论文标题
非理性托里的半线性束方程的长期存在
Long-time existence for semi-linear beam equations on irrational tori
论文作者
论文摘要
我们考虑在d维度非理性的圆环上的半线性束方程，其平稳性n-1，n $ \ ge $ 3和d $ \ ge $ 2。如果$ $ $ $ $ \ ll $ 1是初始基准的大小，我们证明了lifespan t $抽utions的lifespan t $ε$（$ a $ $ $ - n）= 1 + 3 d--1当n均匀而a = 1 + 3 d--1 + max（4--d d--1，0）时，n是奇数时。例如，对于d = 2和n = 3（二次非线性），我们获得了T $ε$ = O（$ε$ -6-），比O（$ε$ -1）好得多，这是本地存在理论给出的时间。圆环的非理性性使两个$ \ sqrt $ $δ$ 2 + 1之间的特征值之间存在差异，从而累积到零，从而促进了高傅立叶模式之间的交换，并在长期内使解决方案的控制变得复杂。我们的结果是通过结合Birkhoff正常形式步骤和修改的能量步骤来获得的。
We consider the semi-linear beam equation on the d dimensional irrational torus with smooth nonlinearity of order n -- 1 with n $\ge$ 3 and d $\ge$ 2. If $ε$ $\ll$ 1 is the size of the initial datum, we prove that the lifespan T$ε$ of solutions is O($ε$ --A(n--2) --) where A $\not\equiv$ A(d, n) = 1 + 3 d--1 when n is even and A = 1 + 3 d--1 + max(4--d d--1 , 0) when n is odd. For instance for d = 2 and n = 3 (quadratic nonlinearity) we obtain T$ε$ = O($ε$ --6 --), much better than O($ε$ --1), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of $\sqrt$ $Δ$ 2 + 1 accumulate to zero, facilitating the exchange between the high Fourier modes and complicating the control of the solutions over long times. Our result is obtained by combining a Birkhoff normal form step and a modified energy step.
