I’ve just uploaded to the arXiv the paper “Global existence and uniqueness results for weak solutions of the focusing mass-critical non-linear Schrödinger equation“, submitted to Analysis & PDE.  This paper is concerned with solutions $u: I \times {\Bbb R}^d \to {\Bbb C}$ to the focusing mass-critical NLS equation $i u_t + \Delta u = -|u|^{4/d} u$, (1)

where the only regularity we assume on the solution is that the mass $M(u(t)) := \int_{{\Bbb R}^d} |u(t,x)|^2\ dx$ is finite and locally bounded in time.  (For sufficiently strong notions of solution, the mass is in fact conserved, but part of the point with this paper is that mass conservation breaks down when the solution becomes too weak.)  Note that the mass is dimensionless (i.e. scale-invariant) with respect to the natural scale invariance $u(t,x) \mapsto \frac{1}{\lambda^{d/2}} u(\frac{t}{\lambda^2}, \frac{x}{\lambda})$ for this equation.  For various technical reasons I work in high dimensions $d \geq 4$ (this in particular allows the nonlinearity in (1) to be locally integrable in space).

In the classical (smooth) category, there is no ambiguity as to what it means for a function u to “solve” an equation such as (1); but once one is in a low regularity class (such as the class of finite mass solutions), there are several competing notions of solution, in particular the notions of a strong solution and a weak solution.  To oversimplify a bit, both strong and weak solutions solve (1) in a distributional sense, but strong solutions are also continuous in time (in the space $L^2({\Bbb R}^d)$ of functions of finite mass).   A canonical example here is given by the pseudoconformally transformed soliton blowup solution $\displaystyle u(t,x) := \frac{1}{|t|^{d/2}} e^{-i/t} e^{i|x|^2/4t} Q(x/t)$ (2)

to (1), where Q is a solution to the ground state equation $\Delta Q + |Q|^{4/d} Q = Q$.  This solution is a strong solution on (say) the time interval $(-\infty,0)$, but cannot be continued as a strong solution beyond time zero due to the discontinuity at t=0.  Nevertheless, it can be continued as a weak solution by extending by zero at t=0 and at $t>0$ (or alternatively, one could extend for $t>0$ using (2); thus there is no uniqueness for the initial value problem in the weak solution class. Note this example also shows that weak solutions need not conserve mass; all the mass in (1) concentrates into the spatial origin as $t \to 0$ and disappears in the limit t=0).

There is a slightly stronger notion than a strong solution, which I call a Strichartz-class solution, in which one adds an additional regularity assumption $u \in L^2_{t,loc} L^{2d/(d-2)}_x$.  This assumption is natural from the point of view of Strichartz estimates, which are a major tool in the analysis of such equations.

There is a vast theory for the initial value problem for these sorts of equations, but basically one has the following situation: in the category of Strichartz class solutions, one has local existence and uniqueness, but not global existence (as the example (2) already shows); at the other extreme, in the category of weak solutions, one has global existence, but not uniqueness (as (2) again shows).

(This contrast between strong and weak solutions shows up in many other PDE as well.  For instance, global existence of smooth solutions to the Navier-Stokes equation is one of the Clay Millennium problems that I have blogged about before, but global existence of weak solutions is quite easy with today’s technology and was first done by Leray back in 1933.)

In this paper, I introduce a new solution class, which I call the semi-Strichartz class; rather than being continuous in time, it varies right-continuously (in both the mass space and the Strichartz space) in time in the future of the initial time $t_0$, and left-continuously in the past of $t_0$.  With this tweak of the definition, it turns out that one has both global existence and uniqueness in this class.  (For instance, if one started with the initial data u(-1) given by (2) at time t=-1, the unique global semi-Strichartz solution from this initial data would be given by (2) for negative times and by zero for non-negative times.)  This notion of solution is analogous (but much, much simpler than) the notion of Ricci flow with surgery used by Hamilton and Perelman; basically, every time a singularity develops, the semi-Strichartz solution removes the portion of mass that was becoming discontinuous, leaving only the non-singular portion of the solution to continue onwards in time.

There are a number of other auxiliary results.  For instance, I show that the mass in the semi-Strichartz class is conserved except at a finite number of “surgery times”, in which the mass drops by at least a fixed amount; in the spherically symmetric case, the mass must drop by at least the mass M(Q) of the ground state; this is consistent with the general philosophy of “quantisation of mass” for these equations.  Outside of the surgery class, one has a Strichartz class solution.  These results are proven by minor variations of the standard Strichartz theory (and in the spherically symmetric case, also use a global well-posedness result for masses below M(Q) of myself, Visan, and Zhang).

When I work in higher dimensions $d \geq 5$ and assume spherical symmetry, I can use a decomposition of spherically symmetric waves into inward and outward components (that I used an earlier paper on NLS, as well as in a paper with Killip and Visan), there are some further results available.  First of all, one can show that the notions of strong solution and Strichartz class solution are equivalent, and that weak solutions of sufficiently small mass are automatically strong.  In particular, strong solutions are unique in this category (this type of result is known as unconditional uniqueness in the literature).  Also, a weak solution is strong precisely when its mass function is continuous, and if the mass is continous, then it is in fact constant.  So one can easily read off the nature of a solution just by inspecting how the mass changes in time.