 Solution

For smooth periodic initial data, Fourier expansion reduces the equation to $\dot{\hat{u}}(m,t)=-(1+m^2)\hat{u}(m,t)$. Therefore define the Fourier multiplier solution operators byTheir Sobolev norm satisfies $\Vert S(t)u_0{\Vert }_{H^s}^2\le e^{-2t}\Vert u_0{\Vert }_{H^s}^2\le \Vert u_0{\Vert }_{H^s}^2$, so they extend to contractive linear operators on $H_{\mathrm{per}}^s$. Multiplication of the Fourier factors gives $S(t+s)=S(t)S(s)$ and $S(0)=I$. Finallyby dominated convergence theorem, dominated by four times the summable norm terms. This proves a strongly continuous contraction semigroup for every requested $s$; the semigroup property extends continuity from zero to other times. For smooth data, termwise derivatives converge and recover the original Cauchy problem for a partial differential equation uniquely.