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I have tried to search for references online but I am unable to do so.

I am looking for references that uses Cartan's method of moving frames to classify heat equations.

Also are there references that compute normal forms for heat equations?

It seems like there are very very few references on this subject, and I could only manage to find two.

The problem here is that the classical heat equation has infinite dimensional symmetry. I wonder if people have considered this as possible obstruction to computing the invariants?

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    $\begingroup$ What do you mean by "classifying heat equations"? Why "equations" in the plural? On what space are you working, and "how many" heat "equations" are there on it? $\endgroup$
    – Alex M.
    Commented Feb 12, 2022 at 15:39
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    $\begingroup$ 1. Classifying heat equations in the sense of Lie, Tresse and Poincaré. 2. Different heat equations in different dimensions. Also there are heat equations of types $u_{t}=u_{xx}$, or $u_{t}=div(\lambda(u)grad(u))+Q$. Hence equations. 3. Jet spaces of 2nd order. I have said, using Cartan's method of moving frames. 4. I don't get your last question. Can you be specific? $\endgroup$
    – fwg
    Commented Feb 12, 2022 at 16:18
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    $\begingroup$ Why the downvotes? $\endgroup$
    – tj_
    Commented Feb 13, 2022 at 9:31

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I'll just add a few more references to what Ben McKay mentioned.

Phillip Griffiths and I wrote a paper on invariants of parabolic equations in 1 space variable (Characteristic cohomology of differential systems, II: Conservation laws for a class of parabolic equations, Duke Mathematical Journal 78 (1995), 531–676. MR1334205) that explains how the method of equivalence can be used to determine invariants of parabolic second order PDE. We use it to, for example, classify up to equivalence the parabolic equations that admit non-trivial conservation laws.

As Ben McKay mentioned, Jeanne Clelland's thesis solves the equivalence problem for parabolic equations in two space variables and uses it to classify the parabolic equations that admit infinitely many independent conservation laws. (Jeanne was a student of mine.)

More recently, another student of mine, Ben McMillan, worked on the equivalence problem for parabolic equations in an arbitrary number of space variables, with a view towards classifying the ones that admit a `large' number of conservation laws. He wrote two papers that are on the arXiv, arXiv:1810.00458 and arXiv:1810.02346, that you may find interesting.

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Jeanne Clelland studied parabolic partial differential equations in her PhD thesis, using techniques of Cartan, including equivalence method. See her web page https://www.colorado.edu/math/jeanne-clelland, and I think her thesis is on the arxiv, but if not it will be in the Duke library. Her thesis was only for 2 space variables.

Riemann was the first to study the equivalence problem for heat equations, specifically linear scalar parabolic equations that are invariant under time translation. The symbol is dual to what we now call a Riemannian metric, hence his motivation for studying Riemannian metrics. The Levi-Civita connection and the curvature tensor are differential invariants of the original heat equation. The heat equation associated to the Laplace--Beltrami operator of the Riemannian metric agrees to leading order with the original heat equation, but they can disagree at lower order.

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  • $\begingroup$ Hi Ben McKay, sorry for the late reply because I was busy studying Jeanne Clelland's paper. Could you provide me reference to the part where Riemann studied equivalence problem for heat equation? I am very interested to know about it! Thanks! $\endgroup$
    – fwg
    Commented Feb 20, 2022 at 7:54
  • $\begingroup$ It was his famous On the hypotheses which lie at the foundation of geometry, emis.de/classics/Riemann. If I remember correctly, it was written for a competition to study invariants of the heat equation. $\endgroup$
    – Ben McKay
    Commented Feb 20, 2022 at 9:14
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The formal theory of PDEs, of which Cartan's method is an approach, is poorly suited for the study of the heat equation, which is why you rarely see it done. The key fundamental properties of a heat or parabolic equation are the existence and uniqueness of solutions given initial data at $t = 0$ and the smoothing of a solution for small positive time. Also, the initial value problem is well-posed only in the positive time direction and not in the negative direction.

None of this can be seen using Cartan's method. For example, in the formal theory, an initial problem is well-posed only if the initial data is on a noncharacteristic hypersurface. For the heat equation, the hypersurface $t=0$ is totally characteristic, so there is no way to construct a solution using Cartan's approach. Additional evidence of this is that the solution is not a smooth function of $t$ at $t = 0$.

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    $\begingroup$ The question is about equivalence of heat equations, not about solvability of heat equations. $\endgroup$
    – Ben McKay
    Commented Feb 12, 2022 at 20:04

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