Abstract
We consider the Cauchy problem for the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle. In particular, we prove global well-posedness of the renormalized 4NLS in negative Sobolev spaces$H^{s}(\mathbb{T})$,$s>-\frac{1}{3}$, with enhanced uniqueness. The proof consists of two separate arguments. (i) We first prove global existence in$H^{s}(\mathbb{T})$,$s>-\frac{9}{20}$, via the short-time Fourier restriction norm method. By following the argument in Guo–Oh for the cubic NLS, this also leads to nonexistence of solutions for the (nonrenormalized) 4NLS in negative Sobolev spaces. (ii) We then prove enhanced uniqueness in$H^{s}(\mathbb{T})$,$s>-\frac{1}{3}$, by establishing an energy estimate for the difference of two solutions with the same initial condition. For this purpose, we perform an infinite iteration of normal form reductions on the$H^{s}$-energy functional, allowing us to introduce an infinite sequence of correction terms to the$H^{s}$-energy functional in the spirit of the$I$-method. In fact, the main novelty of this paper is this reduction of the$H^{s}$-energy functionals (for a single solution and for the difference of two solutions with the same initial condition) to sums of infinite series of multilinear terms of increasing degrees.
Highlights
By following the argument in Guo–Oh for the cubic nonlinear Schrodinger equation (NLS), this leads to nonexistence of solutions for the 4NLS in negative Sobolev spaces. (ii) We prove enhanced uniqueness in
We perform an infinite iteration of normal form reductions on the H s-energy functional, allowing us to introduce an infinite sequence of correction terms to the H s-energy functional in the spirit of the I -method
We introduce an alternative formulation for (1.1) such that (i) it is equivalent to (1.1) in L2(T) but (ii) it behaves better than (1.1) in negative Sobolev spaces
Summary
Of weak solutions in the extended sense in negative Sobolev spaces, where the nonlinearity is interpreted only as a limit of smooth nonlinearities.) Instead, our uniqueness statement should be interpreted as follows; given u0 ∈ H s(T), let u be a solution to (1.5) with u|t=0 = u0 constructed in Theorem 1.1 via this particular version of the short-time Fourier restriction norm method. In a recent work [33], the first author with Sosoe and Tzvetkov established an optimal regularity result for quasi-invariance of the Gaussian measures on Sobolev spaces under the original 4NLS (1.1) by implementing a similar infinite iteration of normal form reductions on the H s-functional for solutions for s.
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