Dynamic Behavior Of Two Kinds Of Fractional Nonlinear Schr（?）dinger Equations

The fractional nonlinear Schr（?）dinger equation is a fundamental model in frac-tional quantum mechanics and widely used in optics,fluid mechanics,quantum mechanics and so on.The thesis is devoted to study dynamic behavior of two kinds of fractional nonlinear Schr（?）dinger equations,mainly including the existence global and blowup of solutions,the s-cattering theory of the global solutions and the concentration property of the blowup solutions.Inspired by Dodson-Murphy^[1],we use a new method to discuss the scattering theory of nonlinear fractional Schr（?）dinger equation with Hartree term.In this way,we no longer need the concentration-compactness principle.On the one hand,there is smoothness loss in the dispersive estimation for fractional Schr（?）dinger equation,which makes it difficult to estimate the short time evolution.On the other hand,the nonlocal property of fractional Laplace operator makes it impossible to obtain its Morawetz estimation directly.At first,we show the scattering criterion depending on radial Sobolev embedding inequality and Hardy-Littlewood-Sobolev inequality.Then,combining Balakrishnan’s formula,the corresponding Morawetz estimation is obtained.And finally we prove the scattering result in a new way.For the inhomogeneous fractional nonlinear Schr（?）dinger equation,we apply the fractional Strauss inequality to get the local virial inequality.Then,a sharp threshold of global existence for L~2-supercritical and L~2-critical is obtained via Gagliardo-Nirenberg inequality.Further-more,by defining a new function space,we establish the corresponding compact embedding relation.The concentration phenomenon of blowup solutions for L~2-critical case is obtain in the end.