2 hope it makes more sense like this

Regarding the use of dimensional analysis to derive asymptotic solutions to PDE, this is dimensional analysis techniques can be seen as just a small part of a certain well-developed theory in the case where the asymptotic solutions are for "self-similar" asymptotic solutions with lots written in e.g. the fluid mechanics literature. I learned most of this in "applied mathematics" courses, for what it's worth. If anyone knows where the stuff below is treated from a higher-tech point of view, I'd love to hear of it.

For example, if one is trying to solve an equation of the form

$\partial_t u(x,t)=F[u]$

where $F$ is some typically nonlinear differential operator, one often believes for physical reasons that a solution of the type

$u(x,t)=(t-t_0)^\alpha H\left(\frac{x-x_0}{(t-t_0)^\beta}\right)$

(called self-similar, as the function $H$ is fixed under certain simultaneous scalings of time and space) is likely. For instance if the system is approaching some kind of singularity at $x=x_0$, $t=t_0$, the length and time scales which are going to dominate will be just those describing the distance to the singularity in space-time.

By using this ansatz of this type, one often finds that the PDE we originally had to solve is reduced to an ODE for the function $H$.

It's important to note that naïve dimensional analysis can only get you $\alpha$ and $\beta$ in a small subset of these asymptotic solutions due to the existence of so-called self-similarities of the second kind!

An idiosycratic but readable book along these lines that I enjoyed is G.I. Barenblatt's Scaling, self-similarity, and intermediate asymptotics. As far as I know, Barenblatt introduced and emphasized this division of "self-similar" intermediate asymptotic solutions into two types.

In self-similarities of the first kind, naïve dimensional analysis works. Consider for instance the self-similar solution $u(x,t)=\frac{u_0}{\sqrt{4\pi t}} e^{-x^2/4Dt}$ to the diffusion equation $\partial_t u-D\partial^2_{xx}u=0$. We can derive this solution by assuming that the solution depends only on the dimensionless combination $x^2/4Dt$, then the PDE can be reduced to an ODE.

In self-similarities of the second kind, non-trivial scalings appear, where the exponents in power laws turn out to be determined by nonlinear eigenvalue problems. Roughly speaking, one has a continuum of $\alpha$ and $\beta$ values which work, and imposing proper boundary conditions yields the ones which are of relevance. I believe you get irrational values of $\alpha$ and $\beta$ by using some "microscopic" length scales from short-distance / short-time cut-offs in your system. (Sorry, I'll fill in details / an example when I get my hands on my copy of the book).

For a recent review with plenty of fluid mechanics examples, see this paper by Eggers and Fontelos.

Intriguingly, this seems to be analogous to dimensional analysis in quantum / statistical field theory, where whenever a phase transition is described by "mean field theory" (i.e. a Gaussian RG fixed point), all the critical exponents governing the behavior near the phase transition can be derived from the dimensional analysis arguments (I think some books say that the scaling dimensions are equal to the engineering dimensions). When the RG fixed point governing the phase transition is nontrivial, then there are anomalous dimensions which appear essentially by the same mechanism as above. Nigel Goldenfeld emphasizes this point of view in his book Lectures on Phase Transitions and the Renormalization Group. You might also find his papers connecting the two subjects interesting, e.g. L.Y. Chen, N. D. Goldenfeld and Y. Oono. The renormalization group and singular perturbations: multiple-scales, boundary layers and reductive perturbation theory. Phys. Rev. E 54, 376-394 (1996).

1

Regarding the use of dimensional analysis to derive asymptotic solutions to PDE, this is a well-developed theory in the case where the asymptotic solutions are "self-similar" with lots written in e.g. the fluid mechanics literature. I learned most of this in "applied mathematics" courses, for what it's worth. If anyone knows where the stuff below is treated from a higher-tech point of view, I'd love to hear of it.

For example, if one is trying to solve an equation of the form

$\partial_t u(x,t)=F[u]$

where $F$ is some typically nonlinear differential operator, one often believes for physical reasons that a solution of the type

$u(x,t)=(t-t_0)^\alpha H\left(\frac{x-x_0}{(t-t_0)^\beta}\right)$

is likely. For instance if the system is approaching some kind of singularity at $x=x_0$, $t=t_0$, the length and time scales which are going to dominate will be just those describing the distance to the singularity in space-time.

It's important to note that naïve dimensional analysis can only get you $\alpha$ and $\beta$ in a small subset of these asymptotic solutions due to the existence of so-called self-similarities of the second kind!

An idiosycratic but readable book along these lines that I enjoyed is G.I. Barenblatt's Scaling, self-similarity, and intermediate asymptotics. As far as I know, Barenblatt introduced and emphasized this division of "self-similar" intermediate asymptotic solutions into two types.

In self-similarities of the first kind, naïve dimensional analysis works. Consider for instance the self-similar solution $u(x,t)=\frac{u_0}{\sqrt{4\pi t}} e^{-x^2/4Dt}$ to the diffusion equation $\partial_t u-D\partial^2_{xx}u=0$. We can derive this solution by assuming that the solution depends only on the dimensionless combination $x^2/4Dt$, then the PDE can be reduced to an ODE.

In self-similarities of the second kind, non-trivial scalings appear, where the exponents in power laws turn out to be determined by nonlinear eigenvalue problems. Roughly speaking, one has a continuum of $\alpha$ and $\beta$ values which work, and imposing proper boundary conditions yields the ones which are of relevance. I believe you get irrational values of $\alpha$ and $\beta$ by using some "microscopic" length scales from short-distance / short-time cut-offs in your system. (Sorry, I'll fill in details / an example when I get my hands on my copy of the book).

For a recent review with plenty of fluid mechanics examples, see this paper by Eggers and Fontelos.

Intriguingly, this seems to be analogous to dimensional analysis in quantum / statistical field theory, where whenever a phase transition is described by "mean field theory" (i.e. a Gaussian RG fixed point), all the critical exponents governing the behavior near the phase transition can be derived from the dimensional analysis arguments (I think some books say that the scaling dimensions are equal to the engineering dimensions). When the RG fixed point governing the phase transition is nontrivial, then there are anomalous dimensions which appear essentially by the same mechanism as above. Nigel Goldenfeld emphasizes this point of view in his book Lectures on Phase Transitions and the Renormalization Group. You might also find his papers connecting the two subjects interesting, e.g. L.Y. Chen, N. D. Goldenfeld and Y. Oono. The renormalization group and singular perturbations: multiple-scales, boundary layers and reductive perturbation theory. Phys. Rev. E 54, 376-394 (1996).