Strong ill-posedness for fractional Hartree and cubic NLS Equations

Norm Inflation for Benjamin–Bona–Mahony Equation in Fourier Amalgam and Wiener Amalgam Spaces with Negative Regularity

“…We plan to address the norm-inflation (the stronger phenomenon than the mere ill-poseness) and even the worst situation of norm inflation with infinite loss of regularity for (1.1) in our future works. We note that recently similar questions we have already addressed for Hartree, nonlinear Schrödinger, BBM and wave equations in [2,3,4]. We also expect to develop well-posedness theory for (1.1) in w p,q space in the future.…”

“…survey article [14]. Recently, in [17] authors have studied KdV in modulation spaces, and in [2,3,4] authors have studied ill-poedesness for wave, BBM and NLS in Fourier amalgam spaces, see also [16,13]. We refer to [1] by Bejenaru-Tao for abstract well-posedness and ill-posedness theory.…”

Wellposedness of NLS in Modulation Spaces

“…We want to remark that results on norm-inflation for nonlinear Schrödinger equations in modulation spaces have been proven in [7], though some of them rule out the cubic case due to the complete integrability. Norm inflation and infinite loss of regularity for fractional Hartree and cubic NLS equations have been investigated in [8]. The proof of our result is inspired by [25].…”

“…be quantitatively wellposed. Then there exist > 0 and C 0 > 0 such that for all f ∈ B D ( ) there is a unique solution u[ f ] ∈ B X (C 0 ) to (8). In particular, u can be written as an X -convergent power series for f ∈ B D ( ),…”