Fixed-point local well-posedness

Consider the cubic equation i times u_t plus u_xx equals kappa times |u|²u on the one-dimensional torus of length two pi. The initial datum belongs to H¹, and kappa is a fixed real coefficient. The free group U(t)=exp(it partial_xx) is unitary on H¹, as its Fourier multipliers have modulus one. Seek u in the Banach space of continuous H¹-valued functions on [0,T], with the supremum norm. This records the initial condition strongly and provides completeness for the fixed-point argument.

In one spatial dimension H¹ embeds into bounded functions and is a multiplication algebra. The cubic map therefore sends H¹ to H¹ and is locally Lipschitz there. A product derivative contains one differentiated factor and two undifferentiated bounded factors, giving a cubic norm bound; factoring a difference gives a quadratic Lipschitz coefficient on a ball of radius R. This is why dimension and regularity are part of the theorem. The identical algebra argument does not work in every H¹ space in higher dimensions.

Write M for the H¹ norm of the data and suppose the nonlinear bound and difference bound use a common constant K, enlarged if necessary. Unitarity and integration over time give an image bound M plus |kappa| T K R cubed. The difference of two images is at most |kappa| T K R squared times their distance. Choose R=2M when M is positive, and choose T so the nonlinear correction is at most M while the Lipschitz factor is strictly below one. Zero data have the zero solution and can be handled separately.

Ball preservation ensures the iteration remains inside the domain where the estimates hold. Strict contraction ensures convergence to one fixed point there. Neither condition should be omitted or substituted for the other. Uniqueness in the full local continuous-H¹ class follows by applying the same difference estimate on sufficiently short subintervals and patching them. Comparing solutions from two nearby initial data adds the free evolution of their initial difference, yielding continuous, indeed locally Lipschitz, dependence on a common sufficiently short interval.

The fixed point is a mild solution continuous in H¹. The equation gives its time derivative in H minus one, because taking two spatial derivatives lowers Sobolev regularity by two. A classical pointwise solution requires more regularity; the fixed-point theorem should not quietly claim it for arbitrary H¹ data. Smooth data can be treated with a persistence-of-regularity argument, showing that higher regularity continues while the relevant control remains available. That argument is an additional estimate rather than a consequence of notation.

The construction is local and works for either sign of kappa. Global existence requires a continuation argument and bounds that prevent the controlling norm from escaping. The torus has no whole-space dispersive decay of the usual kind, but the high-regularity multiplication estimate was enough here. Later Strichartz estimates will replace this simple algebra argument for rougher whole-space data. Comparing those proofs helps identify which ingredient actually supplies smallness: interval length, the linear orbit in a spacetime norm, or the initial norm itself.