On the Cauchy problem for logarithmic fractional Schrödinger equation

Rémi Carles (1) , Fangyuan Dong (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

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.

Domaines: Equations aux dérivées partielles [math.AP]

Rémi Carles, Fangyuan Dong. On the Cauchy problem for logarithmic fractional Schrödinger equation. 2024. ⟨hal-04538854⟩