2.3. Well-posedness on Sobolev space🔗

We have the equality

‖u(t, x)‖_{H^s} = ‖u₀‖_{H^s}

as a consequence of the fact that the solution operator is a Fourier multiplier. By density, we can thus extend the solution operator to a map H^s → C(ℝ, H^2).