and therefore if Im J dxuu = 0 then
(v) J r(x,t)dx = J T(x,0)dx, (L1 ,H~X/2), with r the torsion of the X-curve in the binormal flow.
In the above list we include whithin brackets the Sobolev Hs and Lebesgue Lp spaces which are invariant under the same scaling of the corresponding quantity. Because of (2) we are mainly interested in L1 — H1/2 scalings which are the ones of the (iv) and (v) quantities.
Among the transformations which leave invariant the set of solutions of (1) we have the following ones.
(ii) Dilations: u\ = Xu(Xx, X2t) is a solution if so is u\
In terms of the initial condition we have u\(x,0) — Xuo(Xx).
Therefore the solutions which are invariant with respect to scaling should be related to initial data which are homogeneous of degree — 1:
(iii) Multiplication by a constant of modulus one;
(iv) Galilean transformations: given N e R then uN(x, t) - e~itN2+iNxu(x - 2Nt, t) (7)
is a solution if u is a solution. The corresponding initial conditions are related by the identity uN(x,0) = eiNxu(x,0). (8)
Therefore the solutions which are invariant with respect to the Galilean transformations should verify that elNxu(x, 0) = u(x, 0) for all N. Hence u(x, 0) = aS. In fact if
UN = u for all N we can differentiate in N in both sides to conclude that ir]u(rj,t) — 2tdnu(T],t) = 0.
And from (1) and some simple manipulations  we get a ( |x|2\
But —41 converges as t goes to zero to a constant times the Vi i-function. As a consequence ua has no limit at t = 0 and the IVP:
either has no limit or has more than one solution . 5. Stability
Recall that the equation coming from the binormal flow (5) has an extra term A(t) with respect to equation (9). In fact u(s,t) = a^=eia2/4t, (10)
This suggests that the natural problem related to the ¿-function as initial datum has to be modified as in (10). In fact with S. Gutierrez  and G. Perelman  we have proved that (9) has no solution within the set of self-similar solutions according to scaling.
At the same time a natural question is if the explicit solution (10) of (11) is stable in any sense. The problem suggests different possibilities that we shall try to explain.
The first issue we want to consider is which is the minimal regularity needed to solve the problem f tut + d2xu ± \u\2u = 0
The question is solved if uo is in the Sobolev space Hs, s > 0   . The way of doing it is to find by a fixed point argument a solution to the corresponding integral equation f4
Hence one has to look for good estimates for the solutions of the linear problem
with v the solution of (14). The smallness condition necessary to obtain a fixed point is obtained by taking the time of existence T small enough. But T = Tdluolli2)» and from the L2-conservation law the local result is extended to all t > 0.
It is easy to see using the Galilean transformations , that H3 with s < 0 is not very well adapted to solve (12) (TP, s < 0, is not invariant under translations of Fourier space cf. (8)).
Instead of using Sobolev spaces in a joint work with A. Vargas  we proposed to use the space
This space is not easy to handle. A sufficient condition to belong to X is for example to assume that Uq G C1 and dj - tt\
with a > 1/6. Then we prove  that if uq G X then (12) is locally well posed in time. Notice that if uq = aS then uq verifies (17) with a = 0.
We also obtain a global result following some previous ideas developed by J. Bourgain . For this result we need to be able to decompose uo as u0 = fN + gN-, fN e L2, gNeX,
for all TV > 1. We find  such a decomposition by imposing some extra properties. I think that to find good ways of achieving (18) it is a very interesting question in Fourier Analysis.
Another approach to "decrease" the regularity is to look at tut +d2± M7u = 0, (19)
and to look for selfsimilar solutions with respect to the corresponding scaling. The question is then which is the minimal 7 so that those solutions can be constructed. In a joint work with T. Cazenave and M. C. Vilela we improve previous results obtained by T. Cazenave and F. Weissler  and later extended by F. Planchon  to the classical Hs, s > 0. We extend  Planchon's result to s = 0 in dimensions one and two, and define some alternative spaces which behave as Hs, s < 0, in all dimensions. In n = 1
g the lowest 7 obtained by this method is 7 = -, still far from the cubic o equation 7 = 2.
The spaces introduced in  are quite simple:
Interestingly A. Griinrock  has seen that this type of spaces are quite well adapted to deal with the cubic (and others) non-linearity. He has announced the result of local wellposedness of (12) with u0£Yp p < 00.
Therefore he is "just" missing the initial data we are interested in. Namely uq = aS + bp.v. —.
Recall that as we said before for this type of initial condition the cubic non-linearity should be modified by an extra term as in (11).
Finally let us say a few words of some work in progress done with V. Banica about the stability of the solution
The question is to solve (20) backwards up to time £ = 0. In order to do that we use the conformal transformation u=Vteix2^v(j,11), (21)
and look for solutions of the problem
1 u(s,l) = ui(s), with Vi (s) = a + ei(s). This amounts to solve a2 2a 11 9
From that we obtain
On the other hand given eo = eo(£o) £ L2 we define the operator S(F) = e^-^-eo -i f e^'^l [2a2Re F + 2aFRe F + \F\2 F] —
Jto 2 r
with sup ||F(£)||L2 + ||F||L6 < +00, and I = [t0,t0 + h]. Then given T > 0
for any 1 < £0 < T we can take h < c(a)---j
and use (15) in the classical way to obtain a fixed point of (24), and therefore a solution of (23) for 1 < £ < T and all T.
Going back to (22) we easily obtain the conservation law:
J \vs(t)\2ds - I J(M2 - a2)2ds -\J'J(M2 - a2)2ds J = j[\vs(l)\2-(\v(l)\2-a2)2}ds.
It is also easy to prove that if us(l) e L2 then va(t) € L2. But from
This can be understood as an orbital stability result of the the solution v = a, a < 1/2, because just ||<9s^||l2 remains small.
Therefore the main remaining question is to identify the limit lim Vteix^4te('-, ]).
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