The complete velocity step and periodic projection bounds

Lesson 24 constructed the actual stress cutoffs and smooth energy amplitudes. This chapter proves the complete velocity estimates for those amplitudes: the principal energy, both small corrections, the full increment, spatial and time derivatives, and every source bound through three space-time derivatives.

The proof retains the original physical period, every mean, the amplitude prefactor and all terms in each product derivative. It also constructs the periodic projection kernel, proves its complete finite-exponent bounds, and gives the smooth-input supremum estimate needed for the temporal correction.

The original fields and their norms are in lesson 22. The full residual and oscillatory kernel estimates are in lesson 23. SC and AB labels refer to lesson 24; IK labels refer to lesson 23. VC labels below give the complete square-norm estimates; PL labels give the projection proof; VD labels give every derivative estimate.

The human source is Tristan Buckmaster and Vlad Vicol, Nonuniqueness of weak solutions to the Navier–Stokes equation, arXiv:1709.10033v4. Its original author TeX at 1041–1174 is the immediate comparison, with the parameter, field and amplitude definitions read in the preceding chapters. The exposition supplies the full calculations, including the original amplitude factor at the first step. This is author self-checked exposition; no independent review or novelty is claimed.

Five solved exercises compute exact physical Fourier projections, optimize the kernel cancellation and smooth-input bounds, evaluate the actual projected fourth moment of one temporal pair, and retain the complete initial amplitude and its numerical tail. The stress and energy induction remain necessary for the infinite construction.

VC1. All original parameters and coefficient costs

Keep the original \(L>0\), \(V=L^3\), the rational families, their common integer \(N_\Lambda\), and their positive angular separation \(c_\Lambda\). In this proof use the source's allowed \(b\in16\mathbb N\), \(b\geq512\), and its original restrictions \(\beta_{\rm reg}b^2\leq4\), \(\beta_{\rm reg}b\leq1/40\). In particular its explicit choice \(b=512,\beta_{\rm reg}=2^{-16}\) satisfies these restrictions. Retain

The letter \(\beta\) in the last line is the physical intermittent frequency; it is distinct from the original regularity exponent. Since \(a\) is an integer multiple of \(N_\Lambda\) and \(b\in16\mathbb N\), both \(r\) and \(m\) are positive integers for every \(q\geq0\); the exact period and covering conditions of lesson22 hold. Also \(1<\mu<\Lambda^2\).

Take the already constructed cutoffs and amplitudes of lesson24, on its actual interior time interval. For the source volume use its original \(B=100d,c_0\) and AB12. The arbitrary-volume extension EX6–EX8 has its stated changed constants; the following proof is given with the original source constants, hence uses the volume range of SC12. For \(j\geq1\), define

Here \(C_*,P_j,\overline A_j,\sigma_j\) have the complete definitions AB1, AB18–AB23. These new \(K_j\) are local constants of VC2, not the earlier cutoff constants in AB18. Their values are independent of \(a,q\). The actual full ordered derivative bounds are

The first line follows from AB4, AB21 and AB23. For the second, apply the complete ordered product rule. The two terms with an undifferentiated factor contribute \(2K_0K_N\lambda_q^5\ell^{-N}\). Every other term has cost \(K_jK_{N-j}\ell^{-N}\), with its full binomial multiplicity; use \(1\leq\lambda_q^5\) only to give the common displayed bound. This proves VC3 for all indices, including identical ones and amplitudes that vanish. The actual unsummed derivative formula remains AB2.

VC2. The number of actual nonzero indices and interacting pairs

SC11 gives the finite cutoff limit. Since \(\delta\geq\lambda_q^{-2\beta_{\rm reg}b}\), the original stress bound and all original constants give

The first inequality uses \(\sqrt{1+x^2}\leq1+x\) for \(x\geq0\). The source bounds give \(\varepsilon_R+2\beta_{\rm reg}b\leq3/10\). Next \(2Y<4\lambda_q^{11}\); the ceiling is strictly less than its argument plus one. Finally \(\log\lambda_q\leq\lambda_q\), and \(2\leq2\lambda_q\). This proves every step of VC4.

Each index has six positive directions and twelve signed directions in the explicit family. There are at most \(6i_*\) positive pairs. In an ordered nonopposite interaction the two indices differ by at most one: all other amplitude products are identically zero by SC8, including their derivatives. Thus the number of ordered interactions is at most \(3\cdot12^2 i_*=432i_*\). An absent index at an endpoint only decreases this count. Repeated physical directions in nonadjacent indices create no additional contribution because their actual products vanish identically.

VC3. The complete principal energy and its oscillatory error

Let \(I=(i,\zeta)\), and let \(\mathcal J^+\) be the actual positive-pair labels and \(\mathcal O\) the ordered signed nonopposite interactions just specified. Use \(A_I=a_I^2\), \(\phi_I=\eta_I^2-1\), and the complete nonopposite scalar \(Q_O\) of lesson23. Define

These are the original physical integrals and frequencies. The full trace identity EX12 of lesson24 states \(\|w^{(p)}\|_2^2=3R_{\rm av} +2\sum_{I\in\mathcal J^+}\int A_I\phi_I+2\int Q_O\). Every signed pair and both oscillatory mean types remain.

To avoid confusing the inverse-kernel constant with the stress-cutoff constant SC20, write \(C_N^{\rm IK}\) for the actual constant in IK15 and put

Here \(J\) is a full ordered spatial list of length \(N\), \(\xi_J\) is its coordinate product, and the norm is the full tensor Hilbert norm. The cutoff \(\chi\) is the explicitly defined frequency cutoff IK2, distinct from the stress partition. Its support makes this entire integral finite. The last norm is independent of \(I,t\) by the original integer torus covering and translation; the inequality uses \(\|\eta_I\|_2^2=V\), retaining the original volume.

The full gap estimate EX14 of lesson23 and its nonopposite receiver EX15 give the complete bound

For completeness, the centered factor has its actual frequency gap at least \(\beta\). In the nonopposite scalar term the original dot magnitude is exactly \(s_{I,J}^2\), not an unspecified constant. The product of its two intermittent factors has \(L^1\) norm at most \(V\), by their exact \(L^2\) norms and Cauchy–Schwarz. Its scalar has the half ordered sum from OS3; multiplication by the factor two in the energy identity removes only that half. Thus both constants in VC7 are exact consequences of the earlier proved gap estimate. Pure spatial derivative norms are bounded by the complete space-time norms VC3.

VC4. An explicit universal principal-velocity bound

Enlarge the already specified integer base \(a\), retaining all AB12 thresholds, so that also \(a\geq(10N_\Lambda/c_\Lambda)^{16/(3b)}\). Then the full original separation condition is \(\sigma N_\Lambda r=N_\Lambda\Lambda^{-3/16} \leq c_\Lambda/10\). Every actual nonopposite pair therefore has

Indeed the rational-family separation gives \(|\zeta+\zeta'|\geq2c_\Lambda\), and \(\beta r=\varkappa\sigma N_\Lambda r\). The last strict inequality follows from \(\sqrt3<5\).

Use the actual first-order case of VC7, then VC3–VC4, \(s_{I,J}^2\leq1\) and \(\Psi_1\leq2V\). The full resulting estimate is

In the first centered term, the factors are \(2\cdot2\cdot2\cdot6=48\): the energy coefficient, the gap split, the bound for \(\Psi_1/V\), and the six positive directions. The nonopposite term has \(2\cdot432=864\). The power \(26\) is precisely \(1+5+20\), from the index count, actual product amplitude and original derivative scale. Since \(b\geq512\), the two exponents are at most \(-6\) and \(-486\). The final comparison keeps the complete original \(\delta\) before using the proved bounds for its factors.

All constants in \(C_{\rm vel}\) are already specified and independent of \(a,q\). Add the finite threshold \(a\geq C_{\rm vel}^{20/119}\) to the preceding maximum and round upward to a multiple of \(N_\Lambda\). This actually constructs parameters with \(\mathcal E_1\leq\delta\) for every \(q\geq0\). AB10 and AB13 also give \(3R_{\rm av}=3\rho_0I_0+H\leq2\delta+\delta/800\). The actual full principal velocity consequently obeys

Thus the first original velocity estimate holds with the explicit universal choice \(M=4\) for these constructed families and parameters. No hidden dependence on the source energy profile is put into \(M\); the finite lower threshold for \(a\) retains that dependence explicitly. This proves only the principal estimate at this point.

VC5. The full actual curl and temporal square-norm costs

For a scalar or finite tensor field \(g\), define its actual mixed-derivative cell cost

The mixed-cell inequality DC21 of lesson22 applies to these finite tensor fields as well: the fundamental theorem uses their Euclidean norm, and its triangle and integral inequalities hold for that norm. The periodic multiplier can likewise be vector-valued by using its pointwise norm in each cell integral. The multiplier's \(L^2\) factor still has the exact prefactor \(V^{-1/2}\).

The actual fields \(\eta_I,\nabla\eta_I,\eta_I^2\) all have period \(h\) in each coordinate. Their exact norms from IB18 and lesson22's fourth-moment exercise are \(\sqrt V\), \(\sqrt V\beta\sqrt{r(r+1)}\), and \(\sqrt V Q_r\), respectively. Expand the entire gradient in the curl correction and use the actual \(\varkappa^{-1}\), not \(\Lambda^{-1}\). Since the amplitude and \(\eta_I\) are real and \(|B_\zeta|=1\), \(|v\times B_\zeta|\leq|v|\) for every real \(v\). The full original correction bounds are therefore

The first follows by applying DC21 separately to \(\eta_I\nabla a_I\) and \(a_I\nabla\eta_I\). For the second, the full operator \(\mathbb P\mathbb P_0\) is an orthogonal contraction on the actual periodic \(L^2\) space by lesson2. Apply it first to the complete sum defining \(z\), then the triangle inequality and DC21 with \(a_I^2\) and \(\eta_I^2\). The original positive-pair coefficient is \(1/\mu\), and is not halved again. No projection claim at an unrestricted \(L^1\) or \(L^\infty\) endpoint has been used.

There is also a complete finite moment formula for every spatial and time derivative of the squared field. For nonnegative integers \(u,v\), set

Powers with exponent zero have value one, including the zero mode. The actual Fourier coefficient of \(\eta_I^2\) at its frame index \((j,k,l)\) is \(M_r^{-3}(M_r-|j|)(M_r-|k|)(M_r-|l|)\), with its original time phase. The frame map is injective, has physical spatial length \(\beta\sqrt{j^2+k^2+l^2}\), and time frequency \(\beta\mu j\). Parseval, including the original volume, therefore proves \(\|D_x^u\partial_t^v(\eta_I^2)\|_2=T_{u,v}\). The complete ordered product rule supplies both binomial counts in VC13, and the projection commutes with every derivative and contracts each tensor component in the summed \(L^2\) norm. In particular \(T_{0,0}=\sqrt VQ_r\), exactly matching VC12. No cross-family orthogonality has been assumed.

Finally the original mollification difference from lesson24 and the full increment identity give

Every quantity in this finite bound is constructed and has the explicit derivative bounds of lesson24. The next calculation is to sum their actual index costs and prove the complete small correction scale, then the original \(W^{1,p}\), time, stress and iteration estimates.

VC6. Sum every mixed-cell cost at its original scale

All constants in VC2 are retained. Define the following finite polynomials with their complete binomial counts:

There are exactly \(\binom3j\) spatial multiindices in \(\{0,1\}^3\) of order \(j\). Each of their derivatives is a subarray of the corresponding full ordered tensor. VC3, including its original zeroth term, gives

The factor \(\sqrt V\) is the exact conversion of the actual supremum norm to a physical \(L^2\) bound. The first line uses \(K_{j+1}\), because the gradient in its argument is an additional full derivative. No term with a repeated derivative is lost: each \(\partial^\epsilon \nabla a_I\) is bounded by the full derivative tensor of order \(|\epsilon|+1\). The last line uses the entire derivative of the original square, not the square of a derivative. This proves every displayed summand.

Since \(b\geq512\), \(0<\tau_h\leq L\lambda_q^{-12}\leq L\). All polynomial coefficients are nonnegative, so each \(U_j(\tau_h)\leq U_j(L)\). Keeping first the actual argument and then substituting VC4 into VC12 gives

The first count is the actual twelve signed labels per index and the second the six positive labels. For the curl term \(\sqrt{r(r+1)}\leq\sqrt2r\), because the original integer \(r\geq1\), and \(\beta/\varkappa=\sigma N_\Lambda\). For the temporal term, \(M_r=2r+1\leq3r\) and \((2M_r^2+1)/(3M_r)\leq M_r\), hence \(Q_r\leq3^{3/2}r^{3/2}\). Substituting the original values of \(\sigma,r,\mu\) proves all three exponents in VC17. The earlier lines retain the exact square-root and fourth-moment factors before those bounds are used.

VC7. The source correction scale and the complete velocity increment

Let the source's full correction scale and the specified constant be

Divide each of the three original VC17 terms by the full first line of VC18. Their respective factors, before any bound for the regularity exponent, are

The three exponents are at most \(-17879/40\), \(-1839/40\) and \(-359/40\), respectively, using \(b\geq512\) and \(\beta_{\rm reg}b\leq1/40\). Each is therefore at most the last one. The original factor \(\lambda_1^{-3\beta_{\rm reg}/2}\) is at most one and is retained in the stronger bound. This proves the source's full correction estimate at 1048–1049, with an explicit stronger power and a finite constant whose entire definition is in VC2 and VC15–VC18. It does not use the source's unjustified replacement of the initial \(\delta\) by a uniform upper bound of one.

The original comparison with the increment scale and the mollification cost are

For the first line, substitute \(F_q/\sqrt\delta =\lambda_q^{20-b/8}\) in VC19. The second is the full space-time mollification bound already proved in lesson24, with its actual first moment. The two comparisons are exact before using the parameter inequalities.

All constants are finite and independent of \(a,q\). Enlarge the same original integer base, retaining every previous threshold, by requiring \(a\geq(2C_{\rm corr})^{40/2119}\) and \(a\geq(2\sqrt V\mathfrak m_1)^{40/639}\), then round the full finite maximum upward to a multiple of \(N_\Lambda\). VC20 proves that each displayed ratio is at most one half for every \(q\geq0\). With the actual original new velocity \(u_{q+1}=u_\ell+w\), the entire perturbation and increment satisfy

The first uses all three actual parts of \(w\); the second adds the retained original mollification difference. The strict inequalities use \(49/\sqrt{800}<2\), proved in VC10. Thus the principal, correction and full increment square-norm estimates at author TeX1046–1049 and 1165–1166 all have complete receiving proofs. No conclusion about the remaining \(W^{1,p}\), time, stress or infinite-iteration estimates is inferred from these square-norm estimates.

PL1. The actual operator, frequencies and tensor norms

Let \(L>0\), \(V=L^3\), \(\alpha=2\pi/L\), and \(\mathbb T_L^3=\mathbb R^3/(L\mathbb Z)^3\). The distance \(d(x,0)\) is the shortest Euclidean distance of a representative of \(x\) to \(L\mathbb Z^3\). For \(k=\alpha n\), \(n\in\mathbb Z^3\), use

The gradient projection \(Q\), the divergence-free projection \(P\), and the removal of the mean \(P_0\) are

The inverse Laplacian is zero on the constant mode and has multiplier \(-|k|^{-2}\) otherwise. The two factors of \(i\) give the positive symbol in PL1. These definitions are first made on smooth vector fields and then on \(L^2\). Parseval, with its factor \(V\), gives

Indeed, at every nonzero frequency \(m(k)\) and \(I-m(k)\) are orthogonal projections, and the constant modes of these three operators are, respectively, zero, identity and zero.

The same proof holds for fields in \(\mathbb C^3\otimes E\), where \(E\) is any finite-dimensional Hilbert space: the multipliers act on the first factor and as the identity on \(E\). All norms are the full Hilbert norm, integrated in the original spatial variables. The estimates below have constants independent of the dimension of \(E\). This includes the full ordered derivative arrays used in the velocity construction.

PL2. Construct the kernels without changing the torus

Set \(b(s)=e^{-1/s}\) for \(s>0\), and \(b(s)=0\) for \(s\leq0\). Use the actual cutoff and matrix function

The denominator is positive everywhere. The function \(\chi\) is smooth, equals one on \(|\xi|\leq1\), vanishes on \(|\xi|\geq2\), and lies in \([0,1]\). The support of \(\psi\) lies in \(1/2\leq|\xi|\leq2\). Thus \(F\), extended as zero near zero, is a smooth compactly supported matrix. Define the finite constants, with the full matrix and derivative tensor Hilbert–Schmidt norm,

Integrating by parts three times, with no boundary term because the integrand is smooth and compactly supported, proves

For \(j=0,1,\ldots\), let \(R_j=\alpha2^j\), and periodize at the actual period:

The series and its derivatives converge uniformly, since repeated integration by parts gives arbitrary polynomial decay. Substitution in the Fourier coefficient integral gives \(\widehat K_j(k)=V^{-1}F(k/R_j)\). With convolution \((K_j*f)(x)=\int_{\mathbb T_L^3}K_j(x-y)f(y)\,dy\), the multiplier is therefore exactly \(F(k/R_j)\): the factor \(V\) in the convolution rule cancels the displayed \(V^{-1}\). In particular, \(\int K_j=0\).

Let \(Q_J f=\sum_{j=0}^J K_j*f\). For a nonzero lattice frequency, \(|k|\geq\alpha\), so \(\chi(2k/\alpha)=0\). The sum telescopes:

It is zero at \(k=0\). Hence \(\|Q_J\|_{2\to2}\leq1\), and dominated convergence in the original Parseval series proves \(Q_Jf\to Qf\) in \(L^2\). No change of measure or rescaling of \(f\) has been made.

PL3. The full periodic kernel bounds

Write \(c=1-\sqrt3/2>0\). The integer shell \(\{n:|n|_\infty=m\}\) has \(24m^2+2\) points, and \(|n|\geq m\) there. Consequently

For the numerical bound in the first line, compare the tails of the decreasing functions \(x^{-4}\) and \(x^{-6}\) with their integrals on \([1,\infty)\). This gives \(24(1+1/3)+2(1+1/5)=172/5\).

Choose the nearest representative \(x\in[-L/2,L/2]^3\), and put \(d=|x|>0\). For the \(n=0\) terms of PL7, enlarge the \(j\)-sum to all integers. Split at the last \(j\) with \(R_jd\leq1\). The lower sum is at most \(d^{-3-s}/(1-2^{-(3+s)})\); the upper sum, using \((1+(R_jd)^2)^{-3}\leq(R_jd)^{-6}\), is at most \(d^{-3-s}/(1-2^{-(3-s)})\). These are the two terms in \(D_s\).

For \(n\ne0\), \(|x+Ln|\geq L(|n|-\sqrt3/2)\geq cL|n|\). Apply the same high-frequency bound for every \(j\geq0\), and sum \(\sum_{j\geq0}R_j^{s-3}=\alpha^{s-3}/(1-2^{s-3})\). The remaining lattice sum is at most \(c^{-6}L^{-6}Z_6\). Since \(d\leq\sqrt3 L/2\), its full physical factor satisfies

This proves absolute convergence away from zero of \(K(x)=\sum_{j\geq0}K_j(x)\), including its first derivative, and proves

The same bounds hold for the corresponding finite sums. If an \(L^2\) field \(f\) is supported a positive distance from an open set, dominated convergence in PL7–PL11 gives \(Q_Jf(x)\to\int K(x-y)f(y)\,dy\) on that set. The \(L^2\) convergence in PL8 identifies this integral with \(Qf\) almost everywhere there. Exhaustion by sets at positive distance proves this representation at almost every point outside the support. The kernel is being used only away from its singularity.

PL4. Decompose an actual integrable field

Take \(f\in L^1\cap L^2\) with values in \(\mathbb C^3\otimes E\), and a number \(t>0\). If \(t\leq V^{-1}\|f\|_1\), then any level set has measure at most \(V\leq\|f\|_1/t\). It remains to treat \(t>V^{-1}\|f\|_1\).

Subdivide \([0,L)^3\) into its half-open dyadic cubes. Select all maximal cubes \(B\) whose average of \(|f|\) exceeds \(t\). They are disjoint. Their parents have averages at most \(t\), so

We record why the pointwise conclusion outside these cubes is valid. For an integrable scalar function \(h\), disjoint maximal cubes in each finite subdivision show

Passing through increasing finite subdivisions proves the same assertion for all levels. For a continuous periodic function, the dyadic averages converge to its value, by uniform continuity. Continuous periodic functions are dense in \(L^1\): simple functions reduce this to measurable sets; inner and outer regularity give a compact subset and an open superset with arbitrarily small difference in volume; the continuous function \(d(x,\mathbb T_L^3\setminus U)/(d(x,\mathbb T_L^3\setminus U)+d(x,C))\) approximates the indicator of a nonempty compact \(C\subset U\). An empty compact set uses the zero function; if \(U\) is the whole torus, use the constant one. The torus distances in the displayed formula are continuous, and the denominator is positive. Finite linear combinations finish the density proof.

For continuous \(h_0\) approximating \(h\), the limsup of the difference between a dyadic average of \(h\) and \(h(x)\) is at most the dyadic maximal average of \(|h-h_0|\) plus \(|h-h_0|(x)\). The two level-set estimates are each at most \(2\|h-h_0\|_1/\varepsilon\) for threshold \(\varepsilon/2\). Letting the approximation error tend to zero proves dyadic differentiation almost everywhere. Apply it to \(|f|\). Outside the selected cubes, \(|f|\leq t\) almost everywhere.

Set \(f_B=|B|^{-1}\int_B f\), put \(g=f\) outside their union and \(g=f_B\) on each \(B\), and put \(b_B=(f-f_B)\mathbf1_B\), \(b=f-g\). Jensen's inequality and disjointness give

For the last assertion, \(g\in L^2\) by the preceding bound, so \(b\in L^2\). The summands have disjoint supports and agree with \(b\) there; the squared norm of a tail tends to zero by integrability of \(|b|^2\).

PL5. Cancellation gives the weak estimate

If \(B\) has side length \(\ell_B\) and center \(c_B\), enlarge it to the geodesic ball \(B^*=\{x:d(x,c_B)<2\sqrt3\ell_B\}\). The image of an ordinary Euclidean ball covers \(B^*\), hence

This remains valid when the ball wraps around or covers the torus. For \(x\notin B^*\), write \(r=d(x,c_B)>2\sqrt3\ell_B\), ignoring the boundary of this ball, which has measure zero. For \(y\in B\), the straight segment from \(c_B\) to \(y\) in the original cube has length at most \(\sqrt3\ell_B/2<r/4\). Along it the kernel argument has distance at least \(3r/4\) from zero. PL11 and the zero integral in PL13 give

The integral representation follows from PL11, since these points are separated from \(B\). In nearest representatives centered at \(c_B\), the integral of \(r^{-4}\) over \(r>R\) on the torus is at most the Euclidean integral \(4\pi\int_R^\infty r^{-2}\,dr=4\pi/R\). Use \(R=2\sqrt3\ell_B\) and \(|y-c_B|/R\leq1/4\). With \(B_*=\pi H_1(4/3)^4\), this proves \(\int_{\mathbb T_L^3\setminus B^*}|Qb_B|\leq B_*\|b_B\|_1\).

For finite sums this estimate can be summed outside the union of all \(B^*\). The \(L^2\) convergence in PL13 and the contraction PL3 give an almost everywhere convergent subsequence of their projections. Fatou's lemma and PL13 then give \(\int_{(\cup B^*)^c}|Qb|\leq2B_*\|f\|_1\). On that complement, if \(|Qf|>t\), then either \(|Qg|>t/2\) or \(|Qb|>t/2\). The first set has measure at most \(4t^{-2}\|g\|_2^2\leq32\|f\|_1/t\); the second at most \(4B_*\|f\|_1/t\). Including the enlarged cubes proves the actual weak estimate

This constant exceeds one and therefore includes the earlier case \(t\leq V^{-1}\|f\|_1\). The reasoning used only Hilbert norms and matrix operator norms bounded by the three-dimensional Hilbert–Schmidt norms in PL5. It is uniform in \(E\).

PL6. Prove the full range of finite exponents

Let \(1<p<2\) and first take \(f\in L^p\cap L^2\). For each \(t>0\) split \(f\) into its parts on \(|f|>\eta t\) and \(|f|\leq\eta t\), where \(\eta>0\). The high part is in \(L^1\cap L^2\); the low part is in \(L^2\). PL16 for the first and PL3 for the second yield

Multiply by \(pt^{p-1}\), integrate in \(t\), and interchange the nonnegative integrals. The first inner integral runs from zero to \(|f(x)|/\eta\), and the second from that value to infinity. They give

Both are finite for precisely the stated range. The derivative of the bracket is \(2p\eta^{-p}(-A_{\mathrm w}+2\eta)\), so its unique minimum is at \(\eta=A_{\mathrm w}/2\). Define

For \(p>2\), first take smooth \(f\), so \(Qf\) is smooth by the rapid decay of its Fourier coefficients. In the ordinary physical integral pairing take \(g=|Qf|^{p-2}Qf\). This bounded field is in \(L^2\) and \(L^{p'}\), where \(p'=p/(p-1)<2\). The symbol in PL1 is self-adjoint, so Parseval and the already proved estimate give

The result is immediate if the last norm is zero; otherwise division proves PL17. Smooth periodic fields are dense in every finite \(L^p\): the continuous approximation argument in PL4 applies to \(|f|^p\) after truncation, and convolution of a continuous periodic field with a smooth nonnegative approximate identity converges uniformly. Thus PL17 extends \(Q\) uniquely to \(L^p\). Fourier coefficients are continuous functionals on \(L^p\) on this finite-volume torus, so the extension retains exactly PL1, including the zero mode, and agrees with the \(L^2\) definition.

Jensen gives \(\|\overline f\|_p\leq\|f\|_p\). Since \(Q\overline f=0\) and \(\overline{Qf}=0\), the exact maps and bounds are

At \(p=2\) retain the sharper constants one in PL3. On a zero-mean input the last constant improves to \(1+C_p\). The Fourier symbols also prove commutation with each spatial derivative and each time derivative of a smooth time-dependent field. Apply the dimension-independent proof to the entire ordered derivative array, not just to its individual entries. These assertions extend to distributional derivatives whenever both sides belong to the indicated \(L^p\) space.

PL7. A smooth-input supremum estimate with every frequency retained

The elementary radial integrals are

For the first, substitute \(r=\tan\theta\) in \(4\pi\int_0^\infty r^2(1+r^2)^{-3}\,dr\), obtaining \(4\pi\int_0^{\pi/2}\sin^2\theta\cos^2\theta\,d\theta=\pi^2/4\). For the second put \(u=r^2\), obtaining \(2\pi\int_0^\infty u(1+u)^{-3}\,du=\pi\). Periodization in PL7, followed by substitution on the entire original lattice tiling, now gives for a \(C^1\) periodic field

The second estimate uses \(\int K_j=0\): subtract \(f(x)\) inside the integral. The shortest torus distance between \(x-y\) and \(x\) is at most \(|y+Ln|\) in each term of PL7. The mean value bound along a shortest segment and PL19 then give exactly the second constant, with no missing power of \(L\).

The high-frequency series therefore converges uniformly for \(C^1\) inputs. Its sum agrees with \(Qf\) by PL8. For every integer \(J\geq0\), sum the first estimate over \(0\leq j\leq J\) and the second over \(j>J\). The exact latter sum is \(\sum_{j>J}R_j^{-1}=R_J^{-1}\). Writing \(C_{\mathrm k}=\pi^2A_0/4\), we obtain

The complete mean term in PL18 accounts for the two in the second line. These are estimates for the stated smooth inputs; no unrestricted \(L^\infty\) projection estimate is asserted. The same formulas apply to each full ordered mixed space-time derivative array by using its finite Hilbert space as \(E\). Thus only one further spatial derivative, with its displayed reciprocal physical frequency, is required.

The receiving velocity estimates retain \(\Pi_p=2+C_p\) for finite \(p\), the sharper orthogonal constant for \(p=2\), and the full PL21 expression for smooth supremum norms. These proved maps can now be applied to the actual \(a_I^2\eta_I^2\xi_I\) in the temporal correction.

VD1. Full physical scales and derivative constants

Write \(X=\lambda_q\), retaining the original variables through

The amplitude prefactor \(\delta=\lambda_1^{3\beta_{\rm reg}}X^{-2\beta_{\rm reg}b}\) is retained in VC. Here use precisely its already proved consequences VC2–VC4:

All \(K_j,B_j,C_I\) have the explicit definitions in VC2–VC4. Let \(c_{n,k}\) denote exactly the finite derivative constants \(C_{n,k}\) in lesson 22, LD13–LD16. In particular \(c_{0,0}=1\). Their full formula is

Here \(J_j\) is the finite one-dimensional kernel integral explicitly constructed in LD5–LD11. The letter \(\mathcal B_j\) denotes that derivative constant; it is distinct from the amplitude-square constant \(B_j\) in VD2.

For \(1<p<\infty\), put \(e_p=3/2-3/p\), and retain the full original norm bound before taking a parameter-independent upper bound:

The last line is the proved covering map and finite Fourier kernel estimate LD18–LD20. The middle line uses \(2r\leq M_r\leq3r\), with both signs of \(e_p\) covered by the maximum. The finite-\(r\) numerator in the first line has not been removed from the definition of \(H_p\).

VD2. Expand the actual finite-exponent derivatives

Use \(\|v\|_{W^{1,p}}=\|v\|_p+\|D_xv\|_p\), with the full ordered gradient norm. The actual fields are

The opposite labels share the real amplitude and intermittent factor. Every spatial or time derivative of those factors is real. For a real vector \(v\), \(|v\times B_I|\leq|v|\), since \(B_I=(A_I+iC_I)/\sqrt2\), with \(A_I,C_I\) orthonormal: the squared norm is \((|v\times A_I|^2+|v\times C_I|^2)/2\leq|v|^2\). The same inequality holds after summing all derivative-array entries.

For an individual label abbreviate \(a_0=K_0X^5\), \(a_1=K_1X^{20}\), \(a_2=K_2X^{40}\). The entire product rule in VD5 and VD4 gives the following per-label bounds:

Here \(C_p\) is the actual constant PL17. PL18 is applied to the full spatial derivative array and commutes with the projection. Hölder uses the original \(L^{2p}\) norms of both \(\eta_I\) and \(\nabla\eta_I\). The three terms in the last bracket come respectively from the zeroth derivative, the derivative of \(a_I^2\), and the two identical product-rule terms from \(\nabla(\eta_I^2)\). In the middle bracket, its first two terms are the zeroth derivative; the next three differentiate \(\nabla(a_I\eta_I)\); the last two differentiate the physical carrier. Thus no derivative term is omitted.

The carrier has no time dependence. The full time derivatives are bounded, per signed label, by

The four terms in the second line differentiate the two original factors and the original gradient in every possible way.

VD3. Receive every term in the source scales

To give fully specified constants, define

They are finite and independent of \(a,q\). To check the receiving powers without hiding a product, divide the four principal terms of VD6, after their label count, by \(X^{40}\Lambda r^{e_p}\). After their displayed coefficients are removed the ratios are, in their original order,

For the seven curl terms, in order, they are

For the three temporal terms, \(H_{2p}^2/r^{e_p}\leq h_{2p}^2r^{3/2}\). Thus the full ratios are

For VD7 divide by \(X^{40}\Lambda^{33/16}r^{e_p}\). The principal ratios are \(X^{-19}\Lambda^{-33/16},X^{-34}\); the curl ratios are

Since \(X\geq2\), \(\Lambda=X^b\) and \(b\geq512\), every one of these ratios is at most one. Multiplying back their complete coefficients in VD8 proves

The second original scale is exactly \(X^{40}\Lambda^{33/16}r^{e_p}\), by VD1. These are the full finite-exponent estimates in the source, with the original periods, all derivative terms and the actual temporal projection accounted for.

VD4. Supremum derivatives from the original finite Fourier sums

Let \(D^N=D_{x,t}^N\) denote the full ordered tensor of derivatives in the four original coordinates. For the positive frame \((\zeta,A,C)\), a term of \(\eta\) with \(J=(j,k,l)\in\{-r,\ldots,r\}^3\) has four-frequency

Every coefficient is \(M_r^{-3/2}\). Thus the triangle inequality for the entire derivative tensor gives the exact finite-sum bounds

Here \(|\Omega_J|\leq\beta r\sqrt{3+\mu^2}\leq2T\), since \(\mu\geq1\). The spatial coefficient of \(\nabla\eta\) has norm at most \(\sqrt3 G\). The carrier adds its original \(\varkappa\), and \(\varkappa+2T\leq d_0\Lambda^{33/16}\). Negative labels have the same norms. The equalities for the frequencies in VD10 and the finite sums in VD11 precede all upper bounds.

Set

For this section these are the polynomials evaluated at the displayed physical constant \(d_0\), distinct from the earlier amplitude polynomials of AB. The full product rule, VD2 and VD11 imply

For the first line use \(\|D^ja\|_\infty\leq K_jX^5X^{20j}\) for all \(j\geq0\), then \(X^{20j}\leq\Lambda^{33j/16}\). For the first curl term retain the stronger original \(\|D^j\nabla a\|_\infty\leq K_{j+1}X^{20(j+1)}\). For the second use the last line of VD11. Summing over all labels and using \(M_r^{3/2}\leq3^{3/2}\Lambda^{9/8}\) gives exactly VD13. In particular the extra gradient in the curl term remains spatial; it has not been replaced by a larger time derivative.

VD5. The actual temporal correction in the supremum norm

The square \(\eta^2\) has Fourier coefficients

They are nonnegative and sum to \(M_r^3\); their four-frequencies have the same formula VD10 on this enlarged index range. Each has length at most \(4T\) and spatial length at most \(2\sqrt3G\). Define

For \(f_I=a_I^2\eta_I^2\zeta_I\), the original product rule now gives

Indeed the extra derivative on \(a_I^2\) uses \(B_{j+1}X^{5+20(j+1)}\), and the extra derivative on the Fourier square uses its stated spatial length. This accounts for both terms of the second line.

Choose \(J=\lceil\log_2\Lambda\rceil\) in PL21. Then \(R_J=\alpha2^J\geq\varkappa\) and \(J+1\leq2+\log_2\Lambda\). Apply PL21 to the full derivative array in VD16, sum the positive labels and retain \(\mu^{-1}\). With \(C_{\mathrm k}=\pi^2A_0/4\), the result before any logarithmic upper bound is

The factor \(162=6\cdot27\) comes from the original positive-label count and \(M_r^3\leq27r^3\). The leading \(\Lambda\) is precisely \(r^3/\mu=\Lambda^{9/4-5/4}\). In the gradient term, the division by the original \(\varkappa\) has been retained.

For \(s\geq1\), differentiation of \(s^{-1/8}\log s\) shows its maximum is \(8/e\), attained at \(s=e^8\). Therefore \(\log_2\Lambda\leq8\Lambda^{1/8}/(e\log2)\). Put

Since \(\Lambda\geq1\), every term of VD17 is received in the explicit bound

This uses the proved smooth-input estimate, with its additional spatial derivative and reciprocal physical frequency. It does not apply a nonexistent unrestricted supremum projection bound.

VD6. Construct the base threshold for the full source derivative bound

Use the full norm \(\|v\|_{C^N_{x,t}}=\sum_{|\gamma|\leq N}\|\partial^\gamma_{x,t}v\|_\infty\). There are \(n_j=\binom{j+3}{3}\) multiindices of order \(j\) in four coordinates. Each component derivative is bounded by the full ordered tensor norm, so the sum of their suprema is bounded by \(n_j\) times that norm. No square root is applied to this number of suprema.

For \(N=0,1,2,3\), define five finite constants

In each of VD13 and VD19 the frequency exponent increases with derivative order. Bounding its \(j\leq N\) value by its \(N\) value and summing proves

All five numbers are positive: they increase with \(N,b\), and at \(N=0,b=512\) they are respectively \(186,683,282,181,742\). Retain every preceding threshold on the same integer base \(a\) and add the finite twenty requirements

Round their full maximum upward to a multiple of \(N_\Lambda\). Because \(X=\lambda_q\geq a\), each summand in VD21 is at most \(1/10\), for every \(q\geq0\). Thus their sum is at most \(1/2\).

For completeness also retain the original mollified old velocity. The source hypothesis \(\|u_q\|_{C^1_{x,t}}\leq X^4\) and the original kernel derivative constants \(\mathfrak M_j\) from SC24 give

One derivative is put on the original \(u_q\), and all remaining derivatives on the actual space-time kernel; its \(L^1\) tensor norm is \(\mathfrak M_j\ell^{1-j}\). Consequently, for these four orders, let

The full old norm divided by \(\Lambda^{(3+5N)/2}\) is at most \(U_N^{\rm old}X^{-d_N^{\rm old}}\). Add the four finite thresholds \(a\geq(2U_N^{\rm old})^{1/d_N^{\rm old}}\) to the same maximum. The actual constructed velocities then satisfy

The complete \(C^1\) induction follows at \(N=1\), with precisely the source power \(\Lambda^4\). Together, VC21, VD9 and VD25 establish all the original finite velocity estimates used at this stage. The original stress and energy induction, convergence and nonuniqueness theorem still require their own complete calculations.

The actual kernel geometry and velocity thresholds

VC1–VC10 prove every arrow in this diagram. Both oscillatory means, the original volume and the full parameter prefactor enter the principal bound. The figure source retains its exact formulas.

This is a central slice of the three-dimensional cube and enlarged ball in PL14–PL15, with the stated original period and coordinates. The kernel difference uses the actual zero integral of the local remainder. PL7–PL16 prove the full bound; EX4–EX6 optimize its enlargement without changing the underlying operator.

The twenty entries are the exact values in VD21 at the source choice \(b=512\). Each controls its complete coefficient \(U_{N,s}\). VD22 constructs one finite original base that receives all five terms for all four derivative orders. The complete reproducible source generates the cancellation slice and this exact table.

Human comparison for all three figures: Buckmaster–Vicol, arXiv:1709.10033v4, original velocity proposition. The complete receiving proofs are given in this chapter.

Five solved exercises

Exercise 1. The exact projection of an original Fourier field

Problem. On the original \(\mathbb T_L^3\), take \(k=2\pi n/L\ne0\), \(n\in\mathbb Z^3\), and real vectors \(c,v,w\). Compute \(Q,P,PP_0\) on \(f(x)=c+v\cos(k\cdot x)+w\sin(k\cdot x)\). Keep its constant mode and all physical \(L^2\) factors.

Solution. Define \(v_\parallel=k(k\cdot v)/|k|^2\), \(w_\parallel=k(k\cdot w)/|k|^2\), \(v_\perp=v-v_\parallel\), \(w_\perp=w-w_\parallel\). The original Fourier coefficients are \(c\) at zero, \((v-iw)/2\) at \(k\), and \((v+iw)/2\) at \(-k\). Applying PL1–PL2 to each gives

Each nonconstant phase has zero mean. Its squared sine and cosine integrals are both \(V/2\), and the mixed integral is zero, by integrating the corresponding nonzero lattice exponentials. Consequently

At each frequency the parallel and perpendicular parts are orthogonal; the constant phase is orthogonal to both nonconstant phases. Thus \(\|f\|_2^2=\|Qf\|_2^2+\|Pf\|_2^2\), with exactly the original volume factor. The mean removal costs exactly \(V|c|^2\) in this orthogonal identity.

The actual scalar whose gradient equals \(Qf\) is

Differentiation proves EX1 directly and checks both signs in PL2.

Exercise 2. Optimize the enlargement in the weak estimate

Problem. In PL5, replace the ball radius \(2\sqrt3\ell_B\) by \(\gamma\ell_B\), where \(\gamma>\sqrt3/2\). Derive the resulting complete weak constant, and prove which enlargement minimizes it.

Solution. Put \(h=\sqrt3/2\). Every point of \(B\) is within \(h\ell_B\) of its center. Outside the enlarged ball, the segment in the cancellation argument has kernel distance at least \((1-h/\gamma)r\). The enlarged volume is at most \((4\pi/3)\gamma^3|B|\), including when it wraps around the torus. The same radial integral as in PL5 gives

The factor four multiplying \(B(\gamma)\) retains both \(\sum\|b_B\|_1\leq2\|f\|_1\) and the threshold \(t/2\). At \(\gamma=4h=2\sqrt3\), this is precisely \(A_{\mathrm w}\) in PL16.

Here \(H_1>0\). Indeed \(F\) in PL4 is nonzero. If its inverse transform had zero first derivative everywhere, it would be constant, and its decay would make it zero, contradicting Fourier inversion. Therefore PL5–PL6 imply \(A_1>0\), and PL9 implies \(H_1>0\). Direct differentiation of the full constant gives

The function \(F_h(\gamma)=(\gamma-h)^5/(\gamma+3h)\) increases strictly from zero to infinity on \((h,\infty)\), since

There is therefore one and only one \(\gamma_*>h\) satisfying \((\gamma_*-h)^5=4H_1h(\gamma_*+3h)\). EX5 is negative below it and positive above it. The exact minimum of this full bound is \(A(\gamma_*)\). Substituting this constant for \(A_{\mathrm w}\) in the already proved layer-cake argument gives the corresponding stronger PL17 constant. This optimizes the displayed estimate, without claiming that it is the sharp operator norm.

Exercise 3. Choose the best dyadic split for a smooth input

Problem. For a nonzero \(C^1\) input \(f\), minimize the upper bound for \(Qf\) in PL21 over all integers \(J\geq0\). Include constant inputs and all cases of equality between choices.

Solution. Write \(u=\|f\|_\infty>0\), \(v=\|D_xf\|_\infty\), \(c=C_{\mathrm k}=\pi^2A_0/4>0\), and \(d=\pi A_0\). The bound is

If \(v=0\), the field is constant on the connected torus, so \(Qf=0\) exactly by PL2. The displayed positive upper bound itself is minimized at \(J=0\). If \(q>0\), the first nonnegative difference occurs at

All earlier differences are negative and all later differences are nonnegative, proving the minimum. If \(q=2^{J_*+1}\), the minimum is attained at exactly \(J_*\) and \(J_*+1\); otherwise it is attained only at \(J_*\). This includes the boundary value \(q=2\). When \(0<q<2\) the unique minimizer is zero. A zero input has zero output and every choice gives zero. For \(PP_0f\), PL21 adds the \(J\)-independent term \(2u\), so the same choices minimize that upper bound. The period remains in \(\alpha=2\pi/L\).

Exercise 4. The exact projected fourth moment of one temporal pair

Problem. Keep one original positive intermittent frame and a constant real amplitude \(a=A\). Compute the exact squared \(L^2\) norm of \(z_A=\mu^{-1}PP_0(A^2\eta^2\zeta)\), and the exact squared norm of its spatial gradient. Retain the zero mode and all multiplicities. This is a calculation of one actual field, without asserting that a single pair realizes the whole stress decomposition.

Solution. For the integer triple \(J=(j,k,l)\) in VD14, write

The physical nonzero frequency is \(\beta(j\zeta+kA_{\rm frame}+lC_{\rm frame})\), where \(A_{\rm frame}\) is the frame vector and is distinct from the scalar amplitude \(A\). The original frame map is injective. For \(J\ne0\), the squared norm of the projection of \(\zeta\) is \((k^2+l^2)/s_J\). The zero mode is removed by \(P_0\), and its unprojected coefficient in \(\eta^2\) is exactly one. Parseval gives

To prove the second equality, symmetry under the three coordinate permutations makes the three sums with numerators \(j^2,k^2,l^2\) equal. Their sum is \(\sum_{J\ne0}W_J\). The numerator \(k^2+l^2\) therefore contributes exactly two thirds. The full sum divided by \(M_r^6\) is \(Q_r^2\), by VC13, and the removed zero term is one. No orthogonality between distinct frames has been used.

For the gradient, multiply each term of the first sum by its original physical squared frequency \(\beta^2s_J\). This cancels only its displayed denominator. The exact one-dimensional sums are

They follow by expanding \(2\sum_{j=1}^{M_r-1}(M_r-j)^2\) and \(2\sum_{j=1}^{M_r-1}j^2(M_r-j)^2\), retaining the zero term \(M_r^2\) in the first. Substituting the elementary power sums, or induction on \(M_r\) after taking consecutive differences, gives the displayed polynomials. Fubini in the two remaining indices yields

Thus the actual projection and the removed mean improve the unprojected contraction estimate in a fully quantified case.

Exercise 5. The initial amplitude and its entire numerical tail

Problem. For the source values \(b=512\), \(\beta_{\rm reg}=2^{-16}\), retain \(\lambda_q=a^{b^q}\) and \(\delta_{q+1}=\lambda_1^{3\beta_{\rm reg}} \lambda_{q+1}^{-2\beta_{\rm reg}}\). Compute the initial value, each consecutive ratio, and an upper bound for the entire series of square roots. What happens to its tail as \(a\) grows?

Solution. With \(\theta=\beta_{\rm reg}>0\), the full formulas are

In particular \(\delta_1=a^{1/128}>1\). The original prefactor therefore prevents a uniform assertion \(\delta_{q+1}\leq1\) starting at \(q=0\). The exact ratio proves the convergent numerical-series bounds

For the tail, use its actual first term \(\sqrt{\delta_2}=a^{1/256}R=a^{-1021/256}\), and the same bound on each subsequent ratio. This tail tends to zero as \(a\to\infty\), although the first term tends to infinity. These are statements about the fully specified numerical sequence. Constructing a sequence of solutions realizing the entire iteration still requires the stress and energy induction; the calculation does not assume those steps.

The next equations in the construction

VC21 gives the full square-norm increment. VD9 gives the original spatial and time bounds, and VD25 proves the full velocity induction through three derivatives. The original stress and physical energy must now be estimated using every term of lesson 23 and the energy identity of lesson 24. Those estimates must precede the convergence and nonuniqueness argument.

The separate Albritton–Brué–Colombo, Alpöge–Buckmaster, OpenAI and workbench constructions retain their original equations, domains, force and regularity requirements.