On the Korteweg–de Vries Limit for the Fermi–Pasta–Ulam System

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).

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 $0\leqq s\leqq 1$,

$$\begin{aligned} \Vert l_h S_h^{\pm }(t) f_h \Vert _{C_t([0,T]:H_x^s(h\mathbb {Z}))} \lesssim \Vert l_h f_h \Vert _{H^s(h\mathbb {Z})} \lesssim \Vert f_h\Vert _{H^s(h\mathbb {Z})}. \end{aligned}$$

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,

$$\begin{aligned} \sup _{h> 0, \ f_h\in H^s(h\mathbb {Z}) } \frac{\Vert \theta (t) l_h S_h^{\pm }(t) f_h \Vert _{X_{\pm }^{s,b}}}{\Vert f_h\Vert _{H^s(h\mathbb {Z})}} = \infty , \end{aligned}$$

(A.1)

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

$$\begin{aligned} \Vert \theta (t) l_h S_h^{\pm }(t) f_h \Vert _{X_{\pm }^{s,b}}^2 > rsim&~{}C_b\frac{1}{h^{6b}}\Vert f_h\Vert _{H^s}^2. \end{aligned}$$

(A.2)

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

$$\begin{aligned}&\Vert \theta (t) l_h S_h^{+}(t) f_h \Vert _{X_{+}^{s,b}}^2 \\&\quad =\sum _{m\in \mathbb {Z}}\int _{\gamma _{m,h}+[-\frac{\pi }{h},\frac{\pi }{h})}\left\| \langle \xi \rangle ^s \left\langle \tau - \frac{\xi ^3}{24} \right\rangle ^b {\mathcal {F}}_{t,x} \big (\theta (t) l_h S_h^{+}(t) f_h\big )(\tau ,\xi ) \right\| _{L_{\tau }^2}^2 \mathrm{d}\xi \\&\quad =\sum _{m\in \mathbb {Z}} \int _{-\frac{\pi }{h}}^{\frac{\pi }{h}} \left\| {{\widehat{\theta }}}( \tau )\left\langle \tau - \frac{1}{24}(\xi +\gamma _{m,h} )^3 + s_h^{+}(\xi ) \right\rangle ^b \right\| _{L_{\tau }^2}^2 \\&\qquad \langle \xi +\gamma _{m,h} \rangle ^{2s} \mathcal {L}_h(\xi +\gamma _{m,h})^2 |{{\mathcal {F}}}_h(f_h)(\xi )|^2 \mathrm{d}\xi \end{aligned}$$

for $\gamma _{m,h} := \frac{2m\pi }{h}$. First, let us compute the $L_\tau ^2$ norm. A direct computation gives

$$\begin{aligned} \frac{1}{24}(\xi +\gamma _{m,h})^3 - s_h^{+}(\xi )&=\frac{\pi ^3}{3}\left( \frac{m}{h}\right) ^3+\frac{\pi ^2 \xi }{6}\left( \frac{m}{h}\right) ^2+\frac{\pi \xi ^2}{4}\left( \frac{m}{h}\right) +\frac{\xi ^3}{24}+\frac{1}{h^2}\left( \xi -\frac{2}{h}\sin \left( \frac{h\xi }{2}\right) \right) \end{aligned}$$

and it is easy to verify that

$$\begin{aligned}&\left| \frac{\pi ^2 \xi }{6}\left( \frac{m}{h}\right) ^2+\frac{\pi \xi ^2}{4}\left( \frac{m}{h}\right) +\frac{\xi ^3}{24} \right. \\&\quad \left. +\frac{1}{h}\left( \xi -\frac{2}{h}\sin \left( \frac{h\xi }{2}\right) \right) \right| \lesssim \frac{m^2}{h^3}, \text { for all } \xi \in \mathbb {T}_h, \end{aligned}$$

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 $m \leqq - m_0$,

$$\begin{aligned} - \frac{1}{24}(\xi +\gamma _{m,h})^3 + s_h^{+}(\xi ) > rsim \left( \frac{|m_0|}{h}\right) ^3 \gg 1, \text { for all } \xi \in \mathbb {T}_h. \end{aligned}$$

(A.3)

Using the above-mentioned observation, we have for $m\leqq m_0$

$$\begin{aligned}&\left\| {{\widehat{\theta }}}( \tau )\left\langle \tau - \frac{1}{24}(\xi + \gamma _{m,h} )^3 + s_h^+(\xi ) \right\rangle ^b \right\| _{L_{\tau }^2}^2 \\&\quad \geqq \int _0^\infty |{{\widehat{\theta }}}( \tau )|^2 \left\langle \tau - \frac{1}{24}(\xi + \gamma _{m,h} )^3 + s_h^+(\xi ) \right\rangle ^{2b} \mathrm{d}\tau \\&\quad > rsim \left( \frac{|m_0|}{h}\right) ^{6b}\int _0^\infty |{{\widehat{\theta }}}( \tau )|^2 \mathrm{d}\tau \\&\quad > rsim \left( \frac{|m_0|}{h}\right) ^{6b}, \end{aligned}$$

which implies that

$$\begin{aligned}&\Vert \theta (t) l_h S_h^{+}(t) f_h \Vert _{X_{+}^{s,b}}^2\\&\quad > rsim \left( \frac{|m_0|}{h}\right) ^{6b}\sum _{m\leqq -m_0}\int _{-\frac{\pi }{h}}^{\frac{\pi }{h}} \langle \xi +\gamma _{m,h} \rangle ^{2s} \mathcal {L}_h(\xi +\gamma _{m,h})^2 |{{\mathcal {F}}}_h(f_h)(\xi )|^2 \mathrm{d}\xi \\&\quad > rsim \left( \frac{|m_0|}{h}\right) ^{6b} \int _{-\frac{\pi }{h}}^{\frac{\pi }{h}} \langle \xi +\gamma _{m_0,h} \rangle ^{2s} \left( \frac{4\sin ^2\left( \frac{h\xi }{2}\right) }{h^2(\xi +\gamma _{m_0,h})^2} \right) ^2 |{{\mathcal {F}}}_h(f_h)(\xi )|^2 \mathrm{d}\xi . \end{aligned}$$

Since

$$\begin{aligned} |\xi | \lesssim |\xi +\gamma _{m_0,h}| \lesssim |\gamma _{m_0,h}| \quad \text{ and } \quad \sin ^2\left( \frac{h\xi }{2}\right) \geqq \frac{1}{2} \end{aligned}$$

for all $\xi \in \text{ supp } {{\mathcal {F}}}_h(f_h)$, we conclude that

$$\begin{aligned} \Vert \theta (t) l_h S_h^{+}(t) f_h \Vert _{X_{+}^{s,b}}^2 > rsim&~{}\left( \frac{|m_0|}{h}\right) ^{6b} \int _{-\frac{\pi }{h}}^{\frac{\pi }{h}} \langle \xi \rangle ^{2s} (h\gamma _{m_0,h})^{-4}|{{\mathcal {F}}}_h(f_h)(\xi )|^2 \mathrm{d}\xi \\ > rsim&~{}\frac{|m_0|^{6b-4}}{h^{6b}}\Vert f_h\Vert _{H_h^s}^2. \end{aligned}$$

$\square$

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

$$\begin{aligned} \Vert u_h^\pm \Vert _{X_{h,\pm }^{s,\frac{1}{2}^+}}\leqq M, \end{aligned}$$

for some constant $M > 0$. Then, for ${\mathcal {R}}$ as in (2.1), we have

$$\begin{aligned}&\bigg \Vert \int _0^t S_h^\pm (t-t_1) \nabla _h e^{\pm \frac{t_1}{h^2}\partial _h}h^2\mathcal {R}(t_1) \mathrm{d}t_1\bigg \Vert _{X_{h,\pm }^{s,\frac{1}{2}^+}} \nonumber \\&\quad \lesssim h^{\min \{\frac{5}{4}-s, \frac{3}{4}+s\}^{-}}T^{\frac{3}{4}}M^3 \sup _{|r|\leqq Ch^\frac{3}{2}M}|V^{(4)}(r)|, \end{aligned}$$

(B.1)

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

$$\begin{aligned} \int _0^t S_h^\pm (t-t_1) \nabla _h \left( 2u_h^\pm (t_1)(e^{\pm \frac{2t_1}{h^2}\partial _h}u_h^{\mp }(t_1)) + (e^{\pm \frac{2t_1}{h^2}\partial _h}u_h^{\mp }(t_1))^2 \right) \; \mathrm{d}t_1 \end{aligned}$$

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

$$\begin{aligned} \Vert u_h^\pm \Vert _{C_tH_x^s}\lesssim M \quad \text{ and } \quad \Vert {\tilde{r}}_h\Vert _{C_tH_x^s}\lesssim M. \end{aligned}$$

By (3.12), we estimate the higher-order remainder

$$\begin{aligned} \bigg \Vert \int _0^t S_h^\pm (t-t_1) \nabla _h e^{\pm \frac{t_1}{h^2}\partial _h}h^2\mathcal {R}(t_1) \mathrm{d}t_1\bigg \Vert _{X_{h,\pm }^{s,\frac{1}{2}^+}}\lesssim \big \Vert \nabla _h e^{\pm \frac{t}{h^2}\partial _h}h^2\mathcal {R}(t)\big \Vert _{X_{h,\pm }^{s,-(\frac{1}{2}^-)}}. \end{aligned}$$

Interpolating the dualization of the Strichartz estimates (Corollary 5.3), that is,

$$\begin{aligned} \Vert u_h\Vert _{X_{h,\pm }^{0,-(\frac{1}{2}^+)}}\lesssim \Vert |\nabla _h|^{-(\frac{1}{4}^-)}u_h\Vert _{L_t^{\frac{4}{3}^-}L_x^{1^+}}, \end{aligned}$$

with the trivial identity $\Vert u_h\Vert _{X_{h,\pm }^{0,0}}=\Vert u_h\Vert _{L_t^2L_x^2}$, we have

$$\begin{aligned} \Vert u_h\Vert _{X_{h,\pm }^{0,-(\frac{1}{2}^-)}}\lesssim \Vert |\nabla _h|^{-(\frac{1}{4}^-)}u_h\Vert _{L_t^{\frac{4}{3}}L_x^{1^+}}. \end{aligned}$$

Using this bound and the Hölder inequality, we obtain

$$\begin{aligned} \begin{aligned}&\big \Vert \nabla _h e^{\pm \frac{t}{h^2}\partial _h}h^2\mathcal {R}(t)\big \Vert _{X_{h,\pm }^{s,-(\frac{1}{2}^-)}} \\&\quad \lesssim h^{(\frac{5}{4}-s)^-} \Vert e^{\pm \frac{t}{h^2}\partial _h}\mathcal {R}(t)\Vert _{L_t^{\frac{4}{3}}L_x^{1+}}\\&\quad \lesssim h^{(\frac{5}{4}-s)^{-}}\Big \Vert \big (e^{\pm \frac{t}{h^2}\partial _h}V^{(4)}(h^2{\tilde{r}}_h^*)\big )\cdot \big (e^{\pm \frac{t}{h^2}\partial _h}{\tilde{r}}_h\big )^3\Big \Vert _{L_t^{\frac{4}{3}}L_x^{1}}\\&\quad \leqq h^{(\frac{5}{4}-s)^{-}}T^{\frac{3}{4}}\big \Vert \big (e^{\pm \frac{t}{h^2}\partial _h}V^{(4)}(h^2{\tilde{r}}_h^*)\big )\cdot \big (e^{\pm \frac{t}{h^2}\partial _h}{\tilde{r}}_h\big )\big \Vert _{L_t^\infty L_x^2}\Vert e^{\pm \frac{t}{h^2}\partial _h}{\tilde{r}}_h\Vert _{L_t^\infty L_x^4}^2. \end{aligned} \end{aligned}$$

By unitarity (with the algebra in footnote 1), we remove the translation operator as follows:

$$\begin{aligned} \big \Vert \big (e^{\pm \frac{t}{h^2}\partial _h}V^{(4)}(h^2{\tilde{r}}_h^*)\big )\cdot \big (e^{\pm \frac{t}{h^2}\partial _h}{\tilde{r}}_h\big )\big \Vert _{L_t^\infty L_x^2} \lesssim \Vert V^{(4)}(h^2{\tilde{r}}_h^*)\Vert _{L_t^\infty L_x^\infty }\Vert {\tilde{r}}_h\Vert _{L_t^\infty L_x^2}. \end{aligned}$$

By assumption, we have

$$\begin{aligned} \Vert h^2{\tilde{r}}_h^*\Vert _{C_tL_x^\infty }\leqq \Vert h^2{\tilde{r}}_h\Vert _{C_tL_x^\infty }\leqq h^{\frac{3}{2}}\Vert {\tilde{r}}_h\Vert _{C_tL_x^2}\leqq h^{\frac{3}{2}}\Big \{\Vert u_h^+\Vert _{C_tL_x^2}+\Vert u_h^-\Vert _{C_tL_x^2}\Big \}\leqq Ch^{3/2} M. \end{aligned}$$

Hence, it follows that

$$\begin{aligned} \Vert V^{(4)}(h^2{\tilde{r}}_h^*)\Vert _{L_t^\infty L_x^\infty }\leqq \sup _{|r|\leqq Ch^{3/2} M}|V^{(4)}(r)|<\infty . \end{aligned}$$

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,

$$\begin{aligned} \Vert e^{\pm \frac{t}{h^2}\partial _h}{\tilde{r}}_h\Vert _{L_x^4}\lesssim \Vert e^{\pm \frac{t}{h^2}\partial _h}{\tilde{r}}_h\Vert _{{\dot{H}}^{\frac{1}{4}}} \lesssim h^{-\frac{1-4s}{4}}\Vert {\tilde{r}}_h\Vert _{{\dot{H}}_x^s}\lesssim h^{-\frac{1-4s}{4}}M. \end{aligned}$$

Meanwhile, if $\frac{1}{4}<s\leqq 1$, then

$$\begin{aligned} \Vert e^{\pm \frac{t}{h^2}\partial _h}{\tilde{r}}_h\Vert _{L_x^4}\lesssim \Vert e^{\pm \frac{t}{h^2}\partial _h}{\tilde{r}}_h\Vert _{H_x^s}=\Vert {\tilde{r}}_h\Vert _{H_x^s}\lesssim M. \end{aligned}$$

Therefore, by combining all these results, we complete the proof of (B.1). $\quad \square$