Long Time Behavior Of Ginzburg -Landau Theory In BCS-BEC Crossovering

This article mainly researches in different circumstances,global attractor of Ginzburg-Landau model during BCS-BEC crossover.BCS is a theory which is based on nearly free electron model and is constructed when electronic-phonon effect is very weak up.The theory is used to explain the microscopic of superconductivity of the conventional superconductors.The bose-Einstein condensation state(BEC)refers the phenomenon that a large number of particles in the boson system will be condensed into one or several of the quantum state when the temperature is lower than a certain critical value.With the deepening of the research,the scientists found that fermions and bosons can mutually transform through Feshbach resonance.In other words,There exist across phenomenon between BCS state to BEC state,so that atomic physics has become a frontier research field in many disciplines.In 2006,Machida M and Koyama T described this crossover phenomenon through ginzburg-landau equations as follows:-idut=(-dg2+1/U+a)u+g[a+d(2v-2?)]?+c/4m?2u(1)+g/4m(c-d)??-b|ugg?|2(u+g?),i?t=-g/Uu+(2v-2?)?-1/4m??.(2)However,there are few results on attractors of the ginzburg-landau model in BCS-BEC crossover,and the particularity of the model makes it difficult to study its long-term behavior,we first discuss the special form and then analyze the general situation.and the specific results are as follows.First,considering the special form of the model(g=0,b>0),the degree of nonlinear is two),we decide to use the property of p-laplace operator(lemma 2.2.7)to overcome the difficulties caused by the non-linear term in the estimation,and we finally obtain the global attractor of the initial boundary value problem of the equations(1)-(2)when the coupling coefficient g=0 by combining the prior estimate and Gronwall inequality.Second,we studied the global attractor namely under nonequilibrium nonlinear term index is p when the coupling coefficient g=0,(i.e.g=0,b>0,the degree of nonlinear is p)But the rise of the degree of nonlinear caused that the previous method can't be used well Then by introducing the properties of P-Laplace operator(lemma 2.2.8)and combining Gronwall lemma so that we solved this problem and got the global attractorThird,Feshbach resonance played an important role in the transformation process of fermion atoms to boson molecules.Therefore,we considered researching the global attractor when the direction of divergence changed(i.e.g=0,b<0,the degree of nonlinear term is two).The change in the direction of the divergence makes the ahead method fail.In repeated tests,we found that this difficult problem can be solved by changing the order of the a priori estimate and using the poincare inequality.At the same time,we need to combine the equation |u|2|?u|2=1/4|?|u|2|2+1/4|4?u-u?u|2 and the properties of the quadratic function for overcoming the difficulties brought by nonlinear term.In the end,we found that global attractor still remains even if the scattering direction changes(b<0).Fourth,the general form is applied more widely,so after completing the basis of above work,we tried to study the general form,(g?0,b>0,the degree of nonlinear is two and choosing spacial initial valuse)i.e.the situation of equations(1)-(2)exists global attractor,it is worth mentioned that we can't complete the necessary estimates just like previous work in the case of no add any limites to solution.So this section needs to add some restricted conditions and prior estimates by combining Sobolev embedding theorem,Gronwall's lemma and inner product complex estimate inequality lemma(2.27).we finally obtained the equations(1)-(2)the initial-boundary value problem of global attractor.Fifth,on the basis of the above work,we studied the general form of the model and the equations of shape such as(1)-(2),the index of the nonlinear term is P under the equilibrium state of the global attractorg?0,b>0,the degree of nonlinear is P and choosing spacial initial valuse)i.e.Nonlinear term in the model number is higher,which makes the global existence of absorbing sets less likely to get,although it is possible to use the foregoing method to deal with the estimation problem of nonlinear term.But,but it can't be avoided to bring some new challenges,for example we found that estimates of ?u+g??2p+22p+2 failed to get in the previous work namely in the estimate of ?ut?2.So,we need to find other solutions.In the end,we find that the problem can be solved by the combination of Gagliardo-Nirenberg,Agmon and Gronwall inequality and so on,we found the global attractor of equations(1)-(2)in the non-equilibrium state when the nonlinear exponential is p.Sixth,In order to obtain a better result under the conditions that without adding any restriction to the solution(g?0,modify the equation),the model was further discussed.It was found through discussion that if no restriction was added to the solution,the equation must be modified as follows:d?t-i(a-1/U)?-ig/U?-ic/4m??+ib|?|p?+?g?=f(x,t),(3)?t+??-ig/U?+ig2/U?+i(2v-2?)?-i/4m??=h(x,t).(4)So the end of this article,when the equilibrium state,we research the global attractor of the Ginzburg-Landau equations during the BCS-BEC crossover of been revised mode.The revised model contains external force related to not only space but also time,the model established in this case has wider application.When making prior estimates,we first use the traditional way,and then combine inequality Gagliardo-Nirenberg and Agmon to overcome the difficulties caused by estimation of higher order nonlinear term,so as to simplify the calculation,and the revised model with initial boundary value problem for a global attractor exists.