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Consider 2 simple differential equations: $x'(t)=x(t)^2, x(0)=1$ and $x'(t)=-x(t)^2, x(0)=1$. As $t$ goes from $0$ to $\infty$, the first equation ($x(t)=1/(1-t)$) will lead to a finite-time blow-up, while the second ($x(t)=1/(1+t)$) doesn't. Even though, from looking at the algebra, there is only one difference between a plus and a minus. (Luckily, these admit exact solutions. Real-world problems don't.)

Is it therefore correct to think that the methods of algebra (loosely defined, as opposed to traditional analysis of PDEs) will never be able to help us with this kind of question? So far, the algebraic characterizations of PDEs I have seen include some kind of jet bundles (Gromov's partial differential relations, Vinogradov's theory of diffiety which generalizes algebraic geometry etc.). Vinogradov even managed to define spectral sequences and de Rham cohomology, and his school has used them to compute conservation laws / integrable structures of PDEs. Gromov's geometric h-principle is related to developments in fluid dynamics such as Onsager's conjecture. But that is the extent of my knowledge. All the blow-up proofs for PDEs I have seen seem to be mainly analytical in nature.

If you are familiar with such algebraic theories of nonlinear PDEs, do you know a lead that can help characterize PDE blow-ups?

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    $\begingroup$ Does microlocal analysis fall within your definition of "methods of algebra"? For example, does J-M Bony's paper, "Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires" Annales scientifiques de l’É.N.S. 4e série, tome 14, no 2 (1981), p. 209-246 address your question? $\endgroup$
    – Phil Harmsworth
    Nov 13, 2020 at 8:06
  • $\begingroup$ I am aware of microlocal analysis and propagation of singularities, characterized by wave front sets. It is an analytical and classical point of view in investigating smoothness, in the analysis of PDEs. But so far we are still stumped by the problem of characterizing general PDE blow-ups. I guess I am looking for something new, or obscure to mainstream analysts (otherwise these problems in, say, fluid mechanics, wouldn't still be open). $\endgroup$
    – Iza_lazet
    Nov 13, 2020 at 16:41
  • $\begingroup$ Do Bony’s methods work to characterize the blowup in the explicit simple example in the question? $\endgroup$
    – Vivek Shende
    Nov 13, 2020 at 21:22

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There are classes of PDE's where blow-ups of solutions can be characterised. The examples I am aware of are those of integrable PDE's. To connect to classical algebraic theory, let us focus on so-called finite gap solutions. A typical example of integrable PDE's is given by the $\sinh$-Gordon equations $$\Delta u=\mp \sinh 2u,$$ where $u$ is a real-valued function defined on the complex plane. Doubly periodic finite gap solutions are given by a linear flow: there exists a (real) linear map from the torus determined by the period lattice into the Jacobian of the so-called spectral curve. This linear map has to take values inside a real (or quaternionic) component of the Jacobian in order for $u$ to be real-valued. The solution becomes singular (it has a blow up) where the linear map intersects the $\theta$-divisor inside the Jacobian. For some classes (e.g., the above sinh-Gordon equation with the - sign) one can prove that you will never intersects the $\theta$-divisor (see Hitchin, Harmonic 2-tori in the 3-sphere, Journal Diff Geo, 1990, Proposition 7.15), while for other classes (e.g. a certain $\cosh$-Gordon) equation you necessarily intersectthe $\theta$-divisor (see Babich-Bobenko, Willmore tori with umbilic lines, Duke Journal, 1993).

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  • $\begingroup$ Thanks for the references. Though I guess it is to be expected since integrable PDEs have a lot of structure. It would be great if there were an idea regarding blow-ups that could be generalized to more general PDEs, like partial differential relations. $\endgroup$
    – Iza_lazet
    Nov 12, 2020 at 10:14

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