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I have the following question about short-time existence and uniqueness results for non-linear schrodinger equations (NLSE) where the non-linearity involves a loss of derivatives (in my case, it is a non-local non-linearity involving a loss of two derivatives.) It seems that most current techniques allow some small number of derivatives in the non-linearity, except for a series of papers by Poppenberg, which use Nash-Moser. (Unfortunately he seems to have left math.)

Question: are there any short-time existence and uniqueness results for NLSE with loss of several derivatives for a space of initial conditions smaller than $C^\infty$? I have in mind for example a space introduced by Floer which is a dense subspace of $C^\infty$ that carries the structure of a Banach space whose norm is a combination of all C^k norms. One could imagine doing something similar by combining all Sobolev norms. I would be quite happy with a short-time existence and uniqueness result for initial conditions in some dense subspace of C^\infty.

p.s. I am posting in a second a related question about Floer's Banach space with a symplectic geometry tag.

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Are you looking for an existence theorem that works with only finitely many derivatives? I believe it is possible to use Nash-Moser to start with $C^{k+N}$ (or a Sobolev analogue) initial data and obtain a $C^k$ (or Sobolev analogue) solution. This might also give what you need for the Floer topology. You might want to look at Hormander's version of the Nash-Moser implicit function theorem. –  Deane Yang Aug 20 '10 at 16:50
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up vote 5 down vote accepted

Just a few thoughts. The answer to your question depends on a number of key factors. To focus, let us consider a nonlinear term like $F(D_x^k u)$ and let us work in one space dimension.

1) How large is $k$? If $k\le2$ you can linearize the equation and work in Sobolev spaces. Of course you need some structural assumption on the nonlinearity (otherwise you may take e.g. $F(u_{xx})=\pm 2i u_{xx}+...$, and create all sorts of difficulties). There are some classical works by Kenig, Ponce, Vega on this (see "The Cauchy problem for the quasilinear S.e." around 2002 I think) which more or less give the complete picture from the classical point of view i.e. without trying to push below critical Sobolev etc. So if this is the case what are you exactly looking for? if you prefer to work in smaller spaces, what you need is a 'regularity' result, i.e., if the data are in some smaller space, this additional regularity propagates and the solution stays in the same space for some time. There are some results of this type, in classes of analytical or Gevrey functions; but see below.

2) If $k\ge3$, then the same remark as in (1) applies, you need some strong structural assumptions on the nonlinearity. Indeed, now the poor $u_{xx}$ is no longer the leading term and the character of the (linearized) equation is entirely determined by the Taylor coefficient of $D^ku$ in the expansion of $F$. So then a careful case-by-case discussion is necessary. Unless...

3) unless, and we come maybe closer to your question, you decide to work in MUCH smaller spaces than $C^\infty$. Gevrey classes are roughly speaking classes of smooth functions such that the derivatives of order $j$ grow at most like $j!^s$. For $s=1$ you get analytic functions. For $ 1 < s < \infty $ you get larger classes $G^s$, with quite nice properties (to mention just one, you have compactly supported functions in these classes). For $ 0 < s < 1 $ the classes $G^s$ are rather small, strictly contained in the space of analytic functions. The only reason why $G^s$ for $s<1$ are useful is that you can prove a sort of very general Cauchy-Kowalewski theorem, local existence in Gevrey classes, for any evolution equation $u_t=F(D^k_xu)$, provided $s<1/k$. No structure is required on $F$, only smoothness. Contrary to the appearance this is a weak result, e.g. you can solve locally both $u_t=\Delta u$ and $u_t=-\Delta u$ in $G^{1/2}$ (and globally in $G^s$ for $s<1/2$ ), so you are essentially trivializing the equation and forgetting all of its structure. But if this is what you need I can give you pointers.

EDIT (since this does not fit in the comments): I am a bit rusty on these topics, now I start to remember more details.

1) $s<1$. There is a problem with working in $G^s$ for $s<1$, and it is that this space is unstable for product of functions. So also in this case you need very special nonlinearities to work with. But the linear theory is straightforward and you can find an account maybe here. Also Beals in some old paper developed semigroup theory in Gevrey classes. BTW, if you find a way to handle products this might be quite interesting.

2) $s>1$. The Mizohata school and other japanese mathematicians worked on NLS in Gevrey classes quite a lot, see e.g. this paper

3) The Nash-Moser approach might be useful provided you can prove that the linearized of your operator is solvable in every Sobolev class, with a fixed loss of derivatives. If you want to try this route, the best introduction to the theory I know of is in Hamilton's 1992 (?) Bull. AMS paper. It's very long but extremely readable, give it a try.

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Thanks Piero and Deane! (1) In my particular equation the potential is obtained from an expression involving the fourth derivatives of the square of the wave-function, by solving a second-order Poisson equation. So if $\psi$ is in $H^s$, then $V(\psi)$ is in $H^{s-2}$ for s sufficiently large, So in this sense $k=2$. But $V(\psi)$ depends non-locally on $\psi$ in much the same sense as in the paper by Ginibre-Velo on non-local interaction in non-linear Schrodinger. –  Chris Woodward Aug 20 '10 at 21:48
(ctd) So it seems my problem is closest to (3) in Piero's answer, and in fact it seems what I need to do is try to find for which Gevrey class of initial conditions is good. Do you have a recommended reference involving Gevrey initial conditions with loss of derivatives in non-linear Schrodinger? I would be very happy to show uniqueness for $G^s$ initial condition for some $s > 1$. –  Chris Woodward Aug 20 '10 at 21:48
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