# Periodic BBM NE-1-H2-to-H3

## Setting

You are solving a mathematical proof problem. Assume only the stated starting checkpoint result and establish the stated target result. Check its hypotheses when applying it. Prove any other intermediate claims you need. End with a clear proof, or identify any gap you cannot close.

## Problem

H2: Let \(\mathbb T=\mathbb R/(2\pi\mathbb Z)\), and let \(u\in C(\mathbb R;H^1(\mathbb T;\mathbb R))\) satisfy
\[
(1-\partial_x^2)u_t+\partial_x(u+u^2/2)=0
\]
in distributions. Suppose \(u=C\) on \(\omega\times(0,T_0)\), where \(C\in\mathbb R\), \(T_0>0\), and \(\omega\subset\mathbb T\) is nonempty and open.
Put \(z=u-C\), \(\beta=1+C\), and
\[
w=(1-\partial_x^2)^{-1}(\beta z+z^2/2),
\]
where the inverse is the periodic Fourier multiplier \((1+n^2)^{-1}\), \(n\in\mathbb Z\).

Fix \(x_0\in\omega\) and use the interval coordinate \(x\in[-\pi,\pi]\) on \(\mathbb T\) with both endpoints representing \(x_0\). For \(\alpha>2\), define
\[
H_\beta(s)=\beta s^2/2+s^3/6,\quad
N_\alpha(t)=\int_{-\pi}^{\pi}e^{\alpha x}H_\beta(z(x,t))\,dx,\quad
W_\alpha(x,t)=e^{\alpha x/2}w(x,t).
\]
Then \(N_\alpha\in C^1(\mathbb R)\), and for every \(t\in\mathbb R\),
\[
N_\alpha'(t)=-\frac\alpha2\int_{-\pi}^{\pi}
\left[(\partial_xW_\alpha)^2+(\alpha^2/4-1)W_\alpha^2\right]dx.
\]

Prove H3: Let \(\mathbb T=\mathbb R/(2\pi\mathbb Z)\), and let \(u\in C(\mathbb R;H^1(\mathbb T;\mathbb R))\) satisfy
\[
(1-\partial_x^2)u_t+\partial_x(u+u^2/2)=0
\]
in distributions. Suppose \(u=C\) on \(\omega\times(0,T_0)\), where \(C\in\mathbb R\), \(T_0>0\), and \(\omega\subset\mathbb T\) is nonempty and open.
Put \(z=u-C\), \(\beta=1+C\), and
\[
w=(1-\partial_x^2)^{-1}(\beta z+z^2/2),
\]
where the inverse is the periodic Fourier multiplier \((1+n^2)^{-1}\), \(n\in\mathbb Z\).

The integral over the entire time axis is finite:
\[
\int_{\mathbb R}\|w(t)\|_{L^2(\mathbb T)}^2\,dt<\infty.
\]

Write the full proof to answer.md in the workspace root. Write the statements and hypotheses of any external results you cite to cited_results.md; if none are used, write 'No external results cited.' Identify any gap you cannot close.