Finite time blowup for an averaged three-dimensional Navier-Stokes equation

Sep 06, 2026 03:25 AM - 1 week ago 6

I’ve conscionable uploaded to the arXiv the insubstantial “Finite clip blowup for an averaged three-dimensional Navier-Stokes equation“, submitted to J. Amer. Math. Soc.. The main intent of this insubstantial is to formalise the “supercriticality barrier” for the world regularity problem for the Navier-Stokes equation, which astir speaking asserts that it is not imaginable to found world regularity by immoderate “abstract” attack which only uses precocious bound usability abstraction estimates connected the nonlinear portion of the equation, mixed pinch the power identity. This is done by constructing a modification of the Navier-Stokes equations pinch a nonlinearity that obeys fundamentally each of the usability abstraction estimates that the existent Navier-Stokes nonlinearity does, and which besides obeys the power identity, but for which 1 tin conception solutions that rustle up successful finite time. Results of this type had been antecedently established by Montgomery-Smith, Gallagher-Paicu, and Li-Sinai for variants of the Navier-Stokes equation without the power identity, and by Katz-Pavlovic and by Cheskidov for dyadic analogues of the Navier-Stokes equations successful 5 and higher dimensions that obeyed the power personality (see besides the activity of Plechac and Sverak and of Hou and Lei that besides propose blowup for different Navier-Stokes type models obeying the power personality successful 5 and higher dimensions), but to my knowledge this is the first blowup consequence for a Navier-Stokes type equation successful 3 dimensions that besides obeys the power identity. Intriguingly, the method of impervious successful truth hints astatine a imaginable way to establishing blowup for the existent Navier-Stokes equations, which I americium now progressively inclined to judge is the lawsuit (albeit for a very mini group of first data).

To authorities the results much precisely, callback that the Navier-Stokes equations tin beryllium written successful the shape

$\displaystyle \partial_t u + (u \cdot \nabla) u = \nu \Delta u + \nabla p$

for a divergence-free velocity section ${u}$ and a unit section ${p}$ , wherever ${\nu>0}$ is the viscosity, which we will normalise to beryllium one. We will activity successful the non-periodic setting, truthful the spatial domain is ${{\bf R}^3}$ , and for liking of exposition I will not talk matters of regularity aliases decay of the solution (but we will ever beryllium moving pinch beardown notions of solution present alternatively than anemic ones). Applying the Leray projection ${P}$ to divergence-free vector fields to this equation, we tin destruct the pressure, and get an improvement equation

$\displaystyle \partial_t u = \Delta u + B(u,u) \ \ \ \ \ (1)$

purely for the velocity field, wherever ${B}$ is simply a definite bilinear usability connected divergence-free vector fields (specifically, ${B(u,v) = -\frac{1}{2} P( (u \cdot \nabla) v + (v \cdot \nabla) u)}$ . The world regularity problem for Navier-Stokes is past balanced to the world regularity problem for the improvement equation (1).

An important characteristic of the bilinear usability ${B}$ appearing successful (1) is the cancellation rule

$\displaystyle \langle B(u,u), u \rangle = 0$

(using the ${L^2}$ soul merchandise connected divergence-free vector fields), which leads successful peculiar to the basal power personality

$\displaystyle \frac{1}{2} \int_{{\bf R}^3} |u(T,x)|^2\ dx + \int_0^T \int_{{\bf R}^3} |\nabla u(t,x)|^2\ dx dt = \frac{1}{2} \int_{{\bf R}^3} |u(0,x)|^2\ dx.$

This personality (and its consequences) supply fundamentally the only known a priori bound connected solutions to the Navier-Stokes equations from ample information and arbitrary times. Unfortunately, arsenic discussed successful this erstwhile post, the quantities controlled by the power personality are supercritical pinch respect to scaling, which is the basal obstacle that has defeated each attempts to lick the world regularity problem for Navier-Stokes without immoderate further assumptions connected the information aliases solution (e.g. perturbative hypotheses, aliases a priori power connected a captious norm specified arsenic the ${L^\infty_t L^3_x}$ norm).

Our main consequence is past (slightly informally stated) arsenic follows

Theorem 1 There exists an averaged type ${\tilde B}$ of the bilinear usability ${B}$ , of the shape

$= \int_\Omega m_{3,\omega}(D) Rot_{3,\omega}$

$\displaystyle B( m_{1,\omega}(D) Rot_{1,\omega} u, m_{2,\omega}(D) Rot_{2,\omega} v )\ d\mu(\omega)$

for immoderate probability abstraction ${(\Omega, \mu)}$ , immoderate spatial rotation operators ${Rot_{i,\omega}}$ for ${i=1,2,3}$ , and immoderate Fourier multipliers ${m_{i,\omega}}$ of bid ${0}$ , for which 1 still has the cancellation rule

$\displaystyle \langle \tilde B(u,u), u \rangle = 0$

and for which the averaged Navier-Stokes equation

$\displaystyle \partial_t u = \Delta u + \tilde B(u,u) \ \ \ \ \ (2)$

admits solutions that rustle up successful finite time.

(There are immoderate integrability conditions connected the Fourier multipliers ${m_{i,\omega}}$ required successful the supra theorem successful bid for the conclusion to beryllium non-trivial, but I americium omitting them present for liking of exposition.)

Because spatial rotations and Fourier multipliers of bid ${0}$ are bounded connected astir usability spaces, ${\tilde B}$ automatically obeys almost each of the precocious bound estimates that ${B}$ does. Thus, this theorem blocks immoderate effort to beryllium world regularity for the existent Navier-Stokes equations which relies purely connected the power personality and connected precocious bound estimates for the nonlinearity; 1 must usage immoderate further building of the nonlinear usability ${B}$ which is not shared by an averaged type ${\tilde B}$ . Such further building surely exists – for instance, the Navier-Stokes equation has a vorticity formulation involving only differential operators alternatively than pseudodifferential ones, whereas a wide equation of the shape (2) does not. However, “abstract” approaches to world regularity mostly do not utilization specified structure, and frankincense cannot beryllium utilized to affirmatively reply the Navier-Stokes problem.

It turns retired that the peculiar averaged bilinear usability ${B}$ that we will usage will beryllium a finite linear operation of local cascade operators, which return the shape

$= \sum_{n \in {\bf Z}} (1+\epsilon_0)^{5n/2} \langle u, \psi_{1,n} \rangle \langle v, \psi_{2,n} \rangle \psi_{3,n}$

wherever ${\epsilon_0>0}$ is simply a mini parameter, ${\psi_1,\psi_2,\psi_3}$ are Schwartz vector fields whose Fourier toggle shape is supported connected an annulus, and $= (1+\epsilon_0)^{3n/2} \psi_i( (1+\epsilon_0)^n x)}$ is an ${L^2}$ -rescaled type of ${\psi_i}$ (basically a “wavelet” of wavelength astir ${(1+\epsilon_0)^{-n}}$ centred astatine the origin). Such operators were fundamentally introduced by Katz and Pavlovic arsenic dyadic models for ${B}$ ; they person the fundamentally the aforesaid scaling spot arsenic ${B}$ (except that 1 tin only standard on powers of ${1+\epsilon_0}$ , alternatively than complete each affirmative reals), and successful truth they tin beryllium expressed arsenic an mean of ${B}$ successful the consciousness of the supra theorem, arsenic tin beryllium shown aft a somewhat tedious magnitude of Fourier-analytic awesome manipulations.

If we see nonlinearities ${\tilde B}$ which are a finite linear operation of section cascade operators, past the equation (2) much aliases little collapses to a strategy of ODE successful definite “wavelet coefficients” of ${u}$ . The precise ODE that shows up depends connected what precise operation of section cascade operators 1 is using. Katz and Pavlovic fundamentally considered a azygous cascade usability together pinch its “adjoint” (needed to sphere the power identity), and arrived (more aliases less) astatine the strategy of ODE

$\displaystyle \partial_t X_n = - (1+\epsilon_0)^{2n} X_n + (1+\epsilon_0)^{\frac{5}{2}(n-1)} X_{n-1}^2 - (1+\epsilon_0)^{\frac{5}{2} n} X_n X_{n+1} \ \ \ \ \ (3)$

wherever $[0,T] \rightarrow {\bf R}}$ are scalar fields for each integer ${n}$ . (Actually, Katz-Pavlovic worked pinch a method version of this peculiar equation, but the differences are not truthful important for this existent discussion.) Note that the quadratic position connected the RHS transportation a higher exponent of ${1+\epsilon_0}$ than the dissipation term; this reflects the supercritical quality of this improvement (the power ${\frac{1}{2} \sum_n X_n^2}$ is monotone decreasing successful this flow, truthful the earthy size of ${X_n}$ fixed the power connected the power is ${O(1)}$ ). There is simply a flimsy method rumor pinch the dissipation if 1 wishes to embed (3) into an equation of the shape (2), but it is insignificant and I will not talk it further here.

In principle, if the ${X_n}$ mode has size comparable to ${1}$ astatine immoderate clip ${t_n}$ , past power should travel from ${X_n}$ to ${X_{n+1}}$ astatine a complaint comparable to ${(1+\epsilon_0)^{\frac{5}{2} n}}$ , truthful that by clip ${t_{n+1} \approx t_n + (1+\epsilon_0)^{-\frac{5}{2} n}}$ aliases so, astir of the power of ${X_n}$ should person drained into the ${X_{n+1}}$ mode (with hardly immoderate power dissipated). Since the bid ${\sum_{n \geq 1} (1+\epsilon_0)^{-\frac{5}{2} n}}$ is summable, this suggests finite clip blowup for this ODE arsenic the power races ever much quickly to higher and higher modes. Such a script was so established by Katz and Pavlovic (and refined by Cheskidov) if the dissipation spot ${(1+\epsilon)^{2n}}$ was weakened somewhat (the exponent ${2}$ has to beryllium lowered to beryllium little than ${\frac{5}{3}}$ ). As mentioned above, this is capable to springiness a type of Theorem 1 successful 5 and higher dimensions.

On the different hand, it was shown a fewer years agone by Barbato, Morandin, and Romito that (3) successful truth admits world soft solutions (at slightest successful the dyadic lawsuit ${\epsilon_0=1}$ , and assuming non-negative first data). Roughly speaking, the problem is that arsenic power is being transferred from ${X_n}$ to ${X_{n+1}}$ , power is besides simultaneously being transferred from ${X_{n+1}}$ to ${X_{n+2}}$ , and arsenic specified the solution races disconnected to higher modes a spot excessively prematurely, without absorbing each of the power from little modes. This weakens the spot of the blowup to the constituent wherever the moderately beardown dissipation successful (3) is capable to termination the precocious wave cascade earlier a existent singularity occurs. Because of this, the original Katz-Pavlovic exemplary cannot rather beryllium utilized to found Theorem 1 successful 3 dimensions. (Actually, the original Katz-Pavlovic exemplary had immoderate further dispersive features which allowed for different impervious of world soft solutions, which is an unpublished consequence of Nazarov.)

To get astir this, I had to “engineer” an ODE strategy pinch akin features to (3) (namely, a quadratic nonlinearity, a monotone full energy, and the indicated exponents of ${(1+\epsilon_0)}$ for some the dissipation word and the quadratic terms), but for which the cascade of power from standard ${n}$ to standard ${n+1}$ was not interrupted by the cascade of power from standard ${n+1}$ to standard ${n+2}$ . To do this, I needed to insert a delay successful the cascade process (so that aft power was dumped into standard ${n}$ , it would return immoderate clip earlier the power would commencement to transportation to standard ${n+1}$ ), but the process besides needed to beryllium abrupt (once the process of power transportation started, it needed to reason very quickly, earlier the delayed transportation for the adjacent standard kicked in). It turned retired that 1 could build a “quadratic circuit” retired of immoderate basal “quadratic gates” (analogous to really an electrical circuit could beryllium built retired of basal gates specified arsenic amplifiers aliases resistors) that achieved this task, starring to an ODE strategy fundamentally of the shape

$\displaystyle \partial_t X_{1,n} = - (1+\epsilon_0)^{2n} X_{1,n}$

$\displaystyle + (1+\epsilon_0)^{5n/2} (- \epsilon^{-2} X_{3,n} X_{4,n} - \epsilon X_{1,n} X_{2,n} - \epsilon^2 \exp(-K^{10}) X_{1,n} X_{3,n}$

$\displaystyle + K X_{4,n-1}^2)$

$\displaystyle \partial_t X_{2,n} = - (1+\epsilon_0)^{2n} X_{2,n} + (1+\epsilon_0)^{5n/2} (\epsilon X_{1,n}^2 - \epsilon^{-1} K^{10} X_{3,n}^2)$

$\displaystyle \partial_t X_{3,n} = - (1+\epsilon_0)^{2n} X_{3,n} + (1+\epsilon_0)^{5n/2} (\epsilon^2 \exp(-K^{10}) X_{1,n}^2$

$\displaystyle + \epsilon^{-1} K^{10} X_{2,n} X_{3,n} )$

$\displaystyle \partial_t X_{4,n} =- (1+\epsilon_0)^{2n} X_{4,n} + (1+\epsilon_0)^{5n/2} (\epsilon^{-2} X_{3,n} X_{1,n}$

$\displaystyle - (1+\epsilon_0)^{5/2} K X_{4,n} X_{1,n+1})$

wherever ${K \geq 1}$ is simply a suitable ample parameter and ${\epsilon > 0}$ is simply a suitable mini parameter (much smaller than ${1/K}$ ). To visualise the dynamics of specified a system, I recovered it useful to picture this strategy graphically by a “circuit diagram” that is analogous (but not identical) to the circuit diagrams arising successful electrical engineering:

circuit-1

The coupling constants present scope wide from being very ample to very small; successful practice, this makes the ${X_{2,n}}$ and ${X_{3,n}}$ modes sorb very small energy, but exert a sizeable power connected the remaining modes. If a batch of power is abruptly dumped into ${X_{1,n}}$ , what happens adjacent is astir arsenic follows: for a mean play of time, thing overmuch happens different than a trickle of power into ${X_{2,n}}$ , which successful move causes a accelerated exponential maturation of ${X_{3,n}}$ (from a very debased base). After this delay, ${X_{3,n}}$ abruptly crosses a definite threshold, astatine which constituent it causes ${X_{1,n}}$ and ${X_{4,n}}$ to speech power backmost and distant pinch utmost speed. The power from ${X_{4,n}}$ past quickly drains into ${X_{1,n+1}}$ , and the process originates again (with a flimsy nonaccomplishment successful power owed to the dissipation). If 1 land the full power $= \frac{1}{2} ( X_{1,n}^2 + X_{2,n}^2 + X_{3,n}^2 + X_{4,n}^2 )}$ arsenic a usability of time, it looks schematically for illustration this:

energy-blowup

As successful the erstwhile heuristic discussion, the clip betwixt cascades from 1 wave standard to the adjacent decay exponentially, starring to blowup astatine immoderate finite clip ${T}$ . (One could picture the dynamics present arsenic being akin to the celebrated “lighting the beacons” segment successful the Lord of the Rings movies, isolated from that (a) arsenic each beacon gets ignited, the erstwhile 1 is extinguished, arsenic per the power identity; (b) the clip betwixt beacon lightings alteration exponentially; and (c) location is nary soundtrack.)

There is simply a existent (but remote) anticipation that this benignant of building tin beryllium adapted to the existent Navier-Stokes equations. The basal blowup system successful the averaged equation is that of a von Neumann machine, aliases much precisely a conception (built wrong the laws of the inviscid improvement ${\partial_t u = \tilde B(u,u)}$ ) that, aft immoderate clip delay, manages to abruptly create a replica of itself astatine a finer standard (and to mostly erase its original instantiation successful the process). In principle, specified a von Neumann instrumentality could besides beryllium built retired of the laws of the inviscid shape of the Navier-Stokes equations (i.e. the Euler equations). In beingness terms, 1 would person to build the instrumentality purely retired of an perfect fluid (i.e. an inviscid incompressible fluid). If 1 could someway create capable “logic gates” retired of perfect fluid, 1 could presumably build a benignant of “fluid computer”, astatine which constituent the task of building a von Neumann instrumentality appears to trim to a package engineering workout alternatively than a PDE problem (providing that the gates are suitably unchangeable pinch respect to perturbations, but (as pinch existent computers) this tin presumably beryllium done by converting the analog signals of fluid mechanics into a much error-resistant integer form). The cardinal point missing successful this programme (in some senses of the word) to found blowup for Navier-Stokes is to conception the logic gates wrong the laws of perfect fluids. (Compare pinch the business for cellular automata specified arsenic Conway’s “Game of Life“, successful which Turing complete computers, cosmopolitan constructors, and replicators person each been built wrong the laws of that game.)