# Iterative argument for 'tempered' NLS / NLW

I'm trying to use an iterative argument to prove (local) well-posedness in $H^1$ for the 'tempered' NLS $$i\partial_t u + \Delta u = \frac{|u|^{2\sigma}}{1 + |u|^{2\sigma}}u$$ and the corresponding NLW (same nonlinearity), where $\sigma$ is a positive integer. The usual conservation laws show that local wellposedness at $H^1$ automatically implies global well-posedness.

Supposedly the Galerkin-based method of Ginibre and Velo (in their paper 'The global Cauchy problem for the nonlinear Schrodinger equation revisited') will yield the result (at least for NLS) but I'd prefer to use an iterative argument if possible.

Let's just focus on NLS for now. It appears that I can pull off the argument for small dimensions, but I'm running into trouble once $d$ gets 'large' (about 6 or so; see below). My basic question is: (1) can we get the iterative argument to work for all (large) dimensions, and (2) if not, is there some reason I should expect the iterative local-existence argument to break down even though the global well-posedness result remains true?

The spaces I'd like to run the iteration in are the usual Strichartz spaces $X = S^1([-T,T] \times \mathbb{R}^d)$ and $Y = N^1([-T,T] \times \mathbb{R}^d)$. The crux of the argument is showing that the nonlinearity $N(u) = \frac{|u|^{2\sigma}}{1 + |u|^{2\sigma}}u$ is Lipschitz continuous from $X$ to $Y$, with Lipschitz constant (bounded by) some positive power of $T$. In other words, given $u,v \in X$ with norm at most $R$, we wish to show that $$||N(u) - N(v)||_{N^0} + ||\nabla N(u) - \nabla N(v)||_{N^0} \leq_R T^{\alpha}||u - v||_{S^1}$$ for some $\alpha > 0$. The first term on the left is trivial to estimate: by some simple algebra we can write $|N(u) - N(v)| \leq |u - v|\cdot b$ where $b$ is bounded by some universal constant depending only on $\sigma$, and since $q = \infty, r = 2$ are always Schrodinger-admissible, we obtain $$||N(u) - N(v)||_{N^0} \leq_{\sigma} ||u - v||_{L^1_tL^2x} \leq T ||u - v||_{L^\infty_tL^2_x} \leq T ||u - v||_{S^1}.$$ The second term on the left, $||\nabla N(u) - \nabla N(v)||_{N^0}$, will (after simple algebra) have some terms which can be handled just like the one above (these will be the terms containing a difference of the form $\nabla u - \nabla v$). There will, however, be a new type of term, a typical example of which is $$\frac{|u|^{2\sigma - 2}}{(1 + |u|^{2\sigma})(1 + |v|^{2\sigma})}(|u| + |v|)(|u| - |v|)\nabla u.$$ (In general the $|u|^{2\sigma - 2}$ would be replaced by $|u|^{2\sigma - 2N}|v|^{2N - 2}$ for some $1 \leq N \leq \sigma$, and $\nabla u$ might be $\nabla v$, and if the derivative hits the denominator we'd basically square the top and bottom, but this is largely irrelevant to what follows, I think.) We can trivially estimate the last displayed term as $b \cdot |u - v| |\nabla u|$ where $b \cdot |u - v|$ is bounded in $L^\infty$ by 1.

Now, what we need to do is find some admissible $q$ and $r$ for which we can prove an estimate of the form $$||b |u - v| \nabla u||_{L^{q^\prime}_tL^{r^\prime}_x} \leq_R T^{\alpha} ||u - v||_{S^1}.$$ What I'd like to do is use Holder (treating $b|u - v|$ as one term and $\nabla u$ as another) to increase the $L^{r^\prime}_x$ norm to some $L^{\tilde{r}}_x$ norm for some admissible $\tilde{r}$ (or more generally one admissible exponent for the $\nabla u$ term and potentially a different one for $b |u - v|$) and then use Holder in time to gain a factor of $T^{\alpha}$ and also to match the time exponent to the admissible space exponent from the previous step. One can basically discard the $b$ term in this argument (thus obtaining the required Lipschitz dependence on $u - v$) since $b$ itself is also bounded by 1. Recall that admissibility implies $q,r \geq 2$ so $q^\prime, r^\prime \leq 2$ and so they can in fact be increased to admissible exponents, though trying to do so with Holder runs into problems (see below).

Now, the main issue with this approach seems to be the first step: if $d$ is sufficiently large, there is no choice of exponent $s \in [1,\infty]$ such that applying Holder in $x$ with $s$ and $s^\prime$ gives two admissible exponents $r^\prime s$ and $r^\prime s^\prime$. This is because as $d$ gets large, the maximal admissible $x$-exponent $\frac{2d}{d - 2}$ gets very close to 2, and so the dual $r^\prime$ (for any $r$ admissible) must also be very close to 2, and since either $s$ or $s^\prime$ must be greater than 2, either $r^\prime s$ or $r^\prime s^\prime$ must exceed $\frac{2d}{d - 2}$. (Using Sobolev embedding on the $b |u - v|$ term (recall $u,v \in S^1$) allows one to exceed the maximal admissible exponent slightly on this term, but the gain is not enough to close this part of the argument for large $d$.)

At this point I should note that the boundedness of $b |u - v|$ allows one to bound an integral of the form $\int |b(u - v)|^c$ by one of the form $\int |b(u - v)|^d$ whenever $d < c$, but there is an issue with the exponents $\frac{1}{c}$ and $\frac{1}{d}$ outside the integral (in an $L^c$ or $L^d$ norm); one cannot in general bound the norm $||b(u - v)||_{L^c}$ with $||b(u - v)||_{L^d}$ because replacing the outer exponent $\frac{1}{c}$ by $\frac{1}{d}$ could decrease the value of the expression if the inner integral is less than 1. If one tries to use this idea without replacing the outer exponent, it seems that one arrives at a Holder $\alpha$ bound (not the same $\alpha$ as on $T$ above, just some $\alpha < 1$) on the term we are trying to prove has Lipschitz dependence on $u - v$, which of course does not in general give the required dependence.

So, this is the point at which I'm getting stuck. If one is careful about the ranges of exponents at one's disposal, I think this argument can be made to work up to $d = 5$ or $6$ (maybe 7), but no higher. I'd be interested in any ideas for salvaging the iterative argument in higher dimensions (such as exploiting the $L^\infty$ bound on $b |u - v|$ in some way I have yet to think of). Also I'm interested in the NLW case, for which I have no reference that actually gives a proof (of any sort). Of course, one has different Strichartz estimates in this case (which I doubt make things any easier, as we still have the decreasing-to-2 upper bound for admissible $x$ exponents as $d \to \infty$).

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I would look at the H^1 argument for energy-critical NLS in these notes of Killip and Visan: math.ucla.edu/~visan/ClayLectureNotes.pdf . For s>1 the later paper arxiv.org/abs/0812.2084 of these authors has a good discussion (but there is a limit as to how large s can get when one is in very high dimension). For NLW there is a bit more room with respect to the derivatives and the H^1 theory should be fairly straightforward, though again the higher H^s theory can get very difficult in high dimensino. –  Terry Tao Apr 10 '13 at 19:44