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ON LEBESGUE NONLINEAR TRANSFORMATIONS
Ganikhodjaev, Nasir,Muhitdinov, Ramazon,Saburov, M. Korean Mathematical Society 2017 대한수학회보 Vol.54 No.2
In this paper, we introduce a quadratic stochastic operators on the set of all probability measures of a measurable space. We study the dynamics of the Lebesgue quadratic stochastic operator on the set of all Lebesgue measures of the set [0, 1]. Namely, we prove the regularity of the Lebesgue quadratic stochastic operators.
On Lebesgue nonlinear transformations
Nasir Ganikhodjaev,Ramazon Muhitdinov,M. Saburov 대한수학회 2017 대한수학회보 Vol.54 No.2
In this paper, we introduce a quadratic stochastic operators on the set of all probability measures of a measurable space. We study the dynamics of the Lebesgue quadratic stochastic operator on the set of all Lebesgue measures of the set $[0,1]$. Namely, we prove the regularity of the Lebesgue quadratic stochastic operators.
Chung, J.,Guo, Z.,Kwon, S.,Oh, T. Gauthier-Villars 2017 Annales de l'Institut Henri Poincaré. Analyse non Vol.34 No.5
<P>We consider the quadratic derivative nonlinear Schrodinger equation (dNLS) on the circle. In particular, we develop an infinite iteration scheme of normal form reductions for dNLS. By combining this normal form procedure with the Cole-Hopf transformation, we prove unconditional global well-posedness in L-2(T), and more generally in certain Fourier-Lebesgue spaces FLs,p (T), under the mean-zero and smallness assumptions. As a byproduct, we construct an infinite sequence of quantities that are invariant under the dynamics. We also show the necessity of the smallness assumption by explicitly constructing a finite time blowup solution with non-small mean-zero initial data. (C) 2016 Elsevier Masson SAS. All rights reserved.</P>