= H2 bound for the defocusing cubic wave equation
{c}
{title2=$\|u(t)\|_{\dot H^2}+\|u_t(t)\|_{\dot H^1}\leq CDe^{CEt}$}
For $u_{tt}-\Delta u+u^3=0$ in three spatial dimensions, conserved positive <wave energy> $E$ controls $\|\nabla u\|_2$. The <Sobolev inequality> gives $\|u^2\nabla u\|_2\lesssim\|\nabla u\|_2^2\|D^2u\|_2\lesssim E\|D^2u\|_2$. Differentiated <wave energy estimates> and the <Gronwall inequality> then give $\|u(t)\|_{\dot H^2}+\|u_t(t)\|_{\dot H^1}\leq C D e^{C E t}$, where $D$ bounds the corresponding initial <Sobolev norms>.
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