Blowup
As evening falls, the photographer goes back to the park and finds the body of the man, but he has not brought his camera and is scared off by the sound of a twig breaking, as if being stepped on. Thomas returns to find his studio ransacked. All the negatives and prints are gone except for one very grainy blowup of what is possibly the body.
blowup
The package blowup only defines the user-level macro \blowUp, which can be used to upscale or downscale all pages of a document. It is similar to the TeX primitive \magnification but more accurate and user-friendly.
CHARLOTTE, N.C. -- In a season in which the only thing the Denver Broncos can consistently do, it seems, is raise their own frustration level with one loss after another, a brief sideline blowup between defensive tackle Mike Purcell and quarterback Russell Wilson was caught on camera Sunday.
I can show that there is an open cover of the blowup by schemes of the form $\textSpec C$, where $B \subset C \subset B_g$ for some integrally closed domain $B$ and some $g \in B$, but I don't see why this would imply that $C$ is integrally closed. Intuitively, it seems reasonable that a blowup would be at least as "nice" as the original scheme, but that intuition may have more to do with how blowups are generally used than what they are capable of.
In principle, if the mode has size comparable to at some time , then energy should flow from to at a rate comparable to , so that by time or so, most of the energy of should have drained into the mode (with hardly any energy dissipated). Since the series is summable, this suggests finite time blowup for this ODE as the energy races ever more quickly to higher and higher modes. Such a scenario was indeed established by Katz and Pavlovic (and refined by Cheskidov) if the dissipation strength was weakened somewhat (the exponent has to be lowered to be less than ). As mentioned above, this is enough to give a version of Theorem 1 in five and higher dimensions.
On the other hand, it was shown a few years ago by Barbato, Morandin, and Romito that (3) in fact admits global smooth solutions (at least in the dyadic case , and assuming non-negative initial data). Roughly speaking, the problem is that as energy is being transferred from to , energy is also simultaneously being transferred from to , and as such the solution races off to higher modes a bit too prematurely, without absorbing all of the energy from lower modes. This weakens the strength of the blowup to the point where the moderately strong dissipation in (3) is enough to kill the high frequency cascade before a true singularity occurs. Because of this, the original Katz-Pavlovic model cannot quite be used to establish Theorem 1 in three dimensions. (Actually, the original Katz-Pavlovic model had some additional dispersive features which allowed for another proof of global smooth solutions, which is an unpublished result of Nazarov.)
I consider this unlikely: blowup or lack thereof is going to be decided by the dynamics of the high-frequency component of the solution, whereas questions of energy decay are mostly decided by the dynamics of the low-frequency components (as the result of Schonbek indicates). Note that if a solution has unusually rapid energy decay, then one should be able to modify that solution to one which does not exhibit such decay simply by placing an additional low-frequency component to the initial velocity that is supported sufficiently far away from the rest of the initial data that it does not significantly affect the dynamics of that data (other than by an approximately linear superposition of the original solution with the evolution of the low-frequency component). This suggests that the singularity behaviour of the solution is more or less decoupled from the energy decay behaviour. (Note also that the construction in my paper suggests that blowup can be achieved using arbitrarily small amounts of energy.)
The energy identity controls the total energy of the fluid, integrated over all of space, but it does not control the pointwise energy density of the fluid (or equivalently, the square of the speed of the fluid), because the energy could be concentrated in an arbitrary small ball of space. In particular, in the blowup scenario envisaged here, the energy remains bounded but is being concentrated into smaller and smaller balls, and in finite time one arrives at a singularity in which a finite nonzero amount of energy is supposed to concentrate into a single point, which cannot occur for a smooth solution to Navier-Stokes.
Hello Professor Tao: Is it necessarily so that the energy is finite at a point in the blowup scenario? Assuming that the energy density is infinite in the blowup scenario, I would imagine that one would need to take limits to calculate the actual energy at that point. For example, if r is the radius of a ball surrounding the blowup point, then the mass of the ball is proportional to r cubed. If the velocity blows up proportional to 1/r^n then the energy blows up 1/r^2n. Since the energy at the blowup point is the product of the energy density and the mass of the ball, only in the case of n = 1.5 will the energy be finite at the point. For n1.5 the energy at the point is infinite.
There is a paper (arXiv:1104.3615 or CommMathPhys 312(3)) whose initial data is close to the type you mentioned. It was claimed that the critical case L3-velocity norm blows up in finite time from initial smooth data with compact support (i.e. finite initial energy in R^3). Apart from lack of convincing apriori bounds and a few technical glitches, one assumption made in that paper was velocity field (and pressure) might be split into two parts: one part is linear and governed by the Stokes system, and the rest by the NSEs. Moreover, the separation assumption was considered to hold independent of the size of the initial data, and of time interval ahead possible singularity. No justification and qualification were given. By the well-known NS regularity for small data, the assumption cannot be valid for arbitrary initial data. In general, the claimed out-of-bound condition (Theorem 1.1) at most implies that the assumed flowfield breaks down in finite time; the blowup does not necessarily represent a genuine singularity condition for the NSEqs. (Similar arguments apply to paper arXiv:1508.05313.)
In two of the theorems, the data is carefully selected with an upper bound of the blowup time of 1, but the result does not say anything about generic data. For the first blowup result, which is more stable, if the initial data is supported in a narrow cylinder of width around the origin with a total circulation of , then the blowup time will be bounded by , which is consistent with dimensional analysis and also the Beale-Kato-Majda criterion (vorticity has units of inverse time, while circulation has units of length squared per unit time).
This is discussed in detail in my other blog post -global-regularity-for-navier-stokes-is-hard/ . Basically, the answer is no, because the time of existence provided by the local theory can be arbitrarily small even when there is very little energy left to dissipate, and iteration of the local existence theory could thus conceivably lead to a convergent series which is consistent with finite time blowup. Note also that the equation considered in my paper here also has local existence and regularity as well as an energy dissipation inequality, yet still manages to exhibit solutions that blow up in finite time.
I believe for the specific bilinear operator I use for the blowup example there is a suitable endpoint paraproduct estimate which allows one to obtain this sort of regularity result, though for the general averaged bilinear operators in the class I describe in the paper there may not be such an estimate. On the other hand, I do not know if one can extend the Beal-Kato-Majda criterion (assuming only control of the vorticity) to my equation because it does not have a usable vorticity equation.
We study the blowup behavior for the focusing energy-supercritical semilinear wave equation in 3 space dimensions without symmetry assumptions on the data. We prove the stability in \(H^2\times H^1\) of the ODE blowup profile.
If $b : X' \to X$ is the blowup of $X$ in $Z$, then we often denote $\mathcalO_X'(n)$ the twists of the structure sheaf. Note that these are invertible $\mathcalO_X'$-modules and that $\mathcalO_X'(n) = \mathcalO_X'(1)^\otimes n$ because $X'$ is the relative Proj of a quasi-coherent graded $\mathcalO_ X$-algebra which is generated in degree $1$, see Lemma 70.11.11.
Lemma 70.17.2. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcalI \subset \mathcalO_ X$ be a quasi-coherent sheaf of ideals. Let $U = \mathop\mathrmSpec(A)$ be an affine scheme étale over $X$ and let $I \subset A$ be the ideal corresponding to $\mathcalI_ U$. If $X' \to X$ is the blowup of $X$ in $\mathcalI$, then there is a canonical isomorphism
of schemes over $U$, where the right hand side is the homogeneous spectrum of the Rees algebra of $I$ in $A$. Moreover, $U \times _ X X'$ has an affine open covering by spectra of the affine blowup algebras $A[\fracIa]$.
Lemma 70.17.3. Let $S$ be a scheme. Let $X_1 \to X_2$ be a flat morphism of algebraic spaces over $S$. Let $Z_2 \subset X_2$ be a closed subspace. Let $Z_1$ be the inverse image of $Z_2$ in $X_1$. Let $X'_ i$ be the blowup of $Z_ i$ in $X_ i$. Then there exists a cartesian diagram
Lemma 70.17.7. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcalI \subset \mathcalO_ X$ be a quasi-coherent sheaf of ideals. If $X$ is reduced, then the blowup $X'$ of $X$ in $\mathcalI$ is reduced.
Lemma 70.17.8. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \to X$ be the blowup of $X$ in a closed subspace. If $X$ satisfies the equivalent conditions of Morphisms of Spaces, Lemma 66.49.1 then so does $X'$.
Lemma 70.17.9. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $b : X' \to X$ be a blowup of $X$ in a closed subspace. For any effective Cartier divisor $D$ on $X$ the pullback $b^-1D$ is defined (see Definition 70.6.10). 041b061a72