Abstract
In this paper, we develop dispersive PDE techniques for the Fermi–Pasta–Ulam (FPU) system with infinitely many oscillators, and we show that general solutions to the infinite FPU system can be approximated by counter-propagating waves governed by the Korteweg–de Vries (KdV) equation as the lattice spacing approaches zero. Our result not only simplifies the hypotheses but also reduces the regularity requirement in the previous study (Schneider and Wayne, In: International conference on differential equations, Berlin, 1999, World Sci. Publ, River Edge, NJ, Vol 1, 2, pp 390–404, 2000).
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Notes
The Hamiltonian is derived from (1.1).
These definitions are consistent with the discrete Laplacian \(\Delta _h\), because \((-\Delta _h)\) is the Fourier multiplier of the symbol \(\frac{4}{h^2}\sin ^2(\frac{h\xi }{2})\); thus, \(|\nabla _h|=\sqrt{-\Delta _h}\) and \(\langle \nabla _h\rangle =\sqrt{1-\Delta _h}\).
By the discrete Fourier transform,
$$\begin{aligned} \begin{aligned} {\mathcal {F}}_h\left( e^{a\partial _h}(u_h^2)\right) (\xi )&=e^{ia\xi }\frac{1}{2\pi }\int _{-\pi /h}^{\pi /h}({\mathcal {F}}_hu_h)(\xi -\eta ) ({\mathcal {F}}_hu_h)(\eta )\mathrm{d}\eta \\&=\frac{1}{2\pi }\int _{-\pi /h}^{\pi /h}e^{ia(\xi -\eta )}({\mathcal {F}}_hu_h)(\xi -\eta ) e^{ia\eta }({\mathcal {F}}_hu_h)(\eta )\mathrm{d}\eta ={\mathcal {F}}_h\left( (e^{a\partial _h}u_h)^2\right) (\xi ). \end{aligned} \end{aligned}$$This computation can be extended to any polynomial of finite degree.
They are sometimes called the Bourgain spaces or dispersive Sobolev spaces.
In particular, when \(\lambda = h\mathbb {Z}\), \({{\tilde{u}}}\) (as in Definition 3.14) is defined by
$$\begin{aligned} {{\tilde{u}}}_h(\tau , \xi ) = h \sum _{x \in h\mathbb {Z}} \int _{\mathbb {R}} e^{-it\tau } e^{-ix \xi } u_h(t,x) \; \mathrm{d}\tau . \end{aligned}$$Roughly speaking, in a (k-)multilinear form, one has a frequency relation \(\xi _1 + \cdots + \xi _k = \xi \); thus, the multilinear form vanishes unless the maximum two frequencies are comparable.
One can fix \(b > \frac{1}{2}\) here,and \(\delta > 0\) below such that the argument of the local well-posedness is valid.
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Acknowledgements
This research of the first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (NRF-2020R1A2C4002615). This work of the second author was supported by project France-Chile ECOS-Sud C18E06, the Ewha Womans University Research Grant of 2020, the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2020R1F1A1A0106876811). This work of the third author was supported by the research grant of the Chungbuk National University in 2020 and the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1C1C1005700). Part of this work was complete while the second author was visiting Chung-Ang University (Seoul, Republic of Korea). The second author acknowledges the warm hospitality of the institution.
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Appendices
Appendix A. Failure of the Linear Estimate
In subsection 5.2, we measured the size of linear interpolation of the FPU flows in \(C_t([-T,T]:H_x^s(h\mathbb {Z}))\) in order to approximate the FPU flows by the Airy flows. More precisely, the crucial estimates were that for \(0\leqq s\leqq 1\),
However, such uniform estimates fail if we consider instead the \(X^{s,b}\) spaces associated to KdVs (7.1) as approximation spaces, which means that even though FPU and KdVs are shown to be well-posed in \(L^2\) via \(X^{s,b}\), justification of approximation from FPU to KdVs via \(X^{s,b}\) is nontrivial.
Proposition A.1
Let \(0\leqq s\leqq 1\) and \(b>0\). Then,
where \(X_{\pm }^{s,b}\) is defined as in (5.1).
Proof
We claim that there exist a constant \(C_b > 0\) independent of \(h>0\) such that
for \(f_h \in H_h^s\) satisfying \(\text{ supp } {{\mathcal {F}}}_h(f_h) \subset \{ \xi \in \mathbb {T}_h : |\xi |\geqq \frac{\pi }{2h} \}\). Then (A.1) immediately follows. We prove only (A.2) for the \(+\) case, since the other case can be treated similarly. Using Lemma 5.12, we compute
for \(\gamma _{m,h} := \frac{2m\pi }{h}\). First, let us compute the \(L_\tau ^2\) norm. A direct computation gives
and it is easy to verify that
which indicates that \(\frac{\pi ^3}{3}(\frac{m}{h})^3\) is the dominant part in \(\frac{1}{24}(\xi +\gamma _{m,h})^3 - s_h^{+}(\xi )\). In particular, there exists \(m_0 \gg 1\), independent of h, such that for \(m \leqq - m_0\),
Using the above-mentioned observation, we have for \(m\leqq m_0\)
which implies that
Since
for all \(\xi \in \text{ supp } {{\mathcal {F}}}_h(f_h)\), we conclude that
\(\square \)
Appendix B. Analysis for General Nonlinearities
This appendix is devoted to some estimates for the higher-order remainder term introduced in Section 2 to complete our analysis established in Sections 4 and 7.2 . For any real number \(\rho \in \mathbb {R}\), we write \(\rho ^+\) if there exists a small \(0 < \epsilon \ll 1\) such that \(\rho ^+ = \rho + \epsilon \). Analogously, we use \(\rho ^-\). The main estimate dealt with in this section is as follows:
Lemma B.1
Let \(0 \leqq s \leqq 1\) and \(0<h \leqq 1\) be given. Assume that
for some constant \(M > 0\). Then, for \({\mathcal {R}}\) as in (2.1), we have
where the constant C in supremum depends only on \(\frac{1}{2}^+\).
Remark B.2
As seen in the proofs of Propositions 4.1, M depends on the initial condition. Meanwhile, in the proofs of Propositions 7.3 and 2.2 , M depends not only on the initial condition but also on the local existence time, especially, \(T^{0^-}\). However, owing to \(T^{\frac{3}{4}}\), the right-hand side of (B.1) can be sufficiently small by choosing a suitable time T independent of h.
Remark B.3
Lemma B.1 indeed completes the proof of Proposition 7.3.
Remark B.4
Together with the embedding property (Lemma 3.12 (3)), Lemma B.1 completes the proofs of Propositions 4.1 and 2.2 .
Remark B.5
Lemma B.1 ensures that the higher-order term in (2.8) is indeed the error term as \(h \rightarrow 0\) in the proof of Proposition 2.2. More precisely, in a strong contrast to the quadratic error terms
in the proof of Proposition 2.2 (see also Lemmas 6.3 and 6.4 ), Lemma B.1 ensures that the higher-order term itself in (2.8) can be understood as a strong error term as \(h \rightarrow 0\) in the sense that the smoothness condition on the data is not necessary.
Proof of Lemma B.1
By assumption, we consequently have
By (3.12), we estimate the higher-order remainder
Interpolating the dualization of the Strichartz estimates (Corollary 5.3), that is,
with the trivial identity \(\Vert u_h\Vert _{X_{h,\pm }^{0,0}}=\Vert u_h\Vert _{L_t^2L_x^2}\), we have
Using this bound and the Hölder inequality, we obtain
By unitarity (with the algebra in footnote 1), we remove the translation operator as follows:
By assumption, we have
Hence, it follows that
For \(\Vert e^{\pm \frac{t}{h^2}\partial _h}{\tilde{r}}_h\Vert _{L_t^\infty L_x^4}\), if \(0\leqq s\leqq \frac{1}{4}\), then by the Sobolev inequality, unitarity and Lemma 3.3,
Meanwhile, if \(\frac{1}{4}<s\leqq 1\), then
Therefore, by combining all these results, we complete the proof of (B.1). \(\quad \square \)
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Hong, Y., Kwak, C. & Yang, C. On the Korteweg–de Vries Limit for the Fermi–Pasta–Ulam System. Arch Rational Mech Anal 240, 1091–1145 (2021). https://doi.org/10.1007/s00205-021-01629-4
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DOI: https://doi.org/10.1007/s00205-021-01629-4