well posedness and smoothing effect of Schroedinger Poisson equation

DE LEO, M.; RIAL, D.

JOURNAL OF MATHEMATICAL PHYSICS

American Institute of Physics

Año: 2007 vol. 48 p. 1 - 15

0022-2488

In this work we take under consideration the Cauchy problem forthe Sch--Poisson type equation $i,Dp{t}u=-Dp{x}^2 u+V(u),u-f(|u|^2),u,,$ where $f$ represents a local nonlinearinteraction (we take into account both attractive and repulsivemodels), $V$ is taken as a suitable solution of the Poissonequation : $V=1/2,|x|ast left(mC-|u|^2 ight)$ and $mCinC_c^8$ is the doping profile, or {em impurities}. We show thatthis problem is locally well posed in the weighted Sobolev spaces$mcHs:={i in H^s(R): int (1+x^2)^{1/2},|i|^2 <8}$with $sgeq 1,$ which means : local existence, uniqueness andcontinuity of the solution with respect to the initial data.Moreover, under suitable assumptions on the local interaction weshow the existence of global solutions. Finally, we establishthat for $sgeq 1$ local in time and space smoothing effects arepresent in the solution ; more precisely, in this problem thereis locally a gain of half a derivative.