Finite-time blowup in a specified norm

Let $u(t)$ be a solution for $t<T_*<\infty$ . It has finite-time blowup in the norm $X$ if

\limsup_{t\uparrow T_*}\|u(t)\|_X=\infty.

The one-sided lim sup, norm, and finite endpoint are part of the assertion. Some authors reserve “blowup” for the stronger statement that the norm tends to infinity; the convention should be stated.

Different norms measure different losses

Blowup of a derivative norm need not mean blowup of the velocity itself. Conversely, a bounded spatial $L^2$ norm does not bound the $L^\infty$ norm: increasingly narrow peaks may grow in height while retaining bounded $L^2$ mass.

Breakdown and extension

Failure to extend a classical solution is often called breakdown. A continuation theorem can relate breakdown to the divergence of a specified norm or space-time integral, but that implication is additional mathematical information.

Explicit continuation obstructions

An unbounded sequence approaching a finite space-time point rules out a continuous extension there. A Sobolev embedding can separately rule out continuous extension in a specified Sobolev norm.