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Long-Time Existence for Semi-linear Beam Equations on Irrational Tori
Journal of Dynamics and Differential Equations ( IF 1.4 ) Pub Date : 2021-03-01 , DOI: 10.1007/s10884-021-09959-3
Joackim Bernier , Roberto Feola , Benoît Grébert , Felice Iandoli

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 \(\varepsilon \ll 1\) is the size of the initial datum, we prove that the lifespan \(T_\varepsilon \) of solutions is \(O(\varepsilon ^{-A(n-2)^-})\) where \(A\equiv A(d,n)= 1+\frac{3}{d-1}\) when n is even and \(A= 1+\frac{3}{d-1}+\max (\frac{4-d}{d-1},0)\) when n is odd. For instance for \(d=2\) and \(n=3\) (quadratic nonlinearity) we obtain \(T_\varepsilon =O(\varepsilon ^{-6^-})\), much better than \(O(\varepsilon ^{-1})\), the time given by the local existence theory. The irrationality of the torus makes the set of differences between two eigenvalues of \(\sqrt{\Delta ^2+1}\) 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.



中文翻译:

无理托里上半线性梁方程的长时间存在性

我们考虑d维无理环上的半线性束方程,其光滑非线性为\(n-1 \)\(nge 3 \)\(dge 2 \)。如果\(\ varepsilon \ ll 1 \)是初始基准的大小,我们证明溶液的寿命\(T_ \ varepsilon \)\(O(\ varepsilon ^ {-A(n-2)^- }} \)其中\(A \ equiv A(d,n)= 1+ \ frac {3} {d-1} \)n为偶数且\(A = 1+ \ frac {3} {d-n为奇数时,1} + \ max(\ frac {4-d} {d-1},0)\)。例如,对于\(d = 2 \)\(n = 3 \)(二次非线性),我们得到\(T_ \ varepsilon = O(\ varepsilon ^ {-6 ^-})\),比局部存在理论给出的时间(\(O(\ varepsilon ^ {-1})\)好得多。圆环的非理性使得\(\ sqrt {\ Delta ^ 2 + 1} \)的两个特征值之间的差集累积为零,从而促进了高傅立叶模态之间的交换,并使长时间控制解变得复杂。我们的结果是通过组合Birkhoff法线形式步骤和修改后的能量步骤而获得的。

更新日期:2021-03-01
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