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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(T), s > -1/3, with enhanced uniqueness. The proof consists of two separate arguments. (i) We first prove global existence in H^s(T), 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 non-existence of solutions for the (non-renormalized) 4NLS in negative Sobolev spaces. (ii) We then prove enhanced uniqueness in H^s(T), 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 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.
|Number of pages||80|
|Journal||Forum of Mathematics, Sigma|
|Publication status||Published - 11 May 2018|
- fourth order nonlinear Schrödinger equation
- biharmonic nonlinear Schrödinger equation
- short-time Fourier restriction norm method
- normal form reduction
- enhanced uniqueness
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- 1 Finished
ProbDynDispEq - Probabilistic and Dynamical Study of Nonlinear Dispersive Equations
1/03/15 → 29/02/20
- School of Mathematics - Personal Chair of Dispersive Equations
Person: Academic: Research Active