In the mathematical field of analysis, the Nash–Moser theorem, discovered by mathematician John Forbes Nash and named for him and Jürgen Moser, is a generalization of the inverse function theorem on Banach spaces to settings when the required solution mapping for the linearized problem is not bounded.
The theorem is widely used to prove local existence for non-linear partial differential equations in spaces of smooth functions.
Moser (1966a, 1966b), for instance, showed that Nash's methods could be successfully applied to solve problems on periodic orbits in celestial mechanics in the KAM theory.
However, it has proven quite difficult to find a suitable general formulation; there is, to date, no all-encompassing version; various versions due to Gromov, Hamilton, Hörmander, Saint-Raymond, Schwartz, and Sergeraert are given in the references below.
This will be introduced in the original setting of the Nash–Moser theorem, that of the isometric embedding problem.
In Nash's solution of the isometric embedding problem (as would be expected in the solutions of nonlinear partial differential equations) a major step is a statement of the schematic form "If f is such that
Following standard practice, one would expect to apply the Banach space inverse function theorem.
which coincides with a second-order differential operator applied to f. To be precise: if f is an immersion then
is the scalar curvature of the Riemannian metric P(f), H(f) denotes the mean curvature of the immersion f, and h(f) denotes its second fundamental form; the above equation is the Gauss equation from surface theory.
In the case of uniformly elliptic differential operators, the famous Schauder estimates show that this naive expectation is borne out, with the caveat that one must replace the
; this causes no extra difficulty whatsoever for the application of the Banach space implicit function theorem.
However, the above analysis shows that this naive expectation is not borne out for the map which sends an immersion to its induced Riemannian metric; given that this map is of order 1, one does not gain the "expected" one derivative upon inverting the operator.
is an order-one differential operator on some function spaces, so that it defines a map
One can concretely see the failure of trying to use Newton's method to prove the Banach space implicit function theorem in this context: if
transparently does not encounter the same difficulty as the previous "unsmoothed" version, since it is an iteration in the space of smooth functions which never loses regularity.
For many mathematicians, this is rather surprising, since the "fix" of throwing in a smoothing operator seems too superficial to overcome the deep problem in the standard Newton method.
For instance, on this point Mikhael Gromov says You must be a novice in analysis or a genius like Nash to believe anything like that can be ever true.
[...] [This] may strike you as realistic as a successful performance of perpetuum mobile with a mechanical implementation of Maxwell's demon... unless you start following Nash's computation and realize to your immense surprise that the smoothing does work.Remark.
for an entire open neighborhood of choices of f, and then one uses the "true" Newton iteration, corresponding to (using single-variable notation)
Certain approaches, in particular Nash's and Hamilton's, follow the solution of an ordinary differential equation in function space rather than an iteration in function space; the relation of the latter to the former is essentially that of the solution of Euler's method to that of a differential equation.
The following statement appears in Hamilton (1982): Let F and G be tame Fréchet spaces, let
denotes the vector space of exponentially decreasing sequences in
The laboriousness of the definition is justified by the primary examples of tamely graded Fréchet spaces: To recognize the tame structure of these examples, one topologically embeds
Presented directly as above, the meaning and naturality of the "tame" condition is rather obscure.
The situation is clarified if one re-considers the basic examples given above, in which the relevant "exponentially decreasing" sequences in Banach spaces arise from restriction of a Fourier transform.
Recall that smoothness of a function on Euclidean space is directly related to the rate of decay of its Fourier transform.
"Tameness" is thus seen as a condition which allows an abstraction of the idea of a "smoothing operator" on a function space.
the precise analogue of a smoothing operator can be defined in the following way.
If one accepts the schematic idea of the proof devised by Nash, and in particular his use of smoothing operators, the "tame" condition then becomes rather reasonable.
The fundamental example says that, on a compact smooth manifold, a nonlinear partial differential operator (possibly between sections of vector bundles over the manifold) is a smooth tame map; in this case,