Selecting an annulus and controlling local velocity

The preceding two lessons give a moving-annulus energy estimate and a bound for the total distance its boundaries can travel. We now choose an actual annulus from an actual solution. Its initial vorticity, incoming heat field and two force contributions will satisfy the required energy budget because of the choices we make.

The result also gives a local velocity bound in the fourth power of time. We prove the cutoff div-curl and Fourier estimates that give this bound, then derive the complete localized vorticity heat equation. These are finite estimates on the stated interval. The general large critical-velocity endpoint and the later nonlinear pointwise iteration remain to be proved.

Read Moving annuli and localized vorticity energy and Global nonlinear energy and total speed first. The strategy is compared with Terence Tao's Quantitative bounds for critically bounded solutions to the Navier–Stokes equations, version 2, the annuli-of-regularity argument. Sections 1–7 below supply the complete receiving argument with the original viscosity and force. Section 8 gives precise source locators and five solved exercises.

1. The original solution and its actual global bounds

Work in the original coordinates on \(\mathbb R^3\). Let \(t_b<t_2\), \(L=t_2-t_b\), and \(\nu>0\). Suppose the smooth solenoidal velocity and pressure solve

on \([t_b,t_2]\), with \(u\) and its spatial derivatives bounded in \(L^2(\mathbb R^3)\) on this interval. This smooth class justifies the identities below. None of the resulting constants uses its auxiliary higher-derivative bounds. The actual force decomposition satisfies

In particular \(F_1\leq\|f_a\|_{L^1L^2}+\sqrt L\|f_b\|_{L^2L^2}\). The force in (1.1) stays unchanged throughout the construction.

Retain the actual earlier heat flow and its nonlinear remainder:

Here \(v(t_b)=u(t_b)\) and \(w(t_b)=0\). Ordered derivative tensors carry the Euclidean norm over all their indices. Define the finite Gaussian constants

Sections 1–2 of the preceding lesson prove, from the full heat/Leray kernel and the original energy identity,

Fix \(0<2\delta<L\), put \(t_0=t_b+\delta\), and let \(J_\delta=[t_0,t_2]\), \(T_\delta=L-\delta\). The explicit nonlinear-energy bound is

The first term retains the actual squared norm; it is also at most \(W_0^2/\nu\). The global Fourier identity for the solenoidal field \(w\) gives \(\|z\|_2^2=\|\nabla w\|_2^2\). This equality is global. It will not be applied to a restricted shell.

We will also use the complete total-speed bound (6.3) of the preceding lesson. Here is its exact receiving value. Choose any \(N_*>0\) and put

Use exactly the Fourier symbols and kernel integrals \(c_\varphi,d_1,d_{6/5},d_2,d_\infty,c_p,B_0,K_0\) defined and proved finite in that lesson, equations (3.1)–(3.12). There is no new choice or replacement of those constants. Define

The proved estimate says \(\int_{J_\delta}\|u(t)\|_\infty\,dt\leq\mathcal S\). Similarly \(\int_{J_\delta}\|v(t)\|_\infty\,dt\leq V_\delta\). Every coefficient in (1.8) is independent of the annulus that we will select. In particular, selecting a larger annulus cannot invalidate the speed bound.

2. An actual time and the full initial heat data

The integral of \(\|\nabla w(t)\|_2^2\) over \([t_b+\delta,t_b+2\delta]\) is at most \(M_\delta\). Consequently there is a time \(t_1\) in that interval for which \(\|\nabla w(t_1)\|_2^2\leq M_\delta/\delta\). Otherwise the integral would exceed \(M_\delta\). Continuity in the stated smooth class permits the pointwise choice. Fix this actual time and define

The four heat-derivative orders beyond the field itself will give pointwise control of its first two derivatives. All are included before the annulus is chosen.

2.1. The exact local evaluation estimate

Keep the smooth even function \(\theta\) of equation (4.7) in the moving-annulus lesson: it is one on \([-1/2,1/2]\), zero outside \((-1,1)\), and nonincreasing for positive arguments. Write

For a compactly supported vector or tensor field \(g\in H^2\) and any physical length \(\rho>0\), Fourier inversion and Cauchy–Schwarz give

Here the Fourier convention is \(e^{-2\pi i x\cdot\xi}\). Indeed

Polar coordinates, the substitution \(s=2\pi\rho|\xi|\), and then \(s=\tan\alpha\) reduce the integral to \(\int_0^\infty s^2(1+s^2)^{-2}\,ds=\pi/4\). The other Cauchy factor is the square root of \(\int(1+4\pi^2\rho^2|\xi|^2)^2|\widehat g(\xi)|^2\,d\xi\). Parseval expands it into the three terms in (2.3), including the middle gradient term. The argument uses the Euclidean tensor norm before Cauchy–Schwarz. Approximation in \(H^2\) proves the formula in that space; its integrable Fourier transform supplies the continuous representative used for evaluation.

Consider a shell \(S_0=\{\Lambda^{-m}R\leq|x|\leq\Lambda^mR\}\), where \(\Lambda\geq2\), \(m\geq4\) is an integer, and \(R>0\). Set

The first distance is a lower bound for the distance from \(S_1\) to the complement of \(S_0\). The second has the same role for \(S_2\) inside \(S_1\). The superscript distinguishes it from the Fourier-kernel constant \(d_1\) in (1.8).

Suppose \(2\rho\leq d_0\). For \(x\in S_1\) the cutoff \(\chi_x(y)=\theta(|y-x|/(2\rho))\) is one near \(x\) and supported in \(S_0\). Its derivatives obey \(|\nabla\chi_x|\leq\beta_1/(2\rho)\) and \(|\Delta\chi_x|\leq\beta_\Delta/\rho^2\). The radial derivative is supported where \(\rho\leq|y-x|\leq2\rho\), so the full curvature term \(2\partial_r\chi_x/r\) satisfies the latter bound.

Put \(a_j=\|\nabla^jv(t_1)\|_{L^3(S_0)}\). Apply (2.3) to \(g=\chi_x\nabla^jv(t_1)\), \(j=0,1,2\). The cutoff ball has volume \(8v_3\rho^3\); Hölder converts each local \(L^3\) norm into \(L^2\) with factor \((8v_3\rho^3)^{1/6}\). The exact product gradient and Laplacian give the respective bounds

after removing that common volume factor. The \(\sqrt3\) comes from retaining all three diagonal terms of the Laplacian inside the complete second-derivative tensor. Thus

This is a proved evaluation estimate with two extra derivative orders. No critical first-derivative embedding is used.

2.2. Forward heat propagation across the actual distance

For \(d,S>0\), define

For \(s>0\), splitting the Gaussian exponent into equal halves gives

One half provides the exponential factor at distance \(d\). The remaining kernel is \(2^{3/2}H_{2s}\), whose \(L^{3/2}\) norm is \(2s^{-1/2}/(3\sqrt\pi)\) by the full Gaussian integral. The logarithmic derivative of the time factor is \(-1/(2s)+d^2/(8s^2)\), so its maximum over \(0<s\leq S\) occurs at \(s_*\). Its limit at zero is zero.

Split the heat convolution for \(\nabla^jv(t)\) into its data in \(S_1\) and outside \(S_1\). The first part is bounded by \(B_j\), using positivity and total Gaussian mass one. Hölder and (2.9) bound the second. For all \(t\in[t_1,t_2]\) and \(x\in S_2\), including \(t=t_1\),

The full original field enters the second term. Heat entering from outside the annulus has been estimated explicitly.

3. Choosing all targets before choosing the annulus

Let \(C_1,C_2,C_3\) be exactly the positive stretching constants of equation (5.16) in the moving-annulus lesson. Fix arbitrary \(h,\gamma,\sigma,\lambda_5,\lambda_8, \lambda_9,\rho,R_0>0\) and \(c\geq8(C_3+1)\). Keep \(\Lambda\geq2\), integer \(m\geq4\), and \(N_*>0\) as above. All of these are choices in the original equation.

Let

Choose \(\alpha\) to be the minimum of one and the values \(e_0/c_5\), \(\sqrt{e_0/c_8}\), \(e_0/c_9\) whose denominators are positive. A zero coefficient supplies no restriction. This rule always gives \(\alpha>0\), with

Use \(\alpha\) for this target; the moving tent below is denoted \(\eta\). They are different, fully specified objects. Define

These positive tolerances control the initial density, the time-integrated curl of \(f_a\), and the time-integrated square of \(f_b\), respectively. The number \(D_0\) bounds the full boundary recession computed from the actual speed.

To fix the incoming heat contribution, put

If \(V_{\rm in}=0\), set \(d_{\rm heat}=2\sqrt{S_{\rm heat}}\). If \(V_{\rm in}>0\), define its positive square by

For every \(d\geq d_{\rm heat}\), \(\mathcal T(d,S_{\rm heat})V_{\rm in}\leq\alpha/2\). Indeed \(d^2\geq4S_{\rm heat}\) places the maximum in (2.8) at \(S_{\rm heat}\). If the displayed logarithm is positive, substitution of its lower bound for \(d^2\) proves the claim. If it is nonpositive, the prefactor is already at most \(\alpha/2\). The zero case gives a zero product directly.

Finally choose the actual minimum radius

Each entry has an explicit role: the requested lower radius; the tent height, boundary recession and sphere-flux budget; the initial evaluation-ball margin; and the forward heat margin. These choices precede the shell selection and do not depend on its outcome.

4. One annulus for the initial data and both forces

Take the finite integer and radii

Adjacent shells meet at one sphere and have disjoint interiors. Their boundary spheres have zero three-dimensional measure. The number whose initial density integral is larger than \(\varepsilon_0\) is at most \(\mathcal B/\varepsilon_0\). The number whose \(\int_{J_\delta}\int_{S_k}|f_b|^2\) is larger than \(\varepsilon_b^2\) is at most \(F_b^2/\varepsilon_b^2\). Both assertions follow by summing the nonnegative integrals over the disjoint interiors.

For the other force, set \(a_k^f(t)=\|\nabla\times f_a(t)\|_{L^2(S_k)}\) and \(A_k^f=\int_{J_\delta}a_k^f(t)\,dt\). The needed sum is

To prove it without exchanging an \(L^1\) norm for an \(L^2\) norm, regard \((a_k^f(t))_{k=0}^{N-1}\) as a vector in Euclidean \(N\)-space. For every unit vector \(b\), \(b\cdot\int a^f\leq\int|a^f|\) by pointwise Cauchy–Schwarz. Taking the supremum over \(b\) gives \(|\int a^f|\leq\int|a^f|\). Disjointness gives \(|a^f(t)|^2\leq\|\nabla\times f_a(t)\|_2^2\). Together these prove (4.2). The number with \(A_k^f>\varepsilon_a\) is therefore at most \(F_a^2/\varepsilon_a^2\).

The integer \(N\) exceeds the sum of these three upper bounds for bad shells. At least one index is good for all three tests. Fix it, write its radius as \(R\), and call its shell \(S_0\) in the notation of Section 2. We have proved

One spatial shell works on the whole time interval. No force has been restricted in the equation; (4.3) only estimates its values on that shell.

Set \(A=\Lambda^{-m+2}R\), \(B=\Lambda^{m-2}R\), and \(J=[t_1,t_2]\). The inner shell \(S_2\) is \(\{A\leq|x|\leq B\}\). All its heat maxima in (2.10) now satisfy

Indeed each \(a_j\leq\varepsilon_0^{1/3}\), so (2.7) is at most \(K_\rho\varepsilon_0^{1/3}\leq\alpha/2\). Equations (3.5)–(3.6) make the heat term at most \(\alpha/2\), using \(\nu(t_2-t_1)\leq\nu L\) and \(\|\nabla^jv(t_1)\|_3\leq V_{\rm in}\). The two distance conditions of Section 2 follow directly from the last two entries of (3.6).

5. The complete seven-term energy budget

On the fixed enclosing shell \(S_2\), use the original speed law

Equations (1.7)–(1.8) give \(0\leq q(t)\leq D_0\leq A\). Also \(h\leq A\) and \(B/A=\Lambda^{2m-4}\geq16\). For \(a_0\in[A,2A]\), \(b_0\in[B/2,B]\), define

The positive part in this tent means the maximum of zero and the displayed minimum. All tents are admissible by the preceding geometric bounds. The curvature and all four sphere contributions to its distributional Laplacian stay in \(Y_3=(\nu/2)\int|z|^2\Delta\eta\), as proved in Section 2 of the moving-annulus lesson. Its absolute averaging argument, equation (3.3), selects one pair \((a_0,b_0)\) such that

This selection is compatible with the earlier shell selection: the shell is already fixed, and (5.1) is independent of the two radii being averaged. Moreover the initial tent energy obeys \(E(t_1)\leq h\varepsilon_0\leq e_0\), since pointwise \(|\nabla\times w|^2\leq2|\nabla w|^2\).

For completeness specify every receiving coefficient. All local norms in the next display are on \(S_2\):

In equation (6.10) of the moving-annulus lesson they give

The original force pairings localize to these norms exactly: their factors \(\eta\) and \(\nabla\eta\) vanish outside \(S_2\). In particular this does not replace \(f\) by a cutoff force in (1.1). The inequalities already proved there give, while \(E\leq E_*\),

We now calculate its full budget rather than assume it is small. First (4.4), \(W\leq W_0\), and (4.3) give

Next the actual heat field obeys

The curl and differentiated-curl pointwise inequalities give the \(\sqrt2\) factors. The other step is the direct estimate \(\|g\|_6^2\leq\|g\|_\infty\|g\|_3\), obtained by integrating \(|g|^6\leq\|g\|_\infty^3|g|^3\). Combining (5.8) with (1.5)–(1.6), and integrating the original physical time powers, gives

Both endpoints remain in the first inequalities; only then is \(t_1-t_b\geq\delta\) used. The two force terms satisfy

There are seven contributions: the initial energy, the absolute sphere/curvature heat flux, the three terms of (5.9), and the two terms of (5.10). Each is at most \(e_0\). Gronwall applied to (5.6) therefore bounds the energy by \(7e_0e^{\mathcal A_0}=7E_*/8<E_*\) until a putative first crossing of \(E_*\). Continuity excludes that crossing: at its time the same strict bound would hold. This proves on the whole actual interval \(J\) that

For the second line, integrate (5.6), use the same seven terms and \(\int\mathfrak aE\leq(7E_*/8)\mathcal A_0\), and retain the nonnegative final energy before bounding it below by zero. The last line follows because every point in the displayed common plateau has \(\eta=h\). Its width is at least \(2A\), as also checked in the next section.

We have now derived every smallness input of the finite annular energy argument from the actual solution and force. Formula (4.3) gives a finite explicit upper radius as well as a lower one; no unspecified radius choice remains.

6. Exact local velocity bounds on the selected annulus

Take the fixed open shell

It lies in the common plateau at every time. Moreover \(r_+-r_-=B/2-2A-2D_0-2h\geq2A\), using \(B\geq16A\), \(D_0\leq A\), \(h\leq A\). We may choose the explicit physical length \(d=A/8\); then \(6d<r_+-r_-\). No extra enlargement of \(R\) is needed.

Define a smooth one-sided transition by

Flatness of \(\theta\) at \(1/2\) and \(1\) proves smoothness at both junctions. Set, for \(k=0,1\),

The cutoff \(\zeta_0\) vanishes outside the closure of \(\Omega_0\), equals one on \(r_-+d\leq r\leq r_+-d\), and all its nonzero values lie in \(\Omega_0\). The support of \(\zeta_1\) is in that plateau, and \(\zeta_1=1\) on \(\Omega_2\). The two transition bands in each product are disjoint, so there is no product of two nonzero radial derivatives. The complete radial Laplacian gives \(|\nabla\zeta_k|\leq G_1\) and \(|\Delta\zeta_k|\leq G_2\). In particular its curvature term retains the original lower radius \(r_->0\).

Figure 1. Original radial coordinates for one admissible geometry, \(\Lambda=2\), \(m=4\), \(R=16\), \(A=4\), \(B=64\), \(h=1\), \(D_0=2\). The first panel shows the three heat shells on a logarithmic radial axis. The second shows the common plateau \([11,29]\) and the exact two cutoffs with \(d=1/2\) on a linear radial axis. These are specified geometry and cutoff functions, not sampled solution energies. Equations (2.5) and (6.1)–(6.3) prove the inclusions. Reproducible figure source.

Let

The first two follow directly from (5.11) because the tent is identically \(h\) on this fixed shell.

6.1. The cutoff div-curl identities

For every compactly supported \(H^1\) vector field \(X\), the full Fourier identity gives

Indeed \(|\xi\times\widehat X|^2+|\xi\cdot\widehat X|^2 =|\xi|^2|\widehat X|^2\); all three differentiated quantities have the same \(2\pi\) multiplier. Apply (6.5) to \(X=\zeta w\), using \(\nabla\times X=\zeta z+\nabla\zeta\times w\) and \(\operatorname{div}X=\nabla\zeta\cdot w\). The two-term square inequality and \(|a\times w|^2+|a\cdot w|^2=|a|^2|w|^2\) yield

For the second inequality substitute \(\zeta\nabla w=\nabla(\zeta w)-w\otimes\nabla\zeta\) and apply the same square inequality. The velocity term is essential to this local estimate.

Use \(\zeta_0\) in its second line. It bounds the gradient on the support of \(\zeta_1\), uniformly in time, by \(P_1^2=4Z_0^2+6G_1^2\overline W^2\). Next apply that same line with \(\zeta_1\) to each field \(\partial_kw\). Its divergence is zero and its curl is \(\partial_kz\). Summing the three inequalities and integrating gives

Every derivative of \(\zeta_1\) is supported where \(\zeta_0=1\), so the stated \(P_1\) controls all the receiving gradient terms.

6.2. A full Fourier interpolation calculation

For a compact \(H^2\) vector field \(F\), set \(P=\|\nabla F\|_2\), \(Q=\|\Delta F\|_2\). Split its full Fourier integral at any radius \(R_f>0\). The two Cauchy–Schwarz estimates give

For the low-frequency term, \(\int_{|\xi|\leq R_f}|\xi|^{-2}\,d\xi=4\pi R_f\) and the gradient multiplier is \(2\pi|\xi|\). For the high-frequency term, \(\int_{|\xi|>R_f}|\xi|^{-4}\,d\xi=4\pi/R_f\) and the Laplacian multiplier is \(4\pi^2|\xi|^2\). These retain both parts of the original Fourier integral. If \(P,Q>0\), choose \(R_f=Q/(2\pi P)\); the two terms are equal and therefore

If \(P=0\), the \(L^2\) field is a constant and hence zero. If \(Q=0\), its Fourier transform vanishes off the origin and is an \(L^2\) function, hence zero. Thus both exceptional cases satisfy (6.9). Approximation and (6.8) give the same continuous representative and inequality for every such \(H^2\) field.

Apply this to \(F=\zeta_1w\). The first line of (6.6) gives \(P^2\leq P_0^2:=2Z_0^2+2G_1^2\overline W^2\). The complete product Laplacian is

Here \(\Delta w=-\nabla\times z\) uses the original divergence-free condition. The three-term square inequality and \(|\nabla\times z|^2\leq2|\nabla z|^2\) prove

Finally \(\zeta_1=1\) on \(\Omega_2\). Integrating (6.9), and adding back the actual heat field bounded in (2.10), proves

This is the original local velocity, with all cutoff and heat terms. In particular the fourth root has been taken explicitly. A bound on squared gradient norms cannot be read as the same power bound on this velocity norm.

7. The complete localized vorticity heat equation

The vorticity of (1.1) satisfies \((\partial_t-\nu\Delta)\omega=\nabla\times(u\times\omega+f)\). Choose a smooth spatial cutoff \(\psi\) with support inside \(\Omega_2\) and a smooth time cutoff \(\chi\). Keep \(\Psi(t,x)=\chi(t)\psi(x)\), \(\Omega=\Psi\omega\), and \(\mathcal F_v=u\times\omega+f\). The product rules give exactly

To verify all signs, use \(\nabla\times(\Psi\mathcal F_v) =\Psi\nabla\times\mathcal F_v+\nabla\Psi\times\mathcal F_v\). Also \(-2\nu\nabla\Psi\cdot\nabla\omega =-2\nu\sum_j\partial_j((\partial_j\Psi)\omega) +2\nu(\Delta\Psi)\omega\). The original product Laplacian contributes \(-\nu(\Delta\Psi)\omega\), leaving the positive \(\nu\Delta\Psi\) in (7.1).

At an original starting time \(s_0\in J\), the full Duhamel formula is

The curl on the second line acts on the vector convolution; in components its kernel is the corresponding signed Levi-Civita sum of first Gaussian derivatives. All terms are classical for the stated smooth fields, with the displayed integrals also defining their distributional versions.

If \(\chi(s_0)=0\), the initial term is exactly zero. Otherwise it remains. Any estimate of its derivatives from an initial spatial norm retains the positive physical time \(t-s_0\). In particular the original rough force spaces in (1.2) do not authorize replacing their convolution terms by unstated pointwise bounds on \(\nabla\times f\).

Formula (7.2), with (6.12), is the next exact equation to estimate on a smaller region and a later time interval. It retains the direct force flux and the force in the boundary cross product. Exercise 5 proves the precise heat-potential map between these two terms and the original localized curl force.

8. Source comparison and solved exercises

The human comparison is Tao, arXiv:1908.04958v2, original author file article.tex: lines 660–699 select the initial time and annulus and propagate the linear heat data; lines 700–858 develop annular energy; lines 859–892 pass toward pointwise regularity. The source equation is unforced and has unit viscosity. The earlier two lessons retain visible corrections at their exact source locations. The present lesson supplies the receiving original-force annulus construction and complete local velocity proofs; it does not attribute this expanded argument verbatim to the source.

The initial time and all root powers are explicit here. The pointwise bounds printed on a closed time interval in the last source passage require a positive margin when proved by heat smoothing from initial spatial norms. We have proved the time-integrated velocity estimate (6.12) on the stated interval and the exact heat equation (7.2); a subsequent pointwise iteration is still required. No general endpoint conclusion or novelty claim is inferred from these finite steps.

Exercise 1: why the first force is counted in squares

Prove the square-sum bound (4.2) from finite-dimensional duality. Explain why a linear bound \(\sum_k A_k^f\leq F_a\) cannot be substituted. Check the shell count when all three global quantities vanish.

Solution. For a unit vector \(b\in\mathbb R^N\),

The supremum over unit \(b\) is the Euclidean norm of \((A_k^f)_k\), proving (4.2). The linear substitute is false already for two disjoint shells carrying equal nonzero fields at the same times. For example choose two smooth compact vector fields \(g_1,g_2\), one in each shell, with \(\|\nabla\times g_i\|_2=1\), and a nonnegative nonzero smooth time function \(a(t)\). Such fields exist by taking the curl of a nonconstant compact vector potential and dividing by its nonzero curl norm. Put \(f_a(t,x)=a(t)(g_1(x)+g_2(x))\). Then \(A_1^f=A_2^f=\int a\) while \(F_a=\sqrt2\int a\). Their sum is strictly larger than \(F_a\), whereas their squared sum is exactly \(F_a^2\). If \(\mathcal B=F_a=F_b=0\), (4.1) gives \(N=1\). All three nonnegative shell integrals vanish, so this one shell is good. No division by a zero global quantity occurs.

Exercise 2: verify the Gaussian margin, including zero data

Prove that (3.5) gives \(\mathcal T(d,S_{\rm heat})V_{\rm in}\leq\alpha/2\) for all \(d\geq d_{\rm heat}\). Why is it valid to use the larger time \(S_{\rm heat}\) for the actual heat interval?

Solution. Set \(Q=4V_{\rm in}/(3\sqrt\pi\alpha\sqrt{S_{\rm heat}})\). If \(V_{\rm in}>0\), the distance choice gives \(d^2\geq4S_{\rm heat}\), so (2.8) uses \(s_*=S_{\rm heat}\), and

When \(Q\leq1\), this is at most \(\alpha/2\). When \(Q>1\), (3.5) gives \(d^2/(8S_{\rm heat})\geq\log Q\), with the same conclusion. If \(V_{\rm in}=0\), the left side is zero and the separate definition avoids \(\log0\). The expression \(\mathcal T(d,S)\) is the maximum of a fixed nonnegative function over \(0<s\leq S\), so it is nondecreasing in \(S\). The actual interval has \(\nu(t_2-t_1)\leq S_{\rm heat}\). Thus the same bound holds on the full actual interval, including its initial time where the tail factor tends to zero.

Exercise 3: the full common plateau and the cutoff terms

Verify the width bound in Section 6. Show that every cutoff derivative used for the local velocity lies inside the original common plateau, and derive \(G_2\) without removing the curvature term.

Solution. Since \(B\geq16A\), \(D_0\leq A\), and \(h\leq A\),

For any time \(q(t)\leq D_0\) and any selected \(a_0\leq2A\), \(b_0\geq B/2\), the plateau \([a_0+q+h,b_0-q-h]\) contains \([r_-,r_+]\). The transition bands of \(\zeta_0\) are \((r_-,r_-+d)\) and \((r_+-d,r_+)\); those of \(\zeta_1\) are \((r_-+2d,r_-+3d)\) and \((r_+-3d,r_+-2d)\). The width bound makes each pair disjoint. The latter two are within the region where \(\zeta_0=1\). On any one band the other factor in (6.3) is one, so the complete radial formula gives

Both terms of \(G_2\) are used in (6.11). They also show explicitly why a smaller local shell does not remove the velocity contributions from derivatives of the cutoff.

Exercise 4: the local norm and a genuine time margin

Derive the exponent of time and the fourth root in (6.12). Then give an explicit smooth-field example showing that an \(L^4\)-time maximum bound alone does not control the supremum at every time.

Solution. At each time (6.9) gives \(\|\zeta_1w(t)\|_\infty^4 \leq(4/\pi^4)P_0^2Q(t)^2\). Integrating, using (6.11), and taking the fourth root proves its first line. Minkowski in \(L^4(J)\) and the uniform heat bound give the second, with precisely \(|J|^{1/4}V_0^*\).

For the time-norm assertion, choose a nonzero smooth compact solenoidal field \(b(x)\) supported in the interior of \(\Omega_2\), scaled so \(\|b\|_\infty=1\). Choose a nonzero smooth function \(a\) supported in \((-1,1)\) and an interior time \(t_c\in J\). For sufficiently large integers \(n\), set

The time supports lie in \(J\), and direct substitution gives \(\|g_n\|_{L^4_tL^\infty_x}^4=\int_{-1}^1|a(s)|^4\,ds\), independent of \(n\). Their time suprema are \(n^{1/4}\|a\|_\infty\), which diverge. This example tests the implication between norms; it does not assert a Navier–Stokes solution with fixed force. The equation (7.2) and its actual force terms provide the extra information needed for the next pointwise argument.

Exercise 5: the exact force heat potential

Show how the two force contributions in (7.2) combine into the heat potential of the original localized curl force. Give its derivative bound when that curl has the stated actual derivative norms, retaining the original viscosity.

Solution. Define the full force contribution

The product identity \(\nabla\times(\Psi f)=\Psi\nabla\times f+\nabla\Psi\times f\) and spatial integration by parts in the convolution prove the exact map

For smooth data this is direct. For the force in (1.2), approximation or testing against compact smooth functions proves it as a distributional identity; the derivative on \(f_b\) is not silently made into an \(L^2\) function. The two terms in (8.6) retain exactly that distributional meaning and every boundary contribution.

More quantitatively, suppose for a specified integer \(m\geq0\) that the actual tensor \(g_m(s)=\nabla^m(\Psi\nabla\times f)(s)\) is in \(L^1([s_0,t];L^\infty)\). Commuting derivatives with the exact Gaussian and using its positivity and unit mass gives

This estimate makes no change of viscosity: it uses \(\|H_{\nu(t-s)}\|_1=1\) for the original \(\nu\). An alternative with only a specified spatial \(L^p\) norm retains the complete heat factor. For \(1\leq p\leq\infty\) and its conjugate \(p'\), Gaussian scaling gives

whenever the right side is finite. All these statements are proved operator bounds for the indicated actual inputs. The pointwise force data in (8.8) or (8.9) have not been inferred from (1.2). This exact receiving object is what must be estimated when the nonlinear interior argument continues.

The mathematical text and original figure are dedicated to CC0 1.0. Cited author sources retain their respective terms.