Quasi-invariant modified Sobolev norms for semi linear reversible PDEs
Quasi-invariant modified Sobolev norms for semi linear reversible PDEs
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DOI：
10.1088/0951-7715/23/2/011
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发表时间：
2009-08
期刊：
影响因子：
1.7
通讯作者：
E. Faou;B. Grébert
中科院分区：
文献类型：
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作者：
E. Faou;B. Grébert
考虑一类一般的无限维可逆微分系统。假设线性频率上的非共振条件，我们构造了接近类sobolev范数的几乎不变的伪范数。这允许我们证明，如果初始数据z0的指标s的Sobolev范数足够小（阶为λ），那么解的Sobolev范数在很长的时间间隔内（阶为λ−r， r为任意）以2λ为界。结果表明，该定理适用于包括d维环面上非线性Schrödinger （NLS）方程在内的一类可逆半线性偏微分方程。我们还将我们的方法应用于一个可逆但非哈密顿的耦合NLS方程组。我们还注意到，对于同一类可逆系统，我们可以证明Birkhoff范式定理，这反过来又意味着Sobolev范数上的相同界。然而，我们用来证明拟不变伪范数存在的技术要简单和直接得多。
We consider a general class of infinite dimensional reversible differential systems. Assuming a nonresonance condition on linear frequencies, we construct for such systems almost invariant pseudo-norms that are close to Sobolev-like norms. This allows us to prove that if the Sobolev norm of index s of the initial data z0 is sufficiently small (of order ϵ) then the Sobolev norm of the solution is bounded by 2ϵ over a very long time interval (of order ϵ−r with r arbitrary). It turns out that this theorem applies to a large class of reversible semi-linear partial differential equations (PDEs) including the nonlinear Schrödinger (NLS) equation on the d-dimensional torus. We also apply our method to a system of coupled NLS equations which is reversible but not Hamiltonian. We also note that for the same class of reversible systems we can prove a Birkhoff normal form theorem, which in turn implies the same bounds on the Sobolev norms. Nevertheless the techniques that we use to prove the existence of quasi-invariant pseudo-norms are much more simple and direct.