Initial datum for an evolution equation

Positive definiteness: ∥ x ∥ = 0 \|x\|=0 ∥ x ∥ = 0 iff x = 0 x=0 x = 0 .
Absolute homogeneity: ∥ α x ∥ = ∣ α ∣ ∥ x ∥ \|\alpha x\|=|\alpha|\,\|x\| ∥ α x ∥ = ∣ α ∣ ∥ x ∥ .
Triangle inequality: ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ \|x+y\|\le \|x\|+\|y\| ∥ x + y ∥ ≤ ∥ x ∥ + ∥ y ∥ .

An initial datum for an evolution equation is a prescribed state $u_0$ at a starting time $t_0$ , expressed as $u(t_0,\cdot)=u_0$ . If the solution is initially defined only for $t>t_0$ , this condition is interpreted through a specified limit, for example

\lim_{t\downarrow t_0}\|u(t,\cdot)-u_0\|_X=0

in a chosen normed space $X$ . A pointwise initial condition is another possible convention.

Constraints and sources

The datum must satisfy any required compatibility conditions, such as divergence freedom for an incompressible initial velocity. Initial data and forcing have different roles: data specify the starting state, whereas forcing is a prescribed source acting during the evolution. Zero initial data do not imply a zero solution when forcing is present.