss, not a difference in the paper. The verifier recomputes the algebra: the branches meet at 3 − √3 when ℓ = d = 3, (1.8) solves to ℓ > ℓ0(d) = (2√d − d)/(d − 1 − √d) with ℓ0(3) = √3, and e > 0 of (1.12) is the same as γ < (2+√3)/√3. The theorem also excludes ℓ = 3. The blow-up rates are ‖u‖∞ ~ (T−t)−(rk−1)/rk and ‖ρ‖∞ ~ (T−t)−ℓ(rk−1)/rk. The values r2, r3, r4 are quoted from the review arXiv:2301.10101 and not recomputed. The Annals Part I was not read; this is the earlier arXiv paper.
Derived from the equations, not only read. reye-derive.py starts from radial isentropic Euler, puts in the self-similar ansatz with speed r, reduces it to a two-dimensional autonomous ODE, and computes the eigenvalue ratio k at the sonic point. For γ = 5/3 and d = 3 it gives exactly k = 2, 3, 4 at the three values r2, r3, r4 quoted in the review, so those values and this ODE agree. For ℓ < d, in five cases ((d,ℓ) = (3,0.5), (3,1), (3,2), (2,1), (4,2)), the smaller eigenvalue goes to zero linearly as r rises to the formula (d+ℓ)/(ℓ+√d), so k diverges exactly there. For ℓ > d the same script was not able to resolve the behaviour near the formula’s r+ (the sonic-point root finding is noisy), and nothing is claimed about that branch from it. This is floating point, and it does not prove that smooth self-similar profiles exist. ReyeAlgebra.lean states four algebraic facts about the formula (the branches agree at ℓ = d for every d, the threshold ℓ > √3 for d = 3, and positivity on the r+ branch) and was compiled by the author and all four theorems are clean (standard axioms only); one of them, threshold_three, needed a rewrite because the Mathlib lemma name div_lt_div_iff no longer exists.
The previous section rests on a check with two independent parties: a value quoted in a review (r2, r3, r4 for γ = 5/3) and the same value recomputed here from the equations. The check could have failed, and it did not: the ratio came out 2, 3 and 4. That is the whole content of the check. It is not blind, since the review’s values were known before the ODE was solved, and agreement shows that two routes meet, not that either is right about the fluid. Witness.lean states the elementary fact behind the practice of this book of giving every number two routes (the exact script and Lean, then the author’s own compile). Model: a testimony is one report from each party about the same matter, and the only check the parties can run on each other is whether the reports agree. witness_minimum says that this check can fail for some profile of reports exactly when there are at least two parties and at least two possible reports; with one party it cannot fail, so the report cannot be contradicted by anyone who gave it. majority_three and no_majority_two say that with two disagreeing parties a dispute is detected but not located, and that with three yes/no reports a strict majority always exists. The file compiled clean (all six theorems, standard axioms only). It says nothing about truth, independence of the parties, or the sources it is named after; its header says so.
The word has been used here for three different things. A wave of the critical Schrödinger equation focuses on a point (part 5). A gas converges on a point in the compressible-fluid work of Merle, Raphaël, Rodnianski and Szeftel, which this page has not read. And a hollow hull collapses under outside pressure: the loss of the Titan submersible in June 2023 was a hull failure of that kind, and this page makes no claim about its cause, which belongs to the official investigations. The first two are about equations whose solutions run to a point in finite time. The third is about a structure: a shell that holds until the pressure passes a threshold and then buckles. Nothing here links the three beyond the word, and the one thing they share, a threshold, has a different origin in each.
The question for a structure is how to prevent the collapse. There are two ways. One is to resist: make the wall strong enough to carry the whole load, which is what a crewed hull with air inside at one atmosphere must do. The other is to equalize: let the pressure inside match the pressure outside, so that the wall carries almost nothing. Animals living in the deep sea mostly do the second, by being made of water and having no large gas spaces; electronics on remote vehicles are often immersed in oil for the same reason. This sentence about animals and oil-filled housings is general background and is not sourced or checked on this page.
The wall stress. Cut the sphere along an equator. The pressure presses on the disc of area πR² with total force pπR², and this is balanced by the stress σ acting on the ring of wall of cross-section 2πRt: pπR² = σ 2πRt, so σ = pR/(2t). For the numbers above, 50×106 × 1 / (2 × 0.1) = 250×106 Pa.
The threshold, by an energy balance. Suppose the wall develops a wrinkle w = w0 cos(kx). Three energies per unit area, each proportional to w0², compete. Bending the wall costs D k4 with D = E t3/(12(1 − ν²)); the radial displacement stretches the curved wall, which costs E t/R²; and the compressive stress does work σ t k² as the wall shortens. The wrinkle can grow when D k4 + E t/R² ≤ σ t k², that is when σ ≥ D k²/t + E/(R²k²). The right side is a sum of two positive terms whose product is DE/(tR²) for every k, so by the arithmetic–geometric mean inequality it is at least 2√(DE/t)/R, with equality when k4 = E t/(D R²). Putting in D, 2√(DE/t)/R = E t/(R√(3(1 − ν²))), and then p = 2tσ/R gives pcr = 2E/√(3(1 − ν²)) · (t/R)². □
What this derivation is. The wrinkle is a one-dimensional model standing in for the sphere’s mode. Its minimum agrees exactly with the classical constant, which I checked symbolically, but the agreement is a check against the known value; the sphere’s linearised shell equations are not solved here, so this is not a derivation from shell theory. The lesson that does not depend on the constant: pcr grows like (t/R)², so thickness helps quadratically, and a hull that is thin for its size is buckling-limited long before the material is.
The script checks the force balance in exact arithmetic for the numbers above; checks, by a numerical scan over k, that the minimum of D k²/t + E/(R²k²) is 2√(DE/t)/R and sits where k4 = E t/(D R²), for three sets of values; and checks that the resulting stress equals E t/(R√(3(1 − ν²))), which is 0.6052 E t/R for ν = 0.3. In Lean, sphere_force_balance, wrinkle_stress_lower_bound, wrinkle_stress_attained, zoelly_algebra and zoelly_pressure state the force balance, the minimisation and the substitution; all compiled.
Left out: the shell equations and the proof of the classical constant; imperfection sensitivity; composite materials and fatigue; the design of any real vessel; and the biology. A pressure hull is a design problem with testing and certification, and nothing on this page is guidance for building or diving in one.
A prize citation is a sentence written for a committee; it names an area and a few results, and it says nothing about proofs. What can be checked from it is who received it, when, and for which stated work, as in Part 1. What cannot be learned from it is how the results are proved or how they compare with what came before; for that the papers are the only source, and this page did not read them. The cited papers named on the pages read include, for the 2005 Bôcher Prize, work on blow-up for the critical generalized KdV equation (with Yvan Martel) and for the L2-critical nonlinear Schrödinger equation (with Pierre Raphaël), and for the 2023 prize the papers on the implosion of a compressible fluid and on blow-up for energy-supercritical defocusing nonlinear Schrödinger equations.