29E (Partial Differential Equations)

For smooth periodic initial data, Fourier expansion reduces the equation to

\dot{u} (m, t) = - (1 + m^{2}) u (m, t)

. Therefore define the Fourier multiplier solution operators by

S (t) u_{0} (m) = e^{- (1 + m^{2}) t} u_{0} (m) .

(1)

Their Sobolev norm satisfies

∥ S (t) u_{0} ∥_{H^{s}}^{2} \leq e^{- 2 t} ∥ u_{0} ∥_{H^{s}}^{2} \leq ∥ u_{0} ∥_{H^{s}}^{2}

, so they extend to contractive linear operators on

H_{per}^{s}

. Multiplication of the Fourier factors gives

S (t + s) = S (t) S (s)

and

S (0) = I

. Finally

∥ S (t) u - u ∥_{H^{s}}^{2} = \sum_{m} (1 + m^{2})^{s} ∣ e^{- (1 + m^{2}) t} - 1 ∣^{2} ∣ u (m) ∣^{2} \to 0

(2)

by 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.

Here the initial functions are extended periodically to the real line. Direct differentiation verifies both initial conditions and the equation, and arbitrary repeated differentiation shows the solution is smooth and periodic.

This characteristic derivation also proves finite propagation speed from first principles: the value at

(x, t)

depends only on the two endpoint displacements and initial velocity data in

[x - t, x + t]

. If two data sets agree on that interval, their solutions agree at that point. Equivalently initially supported data have no influence outside the unit-speed characteristic cone, interpreted on the periodic covering line. Multiplication by