[Math] Where do some “energy identities” in PDE theory come from

ap.analysis-of-pdesconservation-laws

There are a lot of very complicated expression that helps us obtain useful estimates for PDEs. To just give one example, the following is one of the Virial identities:
$$
\frac12\frac{d}{dt} \Im \left(\int (x \cdot \nabla u) u^* \right)dx =\int|\nabla u|^2 dx -\epsilon\left(\frac{n}{2}-\frac{n}{p+1}\right)\int|u|^{p+1} dx,
$$
where $u\in H^1, xu\in L^2$ solves the non-linear Schrödinger equation $i\partial_t u + \Delta u = -\epsilon u|u|^2$ in $\mathbb R^n.$

One way to come up with complicated "energy" or "momentum" expressions is to use Noether's theorem, but apparently many identities similar to the one above cannot be easily derived by Noether's theorem. In particular, the above expression can be derived without using any Lagrangians at all.

I often see such complicated identities in texts about PDE theory, stated and proven without any clues about where they come from.

My question is: how do people come up with them in the first place? It would be very helpful if there were an explanation for at least one or two identities like this so that I can see what type of reasoning is involved when we discover them.

Best Answer

In addition to the variational approach based on Noether's theorem, there are other ways to find conservation laws for nonlinear PDE's:

  • The symmetry/adjoint symmetry pair method extracts a conservation law from a bilinear skew-symmetric identity. It involves the following steps: (a) Linearize the given system of PDEs; (b) Find the adjoint system of the linearized system; (c) Find solutions of the linearized system, i.e., local symmetries in characteristic (evolutionary) form of the given PDE system; (d) Find solutions of the adjoint system; (e) For any pair, consisting of a solution for the adjoint system and a local symmetry, construct a conservation law.
  • The multiplier approach (also known as the variational derivative method), generalizes Noether's approach so that it can be applied whether or not the linearized system is self-adjoint (no Lagrangian formulation is needed).
  • Scale invariance can be used to obtain a conservation law from a simple algebraic formula.
  • The GeM software package searches for conservation laws of ordinary and partial differential equations without human intervention.

So yes, there are alternatives to just "mucking around until you see something".

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