Applications Of Infinite Dimensional KAM Theory To Beam Equations

In 1954, KAM theory-named after Kolmogrov, Arnold and Moser-deal with the effect of small perturbations on dynamical systems which admit invariant tori carrying quasi-periodic motion. It is the major scientific results in the 20th century. KAM theory is a vast area of research. In the dynamic stability, it claims that any finite degree of freedom of the Hamilton system is stable in the sense of the measure. In other word, most of the movement is stable in phase space, so it explains that the solar system is stable in the sence of measure. In astronomical, KAM theory proves Trojan and Greek asteroids are stable, while it may seem in the critical position. Classical KAM theory claims, under the non-degenerate condition, the majority of the non-reasont tori of integrable system are persistent under small perturbation. Comparing to the same system without disturbances, torus only have minor deformation. That is, the movement of many phase flow in the phase space are still simple In other word, the movement on the torus is still quasi-periodic, and its frequency to satisfy Diopantine conditions. The important application of KAM is to study the existence of invariant tori and linear stability of Hamilton system. Consider Hamilton system: where, P=P(x, y, zj,zj) is perturbation. If P=0, then Hamilton vector is intergable, the low-dimensional invariant torus is Tn×y0×{0}×{0}, The flow of the torus corresponding to quasi-periodic flow is x=wt+x0, where w is cutting frequency,Ωis normal frequency.In 1965. Melinkov [56] stated that elliptic low-dimensional tori of nearly integrable Hamilton system preserves, if the first Melnikov condition and the second Melnikov condition holds, and the frequency w satisfies the following condition: where k∈Zn,l∈Zm,|k|+|l|≠0. most of the invariant tori will pre-serve, under small perturbations. This result is proved by Eliassion[21], Kuksin[40], Poschel[58] after fifteen years. In the iterative procession, the second Melnikov condition is used to maintain the original form of the same specification. J. Bourgain [7] canceled the second Melnikov condition by Liapounov-Schmit method. But he need to control non-diagonal nature of the inverse operator. Furthermor, using different normal form to improve original KAM method, J.You [14] also got non-diagonal operator. Then, the classcial KAM theory is extended to the infinite dimensional Hamilton system by Kuksin [40], and he got the first result of Hamilton partial dif-ferential equations. Next, Poschel [61] re-used to describe the results under Dirichlet boundary conditions of Schrodinger equation and wave equation.Our aim is to prove that there are invariant tori under positive set of measure. So we need the following supposing:1.(Nondegenerary condition):Mappingξ→w(ξ) is Lipschitz homeo-morphism, For all integer vector (k,l)∈Zn×Z∞,where 15≤|l|≤2, we have ≠0, and2.(Spectral asymptotics):if there is d≥1 andδ0,r>0,and‖·‖denote supremum norm:for W=(X,Y,Z,Z)∈P a,p then the corresponding weighted norm: Suppose X P is analysis in D(s,r),and limited norm is and have the following theoryTheorem 1(infinite KAM theory)If H=N+P satisfy above assump-tions(1)(2)(3) where parameter 0<α≤1,γdepends on n,T,s,then there is Cantor (?)α(?)and a family of Lipschitz embedding mapΦ:Tn×(?)α→P a,p and Lipschitz map w*:(?)α→Rn,such that for anyξ∈(?)α,Φ(Tn×{ξ})is real analytic invariant torus with frequency of w*,and it satisfy the fllowing inequality whereΦ0 is trivial embedding of Tn→(?),c≤r-1 and r depend on the same parameters,Furthermore, In this paper,we mainly use the above infinite dimensional KAM theorem and the improved KAM theorem to prove that beam equations have invariant torus with Dirichlet boundary condition,when the frequency vector is in the infinite dimensional flat space,and we also prove that beam equation with quasi-periodic potential have quasi-periodic solutions.Theorem 2 Consider 1D nonlinear beam equation Let (?)IR+n be a compact subset.Then there exists a positive Cantor set (?),when(ξ1,…,ξn)∈(?)p,for all m>0 but a set of small Lebesgue measure, the above beam equation has a small-amplitude, linearly stable quasi-periodic solution of the form where wi(ξ)=λwi(ξ),λ∈IR,λ≈1,1≤i≤n,With w(ξ)=(w1(ξ),w2(ξ),…,wn(ξ)),w(ξ)=α+ξA,α=(λ1,λ2,…,λn)andTheorem 3 Suppose Hamilton(1)system satisfy(1)-(3)and‖XP‖<∈. Then there is a Cantor set (?)*(?),for each parameterξ∈(?)*,For the Hamilton have quasi-periodic solution,which generated by H=N+P.Theorem 4 Consider 1D nonlinear beam equation with Dirichlet boundary u(t,0)=u(t,π)=utt(t,0)=utt(t,π)=0-∞0, there is Cantor subser (?)*(?),when(ξ1,…,ξn)∈(?)*,then the above equation exists a real analytic equations,linear stability of quasi-periodic solution.Theorem 5 For any 00 andΩ(?)[e,2e]d ,which satisfy measΩ≥(1-3r)ed,such that for any 0<∈<∈*(r) and w∈Ω,ther exists a real analytic canonical transformationΨ∞,which define on D((r0)/2×(?)*) and satisfy the following condition:(i)There exists a constant C,such that where id is mapping unit.(ii)Ψ∞change Hamilton system into:...