Which is the smallest space $X\subset L^{2}$ where the conservation law holds in the norm of $X$?

Question

We formally write the solution of nonlinear Schrödinger equation (NLS) as follows:

$$u(t)= U(t-t_{0}) u_{0}- i \int_{t_{0}}^{t} U(t-\tau) (|u|^{2}u(\tau)) d\tau;$$ where $U(t)= e^{it\Delta} $(free Schrödinger equation), and initial data $u_{0} \in L^{2}(\mathbb R)$[For more details, see this ]. It is well-known (see Theorem 1.1) that for initial data $u_{0}$ in $L^{2}(\mathbb R), $ we have the conservation law in $L^{2}-$ norm: $\|u(t)\|_{L^{2}(\mathbb R)}= \|u_{0}\|_{L^{2}(\mathbb R)}$ for all time $t\in \mathbb R.$

Suppose that $X$ is a Banch space with norm $\|\cdot\|_{X}$ and $X$ is a proper subset of $L^{2}(\mathbb R).$

My Questions :

(1) If we start with initial data in $X,$ then can we expect conservation law associated to the above NLS in the norm of $X,$ that is, $\|u(t)\|_{X}=\|u_{0}\|_{X}$ ?(I feel, this may be difficult, I don't know)

(2)Of course, the answer of (1), depends on $X:$ it therefore I am looking for few examples.Can we find few example of $X$ where conservation law hold in the norm of $X$?

(3) Is it true that the space $L^{2}(\mathbb R)$ is the smallest space where one can have the conservation law hold in the space ?

Motivation: To prove the globe well poseness results for NLS conservation law is useful.

Edit: My motivation comes through this consideration:"If we have a local well posedness in $X$ for cubic NLS, then we can expect global well posedness in $X"$ if we have a conservation law in $X$(I think). Also I am looking for explicitly $H^{1}\subset X \subset L^{2}$ where conservation law hold in $X-$norm. Kindly would you suggest me some $X$ where conservation law hold(So far papers I have been seeing every one has consider conservation in $L^{2}-$norm, I am couriers to know if we can come below $L^{2}$ )

Answer 1

Please note that the cubic NLS in one spatial dimension (which seems to be what you are interested in) is completely integrable and has formally infinitely many conserved quantities. For example, the energy. So while I haven't checked to see whether the combinations of those conserved quantities give additional seminorms, I am highly doubtful that $L^2$ is the smallest space.

Also, the global existence theory for the cubic NLS in 1D is pretty well known (see link above), so I am not sure why you are motivated by such.