Knowlpedia

Theorem

Supremum growth obstructs continuous Sobolev continuation

An embedding into bounded functions turns unbounded supremum norm into failure of continuous Hs extension.

Statement

Suppose u∈C([0,T);Hs(Rn))u\in C([0,T);H^s(\mathbb R^n)), T<∞T<\infty, and s>n/2s>n/2. If

lim sup⁡t↑T∥u(t)∥∞=∞,\limsup_{t\uparrow T}\|u(t)\|_\infty=\infty,

then uu cannot extend continuously in HsH^s to an interval containing [0,T][0,T]. This is a direct consequence of the .

Proof and scope

A continuous map from the compact time interval [0,T][0,T] into HsH^s has bounded HsH^s norm. The embedding then gives a uniform L∞L^\infty bound, contradicting the displayed growth. This establishes an obstruction to extension. It does not prove that every finite maximal lifespan forces supremum growth; that converse needs an actual continuation theorem for the equation and solution class.