Definition of the generic rate of a blow-up

Question

I am reading a paper from van den Berg, Hulsof and King about blow-up solution for the harmonic map heat flow onto the sphere in a radially symmetric domain, this equation is given by, $$\theta_t = \theta_{rr} + \frac{1}{r} \theta_r - \frac{\sin 2\theta}{2r^2},$$ and the solution writes as $u(re^{i\phi}, t) = (e^{i\phi}\sin \theta(r, t), \cos \theta(r, t)).$ They claim (in equation (1.5)) that they found a generic rate for the blow-up, given by $$\lambda(t) \approx \kappa\frac{T - t}{|\log(T - t)|^2}.$$ My question is, what do they mean by generic ? It is not defined anywhere and I couldn't find any source defining it properly. I feel that it's these kind of words that are use by everyone without a proper definition. Anyone could help me with this ?

Answer 1

"Generic" could mean different things depending on the context. This is with no doubt explained in the paper, although not necessarily in a clear and explicit way. It is possible that you have to deduce what that means from what has been proved in the paper (although it is generally good practice to summarize all the results in the introduction in a comprehensible and self-contained way).

I did not find a Theorem that summarizes the results in the introduction, so I cannot tell for sure. However, in the section "Conclusions" they first mention again the "generic" blowup rate, and then they mention some degenerate cases with some non-zero codimension $k$. So, I suspect that with "generic" they mean "for all initial data lying in a given set, except for some data that belong to some manifold of codimension 1". The degenerate cases have in a sense "zero measure" because they lie in a set of codimension 1, in that case it would make sense to call all the other cases "generic".

In the same spirit, e.g., a "generic" point in the Eucledian plane $\mathbb R^2$ has non-zero coordinates, because points with zero coordinates lie in a manifold of codimension 1 (the union of the two axis).

Edit: the discussion at page 1686 seems to confirm my suspicion. Note that "...generic (codimension 0)...".