# Periodic BBM — 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 \(t\in\mathbb R\),
\[
N_\alpha'(t)=-\frac\alpha2\int_{-\pi}^{\pi}
\left[(\partial_xW_\alpha)^2+(\alpha^2/4-1)W_\alpha^2\right]dx.
\]