An annulus and velocity mass uniform across scales

A single frequency event has consequences at many earlier time scales. To add those consequences, their constants must work at every scale being counted. This lesson constructs that uniform family from the original energy, heat and Newton estimates. The final velocity mass is independent of the selected time scale. Disjoint spatial shells then bound the original frequency event.

We use the complete providers in lesson 11, lesson 12, lesson 13, lesson 14, lesson 16, and lesson 18. Their kernels, viscosity, pressure, support and time intervals stay explicit. Every field in this lesson is the original field. The figures show proved bounds and regions, with reproducible sources.

The human comparison is Terence Tao, Quantitative bounds for critically bounded solutions to the Navier–Stokes equations, arXiv:1908.04958v2, original author article.tex 660–892 and 1322–1418. This is independently written course exposition with complete receiving calculations and visible further consequences. No novelty or independent review is claimed.

1. Actual times and all global coefficients

Keep the smooth unforced energy class, original frequency event and all previously constructed constants of lesson 18 (8.1)–(9.3):

The time domain is \([t_0-T_{\rm orig},t_0]\). In particular its bounded higher \(L^2\) derivative norms justify the integrations, but will not enter the constants. The actual event has amplitude at least \(bN_0\) in the original projection \(P_{N_0}\) at \((t_0,x_0)\), where \(b>0\). All the iteration constants, including \(\Lambda_{\rm it},\rho_{\rm it}\), and the finite constants \(K,M,c_F,D_F,D_*\) of lesson 18 are already constructed from the original inputs, independently of the next scale.

For every

use the original heat field

lesson 18 (9.1) proves that the entire interval is inside the original domain, using \(\rho_{\rm it}\leq1/32\). The spatial shells below are centered at the original point \(x_0\); writing \(|x-x_0|\) changes no field.

Keep every kernel constant of lesson 12 and \(C_{j,p}\) of lesson 13 (1.4). Define the following positive numbers independent of \(S\):

Here \(\mathcal B_0,\mathcal K_0,c_\phi\) are exactly the complete kernel constants of lesson 12, written \(B_0,K_0,c_\varphi\) in the receiving display of lesson 13. The speed constant \(\mathcal K_0\) is distinct from the later Gaussian \(K_0\). Also \(S_3=4\sqrt3\), \(g_*=(1-2^{-1/2})^{-1}\), \(v_3=4\pi/3\). All other displayed kernel constants have their complete integrals in that provider.

The original bounds of lesson 13 (1.4)–(1.8), with the actual choice \(N_*=1/\ell_a\), give

Indeed the actual initial nonlinear norm in \(M_\delta\) is bounded by \(W_0\), leaving its full contribution \(\mathsf W^2\ell_a/\nu\). The original heat energy integral retains both endpoints \(\sqrt L-\sqrt\delta=(\sqrt8-1)\ell_a\). In the total-speed formula its six nonzero groups respectively contain \(V_\delta\), \(N_*UT_\delta\), \(N_*^{-1}U\), \(N_*^{-1}H_\delta^2\), \(N_*^{-1/2}H_\delta\sqrt{M_\delta}\), and \(M_\delta\). Each has precisely one factor \(\ell_a\), with \(T_\delta=7\ell_a^2\). The force term and the two global force energy terms are exactly zero because the original force is zero.

Select the actual time \(t_1\in[t_b+\delta,t_b+2\delta]\) by the integral of \(\|\nabla w\|_2^2\), exactly as in lesson 13. Then

The time exists by continuity and the mean inequality; no uncontrolled instantaneous gradient norm is introduced.

2. Separate heat tolerances and a uniform shell count

Write the physical tent height as \(h_{\rm ann}=\mathfrak h\ell_a\), where the number \(\mathfrak h\geq1\) will be chosen below from known constants. For the full original energy inequality choose

retaining the original positive \(C_1,C_2,C_3\) of lesson 11. Set

The event gives \(U>0\), hence these denominators are positive. The three original targets are different physical quantities:

We construct them, rather than assume them.

Use the full nonnegative density of the original fields

Every original term is retained with its displayed positive weight. The bound follows term by term from 1.6 and the exact heat derivative bound at \(t_1-t_b\geq\ell_a^2\).

Take the physical evaluation radius \(\rho=\ell_a\). Define

Here \(\epsilon_a\) is an initial-density tolerance, not a force tolerance or the ball margin used later. The actual heat distance is \(d_{\rm heat}=\ell_a d_c\).

Fix \(m=4\) and \(\Lambda=(100M)^{1/4}\), exactly as in lesson 18 (9.4). Thus \(\Lambda\geq2\) and \(\Lambda^4=100M\). The radius needed by the outward receiver has coefficient

All these numbers, apart from the explicitly displayed \(\ell_a\), are independent of \(S\). The closed shells \(\{\Lambda^{-4}R_k\leq|x-x_0|\leq\Lambda^4R_k\}\) have disjoint interiors. Their boundary spheres have zero volume. The number with density integral greater than \(\epsilon_a\) is at most \(\mathsf B/\epsilon_a<N_a\), by summing the nonnegative density. Choose an actual good shell and write its radius as \(R\). We have

The full two-extra-derivative evaluation formula in lesson 13 (2.7) has the three expressions \(a_j\), \(\rho a_{j+1}+\beta_1a_j/2\), and \(\sqrt3\rho^2a_{j+2}+\beta_1\rho a_{j+1}+\beta_\Delta a_j\). At \(\rho=\ell_a\), each has its common factor \(\epsilon_a^{1/3}\ell_a^{-j}\). Its exterior factor is \((8v_3)^{1/6}/(\sqrt{8\pi}\ell_a)\). Hence

The evaluation balls lie inside \(S_0\), since 2.6 gives the original distance \((\Lambda-1)\Lambda^{-4}R\geq2\ell_a\). The next distance is at least \(\ell_a d_c\). For the heat-tail function of lesson 13 (2.8), direct substitution in its complete Gaussian formula gives \(\mathcal T(\ell_a d_c,8\nu\ell_a^2)= \ell_a^{-1}\mathcal T(d_c,8\nu)\). The positive logarithm choice in 2.5 therefore bounds the original incoming \(j\)-th derivative by \(\alpha_{\mathfrak h}\ell_a^{-(j+1)}/2\). This retains the full global input \(C_{j,3}U\nu^{-j/2}\ell_a^{-j}\) for each \(j\). Monotonicity in the upper heat time covers the actual time \(\nu(t_0-t_1)\leq8\nu\ell_a^2\). Together with 2.8 this proves 2.3 on \(S_2=\{A\leq|x-x_0|\leq B\}\), where \(A=\Lambda^{-2}R\) and \(B=\Lambda^2R\), throughout \([t_1,t_0]\).

3. The complete original energy and local velocity

Use the actual speed and tent of lesson 13 (5.1)–(5.2), now with \(h_{\rm ann}\) and the choices 2.1. Its full recession is at most

Retain the actual sphere and curvature term \(Y_3=(\nu/2)\int|z|^2\Delta\eta\). The absolute-flux averaging proof chooses the two original radii \(a_0\in[A,2A]\), \(b_0\in[B/2,B]\) so that

The heat cross terms have the exact original upper bounds

All time integrals here are over \([t_1,t_0]\); the local norms are exactly lesson 13 (5.4). The powers of \(\ell_a\) in the three right sides are respectively \(1-3+2=0\), \(1-4+1+2=0\), and \(1-2+2-1=0\). Each is at most \(e(\mathfrak h)\) by 2.2. The two original force contributions are exactly zero.

The complete original coefficient is

Its integral obeys

In particular the earlier coarse \(2L\) estimate has been recalculated from the actual \(V_1^*\), not omitted. The full differential inequality remains \(E'+(\nu/2)D+(1/4)Y_2\leq\mathfrak aE+|Y_3|+\mathfrak d\). There are five nonzero budget terms and two zero force terms. Thus its original seven-term bound is still valid: \(E\leq7e(\mathfrak h)e^{\mathcal A(\mathfrak h)}=7E_*/8<E_*\). The integrating-factor argument and continuity exclude a first crossing, exactly as in the provider. Keeping this common bound preserves every later estimate.

Put

Here \(b_1,b_2\) are the original cutoff derivative maxima. Integration of the energy inequality, retaining its nonnegative final energy, gives \(Z_0^2\leq E_c/(\mathfrak h\ell_a)\) and \(Z_1^2\leq D_c/(\mathfrak h\ell_a)\). The exact local cutoffs have \(G_1\leq g_1/(\mathfrak h\ell_a)\), \(G_2\leq g_2/(\mathfrak h^2\ell_a^2)\). Consequently the original three terms of \(Q_0^2\) are bounded by

Similarly \(P_0^2\leq P_c/(\mathfrak h\ell_a)\) and \(P_1^2\leq P_{1c}/(\mathfrak h\ell_a)\). The complete Fourier interpolation bound, including the added heat field, and the actual local \(L^2\) vorticity input give

Indeed the heat terms before comparison are \(\vartheta_a^{1/4}\alpha_{\mathfrak h}\ell_a^{-1/2}\) and \(\sqrt{2v_3}\mathfrak h^{3/2} \alpha_{\mathfrak h}\ell_a^{-1/2}/32^{3/2}\). Using \(\alpha_{\mathfrak h}\leq\mathfrak h^{-2}\), \(\mathfrak h\geq1\), \(6\leq\vartheta_a\leq7\) gives precisely 3.6.

4. Four pointwise bounds with their original powers

Use the exact balls and time cutoffs of lesson 14:

where \(k_1=1024b_1\), \(k_2=1024^2b_2+(524288/3)b_1\). For any positive Gaussian pair \(Q_0,Q_1\) define the full expression

Take \(K_0=\|H_1\|_{12/11}\), \(K_1=\sum_j\|\partial_jH_1\|_{12/11}\), and the exact \(J_0,J_1,J_{2,\alpha},J_\nabla\) of lesson 14 (3.2), (5.3), with \(\alpha=1/4\). Set

Substitute 3.6–4.1 in each of the four terms of lesson 14 (2.7). Their \(\ell_a\) powers are \(1/4-1/2=-1/4\), \(-1+3/4=-1/4\), \(-2+7/4=-1/4\), and \(-1+5/4-1/2=-1/4\). Their remaining positive factors are bounded by 4.2 using \(\mathfrak h\geq1\) and \(6\leq\vartheta_a\leq7\). The same argument applies to the full Hölder expression (3.3). The original six steps and the seventh Hölder step therefore give

All force heat potentials are exactly zero for 1.1.

The exact Newton formula retains its main curl field, both boundary fields, the principal value Hessian and its point mass \(-\delta_{ij}\delta_0/3\). Its complete bounds in lesson 14 (4.4)–(4.8) yield the following constants:

For the velocity, the full optimized main term has factors \(Z_\infty^{1/3}Z_2^{2/3}\); the two boundary terms together have \(L_1W_0/\sqrt{\pi\epsilon}\). Their height powers are \(\mathfrak h^{-1/2}\) and \(\mathfrak h^{-3/2}\); the added heat field has \(\mathfrak h^{-2}\). All have physical power \(\ell_a^{-1}\). For the gradient the five original terms have height powers \(-1/4,-1,-1/2,-5/2,-2\), respectively, and all have physical power \(\ell_a^{-2}\). The logarithm is exactly \(\log(2\rho_b/\epsilon)=\log64\). These substitutions prove every summand of 4.5, including the heat contribution to \(Z_\alpha=H_7+\sqrt2V_2^*\epsilon^{1-\alpha}\).

To keep every term in the differentiated vorticity equation, put

Indeed the exact products of lesson 14 (5.2) obey

The mixed nonlinear height exponent is \(3/8+1/16+1/2=15/16\); its physical exponent is \(3/4+1/2+2=13/4\). The other nonlinear term has height exponent \(1\), and the second cutoff term has exponent \(7/4\); both are retained in 4.6 before comparison by \(\mathfrak h\geq1\). The differentiated heat integral contributes \(\ell_a^{1/4}\) to the first two products and \(\ell_a\) to the remainder. Thus all have final physical power \(\ell_a^{-3}\). This proves the last line from the full original derivative formula.

Every constant in 3.5–4.6 was fixed using only \(\mathfrak h\geq1\). We can now make the noncircular choice

Use this actual number in 2.2–2.7 and in all the physical cutoffs above. They supply, on the original shell \(\mathcal K\),

The interval is \([t_1+9T/16,t_0]\). Its left endpoint is \(t_0-7T/16\leq t_0-42T_a/16<t_0-T_a\). Therefore it contains the entire interval required by FT.

5. Original final-time mass and fixed enclosing shells

The actual inner and outer radii remain

The choice \(r_{0c}\) proves \(r_-\geq2\sqrt{\nu T_a}\) and \(20r_-\geq D_*\sqrt S\). All complete time, recession and local cutoff margins of lesson 18 (9.5)–(9.10) hold because \(h_{\rm ann}\leq A\), \(q\leq A\), and \(B/A=100M\). In particular the regular shell endpoints obey \(k_-\leq141A/32\) and \(k_+\geq B/2-77A/32\). Thus the two weighted estimates are applied to the same original equation and the same actual common space-time region as in lesson 18. The replacement annulus selection has proved every one of their inputs, including the four bounds just obtained, without assuming uniform radius or mass.

Use the complete positive coefficients of lesson 18 (12.9):

Here \(H,K,C_I,C_t,C_J,M\) are their full terminating choices in lesson 18; the height number \(\mathfrak h\) is a different constant. The enhanced Gaussian covering coefficient of lesson 18 Exercise 3 can replace \(c_Y\) by \(6^{3/2}e^{-5/4}c_Y\); the proof below already holds for the original \(c_Y\).

Let \(\sigma_G=1/\sqrt{32}\) and define

The complete finite proof in lesson 18 gives

The last comparison follows from the actual radius upper bound in 2.7: \(a\leq\overline a\). The exact shell volume remains \(v_3(\beta_s^3-\alpha_s^3)R^3\). Since \(M\geq128\), its endpoints are strictly ordered. Consequently there is an actual point \(x_*\) of the closed mass shell with

This follows by taking the continuous maximum and comparing its square times the full shell volume with 5.4. No assertion about the location of its maximum outside this shell is needed. The derivative neighborhood of radius \(\delta_F\) is inside \(\mathcal K\), by lesson 18 (14.2); 2.7 gives \(\delta_F\geq d_c^{\rm curl}\ell_a\).

Keep the actual radial smooth bump \(\varphi\) of lesson 18 (13.1)–(13.3): it is nonnegative, supported in the unit ball, has integral one, \(\mu=\int|y|\varphi(y)\,dy\in(0,1)\), and \(C_{3,\varphi}=\|\nabla\varphi\times e\|_{3/2}>0\) for every unit \(e\); radial symmetry makes the latter independent of \(e\). Choose

The positive remainder is at least \(m_c/3\). The chosen ball of radius \(r\) lies in the actual derivative neighborhood, so the original gradient bound gives

The exact curl integration by parts is

The compact support removes the boundary term. Hölder and the original Jacobian \(dx=r^3dy\) bound the absolute right side by \(r^{-2}C_{3,\varphi}(\int_{B(x_*,r)}|u(t_0,x)|^3dx)^{1/3}\). Cubing, and retaining \(r^6=r_c^6\ell_a^6\) and the mean cube \(\ell_a^{-6}(m_c-\mu r_c/\sqrt\nu)^3\), proves a lower bound \(L_c\), independent of \(S\), for the original velocity integral on that ball.

6. Disjoint shells and the frequency bound

The ball just constructed lies in the original annulus \([a_\sharp-\delta_F,b_\sharp+\delta_F]\). Both its endpoints have explicit fixed envelopes:

For the inner endpoint, positivity follows from \(\alpha_s-\Lambda^{-2}/4 =\Lambda^{-2}(10(1-1/\sqrt{32})-1/4)>0\); use \(1/\sqrt{32}<1/4\). The original radius lower bound gives that inner envelope. The original upper bound gives the outer envelope. Finally \(\ell_a=2\sqrt K\sqrt S\), which proves the displayed physical endpoints without changing the integration variable.

Set

If the scale interval is nonempty, take \(J=\lfloor\log(S_{\max}/S_{\min})/\log q_{\rm shell}\rfloor\) and the actual scales \(S_j=S_{\min}q_{\rm shell}^j\), \(0\leq j\leq J\). Each lies in 1.2. The outer radius of the \(j\)-th envelope equals the inner radius of the next, because \(A_S\sqrt{q_{\rm shell}}=B_S\). Their interiors are disjoint; their shared boundary spheres have zero volume. Summing 6.1 thus gives \((J+1)L_c\leq\|u(t_0)\|_3^3\leq U^3\). The floor inequality \(\log(S_{\max}/S_{\min})/\log q_{\rm shell}<J+1\) implies

If the scale interval is empty, it gives directly \(T_{\rm orig}N_0^2<K\Lambda_{\rm it}/\rho_{\rm it}\), which is stronger than 6.3. The equality \(S_{\max}=S_{\min}\) is included in the nonempty case, with one annulus. All constants in 6.3 are finite functions of the original already specified \(\nu,U,b\) and fixed kernel choices; none uses \(S,t_1,R\), \(N_0\), or an auxiliary higher-derivative norm.

7. Propagating the stronger covering coefficient

The stronger coefficient from lesson 18 Exercise 3 has an exact receiver here. Keeping every annular, time and geometric constant just constructed, define

The original choice of \(H\) still absorbs the Gaussian error because its full error coefficient decreases, as proved in that exercise. All the annular constants chosen before the endpoint mass are therefore unchanged. The same complete mass, curl and disjoint-shell proofs apply with the displayed starred values.

For clarity, the inequality in the velocity constant is not inferred just from an enlarged radius. Put \(g=\mu/\sqrt\nu>0\). The derivative of \(r^6(m-gr)^3\), where its mean is positive, is \(3r^5(m-gr)^2(2m-3gr)\). Thus the maximum on \(0<r\leq d_c^{\rm curl}\) with positive mean occurs at \(\min(d_c^{\rm curl},2m/(3g))\). At any fixed positive old radius the expression is strictly increasing in \(m\). Hence \(L_c^*\geq L_c\), with strict inequality whenever \(c_F^*>c_F\); equality holds if \(c_F^*=c_F\). Since \(q_{\rm shell}>1\) and \(U>0\), the frequency-event upper bound decreases strictly in the former case. The other exercise's enlarged derivative neighborhood changes both its integration region and later shell spacing, so no automatic improvement of this last bound is asserted from that enlargement.

Figure. The upper panel retains the original mass, shell volume, derivative bound, radius and Jacobian in 5.3–6.1. The lower panel uses the displayed logarithmic radial coordinate only to draw the exact original annuli in 6.1–6.3. The first three shells are shown when \(J\geq2\); truncate at \(J\) otherwise. Shared boundary spheres have zero three-dimensional volume. This is a diagram of proved bounds and supports, not a sampled fluid solution. Reproducible figure.

This proves a uniform bound for every original frequency event in the stated smooth unforced energy class. It is the needed annulus and shell-sum consequence. Passing from this event bound to the full continuation assertion requires the original frequency reconstruction and the relevant local existence/continuation argument. That final implication is not claimed merely from the word “uniform.”

The separate heat tolerances, weighted original initial density, recalculated energy coefficients, all six heat steps, both Newton boundary fields and the differentiated vorticity terms above give the full comparison missing from a bare scaling assertion. No novelty or independent review is claimed.

8. Five exercises with complete solutions

Exercise 1: count overlapping shells exactly

Keep \(A_S,B_S,L_c\) from Section 6. Choose any \(q>1\), rather than the disjoint choice, and use all scales \(S_j=S_{\min}q^j\) that lie at most \(S_{\max}\). Determine the largest possible overlap away from boundary spheres. Prove the resulting event bound and compare the exponent cost.

Solution. Write

For a radius \(r>0\) not on a shell boundary, membership in shell \(j\) is exactly

This open interval has length \(h_q\). If \(k\) integers lie inside it, their largest minus smallest is at least \(k-1\) and strictly less than \(h_q\). For nonintegral \(h_q\) this gives \(k\leq\lfloor h_q\rfloor+1=\lceil h_q\rceil\); for integral \(h_q\) it gives \(k\leq h_q\). The finite index range gives the additional bound \(k\leq J_q+1\). These bounds are attained on an open interval of radii: choose a consecutive block of \(\min(m_q,J_q+1)\) indices and place the endpoints of the open interval just outside that block. Its length is greater than the block's span. Varying \(r\) translates both endpoints continuously without changing their difference. Thus the exact essential maximum multiplicity is \(\min(m_q,J_q+1)\). Boundary spheres have zero volume.

Sum the original integrals, retaining their actual center:

The same strict floor inequality used in Section 6 gives

The empty scale interval has the earlier stronger bound. Since \(m_q\log q\geq2\log R_s\), no \(q\) improves this exponent cost. Equality holds exactly when \(h_q\) is an integer, that is, \(q=R_s^{2/m}\) for some positive integer \(m\). The disjoint choice is \(m=1\). The statement concerns this uniform mass-and-multiplicity estimate; it does not forbid stronger information about the actual velocity distribution.

Exercise 2: retain a general velocity exponent

Replace the final \(L^3\) Hölder step by \(L^p\), with \(1<p<\infty\). Keep the same original mass point, actual derivative neighborhood and bump. Find the best radius in the resulting bound. Show exactly why \(p=3\) removes the physical scale factor.

Solution. Put \(p'=p/(p-1)\) and

The bump is the same radial compact smooth function, so its constant is finite and invariant under rotation of the unit vector \(e\). It is positive since a nonconstant radial bump has a nonzero tangential cross product on a set of positive volume. At \(r=s\ell_a\), \(0<s\leq d\), the unchanged mean and signed integration by parts give

We use this only while \(m-gs>0\). Hölder and the full Jacobian \(dx=r^3dy\) prove the last inequality; the first mean is positive, so taking its absolute upper bound is valid. Consequently

The derivative of the numerator, on its positive interval, is

It is positive and then negative on opposite sides of the single zero. The actual maximizing radius is therefore

For \(p=3\), these are precisely \(r_c\) and the lower bound \(L_c\) of (5.6). For general \(p\), the full scale factor is \(\ell_a^{3-p}=(2\sqrt K)^{3-p}S^{(3-p)/2}\). It is constant over all \(S>0\) exactly when \(p=3\). No constant-scale \(L^p\) mass is inferred at another exponent.

Exercise 3: propagate a stronger vorticity mass quantitatively

Suppose the final vorticity mass coefficient is multiplied by \(\eta\geq1\), with every selected geometric and derivative constant unchanged. Determine the exact new optimized velocity coefficient and its relation to the old one, including the transition at the derivative-neighborhood radius.

Solution. Use \(d,m,g\) of (8.5) and \(C=C_{3,\varphi}\). The full shell volume is unchanged, so the new vorticity point bound is \(m_\eta=\sqrt\eta\,m\). The exact formula is

If \(2\sqrt\eta\,m/(3g)\leq d\), substitution retains all factors and gives

If that inequality is reversed, the exact value is instead

The two formulas coincide when \(\eta=(3gd/(2m))^2\). When the original optimizer is strictly inside its permitted neighborhood, this transition value exceeds one; otherwise the capped formula applies for all \(\eta\geq1\). Equality of the original two radii gives transition at one.

At the old positive optimizer \(s_1\),

Its final summand is positive when \(\eta>1\). Evaluating the new objective at that old admissible radius and then maximizing therefore proves \(L(\eta)>\eta^{3/2}L(1)\) for \(\eta>1\), with equality \(L(1)=L(1)\) at \(\eta=1\). If both optimizers are uncapped, the exact ratio is \(\eta^{9/2}\). If both are capped, the ratio is \([(\sqrt\eta\,m-gd)/(m-gd)]^3\). Formula (8.10) covers the crossing case without removing either branch. For Section 7 take the actual \(\eta=c_F^*/c_F\). The event exponent is rebuilt with \(L(\eta)\); it is never estimated from a radius increase alone.

Exercise 4: use the actual bump radius in the enclosing shells

The envelope in Section 6 uses the full available derivative neighborhood even when the actual bump ball is smaller. Keep its proved mass \(L_c\), and construct the exact uniform envelope using its actual radius. Compare the event bound.

Solution. Set

For every actual scale, the mass point lies between \(\alpha_sR\) and \(\beta_sR\), and the ball radius is \(r_c\ell_a\). Both triangle inequalities and both actual bounds for \(R\) prove containment in

The inner endpoint is positive because \(r_c\leq d<A_0\). In original time-scale coordinates define

The same original velocity integral is at least \(L_c\). The original Section 6 inner endpoint before multiplication by \(2\sqrt K\) is \(A_0-d\), at most \(A_0-r_c\). Its outer endpoint is \(B_0+\Lambda^{-2}\overline R_c/4\), at least \(B_0+r_c\). Thus \(\widetilde q\leq q_{\rm shell}\), strictly if \(r_c<d\) or \(\overline R_c>R_c\). Repeating the complete disjoint-shell sum gives

It is strictly stronger under exactly the preceding sufficient conditions. When both are false, both endpoints agree. In the actual construction, \(\mathsf B>0\) and \(\epsilon_a>0\), so \(N_a=1+\lceil\mathsf B/\epsilon_a\rceil\geq2\). Since \(\Lambda>1\), (2.7) gives \(\overline R_c>R_c\). The improvement is therefore strict for every nonzero instance of the constructed family. All statements hold with the starred \(m_c^*,r_c^*,L_c^*\) as well, since their neighborhood and annulus are unchanged.

Exercise 5: optimize shell width together with the bump mass

Use the smaller envelopes of Exercise 4, but allow the actual bump radius to vary. Prove that the best mass radius need not give the best frequency bound. Give an exact one-variable minimization with a proved attained minimum.

Solution. Keep \(A_0,B_0,d,m,g,C\) from the preceding exercises, and let \(s_0=\min(d,2m/(3g))\). For \(0<s\leq s_0\), the full earlier proof applies to the actual radius \(s\ell_a\) and gives

The original shell sum proves this for every admissible \(s\); its empty and singleton cases are unchanged. Because \(d<A_0<B_0\), all displayed factors are finite and positive on this interval. As \(s\downarrow0\), its logarithm tends to \(\log(B_0/A_0)>0\), while its denominator tends to zero. Thus \(\Phi(s)\to+\infty\). Choose \(\epsilon\in(0,s_0)\) so that \(\Phi(s)>\Phi(s_0)\) for \(0<s<\epsilon\). Continuity on the compact interval \([\epsilon,s_0]\) gives an attained minimum \(s_*\in(0,s_0]\).

The exact logarithmic derivative is

An interior minimizer satisfies this equation with right side zero. If \(s_0=2m/(3g)\leq d\), the last two terms cancel at \(s_0\), leaving a strictly positive derivative. There is then a smaller radius with a strictly smaller frequency-bound exponent, so the minimizer is interior. This proves that maximizing mass alone is insufficient in this case. No uniqueness of the minimizer is assumed. If the neighborhood caps the radius earlier, a minimizer can instead occur at its endpoint.

For any admissible radius above \(2m/(3g)\) with positive mean, its mass objective decreases and its shell ratio increases; it therefore has larger \(\Phi\) than at \(2m/(3g)\). Radii at or beyond \(m/g\) have no positive mean estimate. The minimization already covers every potential improvement obtained by this bump argument. Using \(s=s_*\) in (8.18) gives the proved best bound within this exact family. All constants remain functions of the original \(\nu,U,b\) and specified kernels, independent of the selected scale \(S\). The same construction applies to the actual strengthened mass of Section 7 by replacing \(m\) with \(m_c^*\).

9. What the calculation supplies

The original scale family now has a definite annulus at every admissible scale, a uniform radius upper bound, and positive final-time velocity mass independent of that scale. The original frequency-event bound follows with all physical constants retained. The source's scale comparison is therefore an actual map between the original estimates and the receiving shell sum.

The five exercises prove the exact overlap count, the full \(L^p\) curl receiver and its critical exponent, quantitative propagation of stronger vorticity mass, a smaller uniform envelope and a joint radius-and-shell optimization. They retain the preceding argument and identify each stronger consequence separately.

The next lesson uses the frequency bound in the full original enstrophy identity, including every heat cross term, and supplies positive-time smoothing and continuation. The later forced-flow constructions and their terminal regularity requirements still have their own arguments.