= Initial incompatibility with a Neumann boundary condition
Initial data whose normal <derivative> is nonzero at a wall cannot satisfy homogeneous <Neumann boundary conditions> continuously all the way to the initial time. For example, $-\operatorname{erf}(ay)$ has wall <derivative> $-2ae^{-a^2}/\sqrt\pi$ at $y=\pm1$, which is nonzero for finite $a>0$. A parabolic <mild solution of an abstract Cauchy problem> can still take those initial data in an integral or square-integrable sense and satisfy the boundary condition for every positive time. An exactly compatible smooth profile is needed for a classical solution on the closed time interval.
Back to article page
