Study Of Chern-Simons-Schr(?)dinger System:Existence And Instability Of Standing Waves

Nonlinear Schr¨odinger equation,as a basic model in quantum mechanics,reveals the laws of motion of microscopic particles in the material world.This thesis mainly studies the existence and instability of standing waves for a nonlinear Schr¨odinger equation,which is derived from the Chern-Simons gauge field in the Minkowski space-time R¹⁺².The thesis is divided into five chapters.In Chapter 1,we introduce the background and research status of the Chern-Simons-Schr¨odinger system,and state the main conclu-sions of this thesis.In Chapter 2,we list some basic theories and notations.In Chap-ters 3-5,we study the well-posedness,existence and instability of standing waves for the Chern-Simons-Schr¨odinger system.In Chapter 3,we discuss the well-posedness and blow-up behavior of the Chern-Simons-Schr¨odinger system in the energy space¹（R²）.Applying the energy method,by the energy estimate,Strichartz estimate,fractional Sobolev embedding and fraction-al Leibniz formula,we prove the existence and uniqueness of the local solutions for the Chern-Simons-Schr¨odinger system in¹（R²）.Making use of the Gagliardo-Nirenberg inequality and the diamagnetic inequality,we establish two invariant evolving manifold-s generated by the Chern-Simons-Schr¨odinger system.Then,we obtain a criterion for the existence of global solutions or blow-up solutions of the Chern-Simons-Schr¨odinger system.In Chapter 4,we study the existence of standing waves for the Chern-Simons-Schr¨odinger system.In the complex-valued Sobolev space¹（R²）,we construct the min-imization problem on a constrained manifold of variational functional for the related ellip-tic equation,and establish the correspondence between the constrained minimizer and the minimal energy solutions of the related elliptic equation.For the class of complex-valued Sobolev functions and Chern-Simons gauge field,we derive a new class of functional in-equalities,which is used to describe the geometric structure of the variational functional and the related properties of constrained manifolds carefully.Through refined calculation and analysis,we apply the concentration compactness lemma to explore the compact-ness of approximate solution sequences,and prove the existence of constrained minimizer.Then,for any given frequency,we obtain the existence of minimal energy solutions for the related elliptic equation problem of the Chern-Simons-Schr¨odinger system.In Chapter 5,we prove the strong instability of standing waves for the Chern-Simons-Schr¨odinger system.By using the integral estimation method of the elliptic equations,we prove that,in the complex-valued Sobolev space¹（R²）,the solutions to the relat-ed elliptic problem of the Chern-Simons-Schr¨odinger system are integrable.Then,we further investigate the variational characteristics of the related elliptic problem for the Chern-Simons-Schr¨odinger system in¹（R²）,and construct a invariant set based on the variational characteristics.Finally,making use of the invariant set,we prove that there are blow-up solutions near the standing waves,and obtain that the standing waves of the Chern-Simons-Schr¨odinger system are strongly unstable.