Research On Attractor Theory Of The Partial Differential Equation Model With BCS-BEC Crossover

In this paper,a partial differential equation model describing the crossover process of BCS-BEC is studied.Employing the properties of P-Laplace operator,the related knowledge of quadratic form function,the ingenious combination of different equations and various forms of inequality,the attractor problem of partial differential equation model with external force is considered in the special case(i.e.g=0);in general case(i.e.g?0)and coupling coefficient b<0,the attractor problem of the modified partial differential equation model and when the degree of nonlinear term changes from 2 to p,the attractor problem of the general modified partial differential equation model are discussed.the specific arrangements are as follows:The first chapter introduces the research background,current situation and main research contents of this paper.In the second chapter,we give the basic lemma,inequality and the relevant knowledge of attractors which are often used in this paper.In the third chapter,the global attractor problem of Ginzburg-Landau equations with external force in BCS-BEC crossover model is introduced.In view of the fact that most scholars have only explored the case that the external force term is related to x,we extend it appropriately here to explore the situation where the external force term is related to both x and t,the existence of global attractors for the following equations is obtained when b>0 and |dr|?(?)di.-idu,(x,t)=-[1/U-a]u(x,t)+c/4m?u(x,t)-b|u(x,t)|2u(x,t)-idf(x,t),i?Bt(x,t)=(2v-2?)?B(x,t)-1/4m??B(x,t),u(x,0)=u0(x),?B(x,0)=?B0(x),x??,u(x,t),?B(x,t)=0,on(?)?×[0,?).where ? is a bounded region of Rn,t?0,the complex valued functions u(x,t)and?B(x,t)are fermion pair field and condensed boson field,respectively,coupling coefficients a,b,c,m,U are real numbers,? is the chemical potential,2v is the threshold energy of the Feshbach resonance,d is generally complex number,let d=dr+idi,|d|2=dr2+di2,external force term f(x,t)is real-valued function,it is uniformly bounded with respect to t.In the fourth chapter,we discuss the global attractor of the modified Ginzburg-Landau theory in the BCS-BEC crossover model.We study the case of b<0,which makes the method of b>0 that most scholars used to study is no longer applicable.As a result,the nonlinear term cannot be removed directly,?U(X,t)?L2(?)2and ??L2(?)2 cannot be estimated.Therefore,we use the properties of quadratic form function to remove the nonlinear term.Then,on the basis of estimating ??u(x,t)?L2(?)2,and ???(x,t)?L2(?)2employing Poincare's inequality,we complete the estimation of ?u(x,t)?L2(?)2and ?(?(x,t)?L2(?)2,the global attractor problem on semigroups generated by the following equations is discussed when b<0 and 3di2?d2r2.-idut(x,t)=[dg2+1/U+a]u(x,t)+g[a+d(2v-2?)]?B(x,t)+c/4m?u(x,t)+g/4m(c-d)??B(x,t)-b|u(x,t)+g?B(x,t)|2(u(x,t)+g?B(x,t))-idf(x,t),i?Bt(x,t)=-i??B(x,t)-g/Uu(x,t)+(2v-2?)?B(x,t)-1/4m??B(x,t)+ih(x,t),u(x,0)=u0(x),?B(x,0)=?B0(x),x??,u(x,t)=0,?B(x,t)=0,on(?)?×[0,?)?. where ?>0 is a damping parameter,external force term f(x,t)and h(x,t)are real-valued functions,it is uniformly bounded with respect to t.In the fifth chapter,we study the global attractor problem of the modified two orbits Ginzburg-Landau equations in the nonequilibrium state.Here the degree of nonlinear term in the equations is no longer limited to constant 2,but p?2di(di+|d|)/dr2,we use the knowledge of matrix and the linear combination of different equations to prove the existence of global attractor for Ginzburg-Landau equations with external force when p?2di(di+|d|)/dr2,the specific equation are as follows:-idut(x,t)=[-dg2+1/U+a]u(x,t)+g[a+d(2v-2?)]?B(x,t)+c/4m?u(x,t)+g/4m(c-d)??B(x,t)-b|u(x,t)+g?B(x,t)|p(u(x,t)+g?B(x,t))-idf(x,t),i?Bt(x,t)--i??B(x,t)-g/U(x,t)+(2v-2?)?B(x,t)-1/4m??B(x,t)+ih(x,t),u(x,0)=u0(x),?B(x,0)-?B0(x),x??,u(x,t)=0,?B(x,t)=0,on(?)? ×[0,?)?.