Global well-posedness of the periodic cubic fourth order NLS in negative Sobolev spaces
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Colleges, School and Institutes
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 Hs(핋) , s>−1/3 , with enhanced uniqueness. The proof consists of two separate arguments. (i) We first prove global existence in Hs(핋) , s>−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 Hs(핋) , s>−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 Hs -energy functional, allowing us to introduce an infinite sequence of correction terms to the Hs -energy functional in the spirit of the I -method. In fact, the main novelty of this paper is this reduction of the Hs -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.
|Number of pages||80|
|Journal||Forum of Mathematics, Sigma|
|Publication status||Published - 11 May 2018|