scholarly journals Norm-inflation with Infinite Loss of Regularity for Periodic NLS Equations in Negative Sobolev Spaces

2017 ◽  
Vol 145 (4) ◽  
pp. 623-642 ◽  
Author(s):  
Rémi Carles ◽  
Thomas Kappeler
Keyword(s):  
Sobolev Spaces ◽  
Loss Of Regularity ◽  
Negative Sobolev Spaces

scholarly journals GLOBAL WELL-POSEDNESS OF THE PERIODIC CUBIC FOURTH ORDER NLS IN NEGATIVE SOBOLEV SPACES

2018 ◽  
Vol 6 ◽  
Author(s):  
TADAHIRO OH ◽  
YUZHAO WANG
Keyword(s):  
Fourth Order ◽  
Sobolev Spaces ◽  
Energy Functional ◽  
Well Posedness ◽  
Initial Condition ◽  
Energy Functionals ◽  
Fourier Restriction ◽  
Fourier Restriction Norm Method ◽  
The Difference ◽  
Negative Sobolev Spaces

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.

scholarly journals ENERGY ESTIMATES FOR WEAKLY HYPERBOLIC SYSTEMS OF THE FIRST ORDER

2005 ◽  
Vol 07 (06) ◽  
pp. 809-837 ◽  
Author(s):  
MICHAEL DREHER ◽  
INGO WITT
Keyword(s):  
Sobolev Spaces ◽  
Upper Bound ◽  
Hyperbolic Systems ◽  
Elliptic Boundary ◽  
Cauchy Data ◽  
Energy Estimates ◽  
Elliptic Boundary Value ◽  
First Order ◽  
The Cauchy Problem ◽  
Loss Of Regularity

For a class of first-order weakly hyperbolic pseudo-differential systems with finite time degeneracy, well-posedness of the Cauchy problem is proved in an adapted scale of Sobolev spaces. These Sobolev spaces are constructed in correspondence to the hyperbolic operator under consideration, making use of ideas from the theory of elliptic boundary value problems on manifolds with singularities. In addition, an upper bound for the loss of regularity that occurs when passing from the Cauchy data to the solutions is established. In many examples, this upper bound turns out to be sharp.

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