Birkhoff normal form for partial differential equations with tame modulus
Birkhoff normal form for partial differential equations with tame modulus
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DOI：
10.1215/s0012-7094-06-13534-2
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发表时间：
2006-12-01
影响因子：
2.5
通讯作者：
Grebert, B.
中科院分区：
文献类型：
--
作者：
Bambusi, D.;Grebert, B.
证明了Hamilton偏微分方程的一个抽象Birkhoff范式定理。该定理适用于非线性满足一个性质的半线性方程，我们称之为驯服模。这一性质与Moser的经典驯服不等式有关。在非共振的情况下，我们推断，任何小振幅的解决方案仍然非常接近环面很长一段时间。我们还发展了一个将抽象理论应用于一维偏微分方程的一般方案，并利用它研究了具有不同边界条件的具体方程（非线性波动方程、非线性薛定谔方程）。最后给出了d维环面上NLS方程的一个应用。在所有情况下，我们推导出的增长上的高Sobolev规范的界限。特别地，我们得到了解存在时间的下界。
We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations (PDEs). The theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions, and we use it to study some concrete equations (nonlinear wave (NLW) equation, nonlinear Schrodinger (NLS) equation) with different boundary conditions. An application to an NLS equation on the d-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular, we get lower bounds on the existence time of solutions.