# Periodic BBM NE-1-H0-to-H1

## Setting

You are solving a mathematical proof problem. Establish the stated target result. Check the hypotheses, logical steps, and quantifiers in your argument as a careful referee would. End with a clear proof, or identify any gap you cannot close.

## Problem

Prove H1: 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 \((1+n^2)^{-1}\), \(n\in\mathbb Z\).

For every \(x\in\omega\) and \(t\in\mathbb R\),
\[
z(x,t)=w(x,t)=\partial_xw(x,t)=0.
\]

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.