On the Cauchy problem for logarithmic fractional Schrödinger equation
Résumé
We consider the fractional Schrodinger equation with a logarithmic
nonlinearity, when the power of the Laplacian is between zero and
one. We prove global existence results in three different functional
spaces: the Sobolev space corresponding to the quadratic form domain
of the fractional Laplacian, the energy space, and a space contained
in the operator domain of the fractional Laplacian. For this last
case, a finite momentum assumption is made, and the key step
consists in estimating the Lie commutator between the fractional
Laplacian and the multiplication by a monomial.
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