Leading-profile stress and exact moment restoration

The following arguments establish the coordinate, stress, moment and finite-dimensional correction identities used in the leading-profile construction. The leading-profile construction, including all six properties of Theorem 4.6, is supplied by V0–V7 and the O/H/J/Q proofs cited there. D0–D6 construct the first positive-order axis correction and prove the fixed-interval solution step. The global positive-order base is constructed in Z0–Z8. The complete theorem and its proof dependencies are stated in the theorem construction.

Source: OpenAI, Finite Time Blowup for Navier–Stokes, original PDF, SHA256 0e779481c4da40bd28d1e642e1d8ca57447d129610df28dfa5a11e9af8ae228f. The source locators are original pages 24–31, 34–35 and 126–128: (4.1)–(4.23), Lemma 4.7, Lemmas A.1–A.2 and Corollary A.3.

G0. Coordinates, regularity and incompressibility

Keep the original parameters and variables

For each \(\tau>0,z\in\mathbb R\), the equation \(q-z^2q^{2h}=\tau\) has exactly one solution \(q>|z|^{1/D}\). Indeed its left side vanishes at the lower endpoint, tends to infinity, and has derivative \(L\ge1-2h>0\) on that half-line. When \(z=0\), the endpoint is zero and the same argument applies. Thus \(|\eta|<1\). The implicit function theorem gives smooth \(q,\eta\) at every \(t<1\). Differentiation gives

The identity \(L-2D\eta^2=d\) proves the middle axial formula. For \(\mathcal D_X=X\partial_X\), any real \(b\), and a smooth profile \(f\), the chain rule therefore gives

All endpoint statements in \(\eta\) mean continuous one-sided derivatives.

Retain the fixed amplitude factor \(\mathsf C>1\) and set

The axis data are the smooth functions \(F=\phi/\mathsf C,U,v_0,\Pi\), with \(\phi>0\), on each finite closed \((X,\eta)\) rectangle. This does not require \(E\) to be smooth as a function of \(X\) at zero. In Cartesian coordinates the exact field is

These expressions prove smoothness across the axis for \(t<1\).

The radial average is \(\mathcal A_Xf=\int_0^1f(aX,\eta)\,da\). Differentiation under this fixed finite integral proves its smooth extension with value \(f(0,\eta)\). Since \(A+D=1\), incompressibility is exactly

Integrate from zero, use \(V_0(0,\eta)=0\) and \(\int_0^XxU_X\,dx=XU-\int_0^XU\,dx\). This gives the original formula, including its sign,

The second identity equates the coefficients \(\sqrt{2X}\Pi_X\) and \(E^2/\sqrt{2X}\) of \(\partial_rp^{(0)}\) and \((u_\theta^{(0)})^2/r\), both with factor \(q^{-2A-1/2}\). It cancels precisely those radial terms. The remaining radial terms are retained in G1.6 below.

G1. Exact stress, including both kinds of viscosity

Define the original quantities

Using G0.3 and \(V_0L=X(2\eta U+W-1)\), direct collection of the three derivatives proves the complete material-derivative formula

For \(b=-h,f=H\), this is \(-q^{-h-1}HS_q/L\). For \(b=-A,f=U\), adding \(\partial_zp^{(0)}=q^{-A-1}Z_{-2A}\Pi\) gives \(-q^{-A-1}S_n/L\). The equality of pressure powers uses \(2A+D=A+1\).

The two radial equations have the unique axis-regular solutions

Indeed the integrating factors are \(XH\) and \(X\). Their homogeneous constants would give \(X^{-2}\) and \(X^{-1}\) singularities, since \(H=2X\phi/\mathsf C\) and \(\phi(0,\eta)>0\). The displayed integrals are smooth and have values \(S_q(0,\eta)/2\) and \(S_n(0,\eta)\) at zero.

For \(r>0\), write \(D_\theta=\partial_r+2/r\), \(D_z=\partial_r+1/r\). The inviscid angular residual is \(-q^{-A-1}ES_q/L\), because angular momentum must first be divided by \(r\). Its negative radial primitive is

Here \(\rho\,d\rho=q\,dx\), \(\rho=\sqrt{2qx}\), and \(\int_0^X\sqrt{2x}ES_q\,dx=XHQ_s\); thus all powers and factors of two are retained. The axial calculation similarly gives \(r^{-1}\int_0^r\rho q^{-A-1}S_n/L\,d\rho=q^{-A-1/2}XN_s/(L\sqrt{2X})\).

Put

The radial viscous operators are exactly \(D_\theta(\partial_ru_\theta-u_\theta/r)\) and \(D_z\partial_ru_z\). Subtracting them from the inviscid residual proves the last line and the stress sign. In particular, \(p_s\) is the integrated inviscid contribution, not the pressure. Axial viscosity has not vanished: its exact profile is \(\partial_z^2(q^{-A}f)=q^{-A-2D}Z_{-A-D}Z_{-A}f\), for \(f=E,U\).

For completeness set \(B_r=V_0/\sqrt{2X}\), so \(u_r^{(0)}=q^{-1/2}B_r\). After the centrifugal-pressure cancellation, the full radial residual is

This follows from G1.2 and \((\partial_{rr}+r^{-1}\partial_r-r^{-2})(q^{-1/2}B_r)=q^{-3/2}(2X(B_r)_{XX}+2(B_r)_X-B_r/(2X))\). It retains every radial term. Its first power is larger than the cancelled leading power \(q^{-3/2-2h}\) by \(q^{2h}\); its axial-viscosity power differs by \(q^{4h}\). In G1.5 the axial viscosity differs from the leading tangential power \(q^{-A-1}\) by \(q^{2h}\). These are exact fixed-profile comparisons, not physical derivative estimates.

The stress vanishes exactly when \(Q_s=-2LF_X/F\) and \(N_s=-2LU_X\). Substitution in G1.3 gives

Conversely these equations make the displayed expressions for \(Q_s,N_s\) solve their radial equations and be smooth at the axis. Uniqueness in G1.3 then proves zero stress. Thus the equivalence uses the actual axis constants.

For any original viscosity \(\nu>0\), retain \(y=x_\nu/\sqrt\nu\), \(u_\nu(x_\nu,t)=\sqrt\nu u(y,t)\), \(p_\nu=\nu p(y,t)\), and \(T_\nu=\nu T(y,t)\). Direct differentiation gives \(\mathcal R_\nu(u_\nu,p_\nu)=\sqrt\nu\mathcal R_1(u,p)(y,t)\), \(D_{j,\nu}T_{j,\nu}=\sqrt\nu D_jT_j\), and \(\nu\partial_{z_\nu}^2u_{j,\nu}=\sqrt\nu\partial_z^2u_j\). G1.5 therefore retains exactly the same physical stress identity at every \(\nu\). The original coordinates are \(X_\nu=r_\nu^2/(2\nu q_\nu)\), \(\eta_\nu=z_\nu/(\sqrt\nu q_\nu^D)\), with \(q_\nu-z_\nu^2q_\nu^{2h}/\nu=\tau\). No viscosity factor is discarded.

G2. All five cumulative moments and the pressure datum

Use the original moments and prescribed axis pressure:

The factor \(1/2\) in \(S\) and the weight \(1/(2x)\) in \(C_p\) remain distinct. Since \(XW=X-2D\eta M-dM_\eta\), \(\partial_X(XW)=1-2D\eta U-dU_\eta\). Integration by parts in \(-WXH_X\) yields

The lower boundary vanishes because \(XWH=O(X^2)\). For the axial terms other than pressure, the identical integration by parts gives

Indeed the coefficient of \(M\) is \(1-A=D\), that of \(\int U^2\) is \(2(A-D)\eta=4h\eta\), and the two copies of \(-d\int UU_\eta\) combine to the derivative shown. No axial boundary value has been omitted: \(XWU=O(X)\) at zero.

Writing \(K_E=\int_0^XE^2/2\,dx\), integration of \(x\Pi_x=E^2/2\) gives \(\int_0^X\Pi\,dx=X\Pi-K_E\). Hence the entire pressure contribution is

The axis pressure is included in \(X\Pi\). Combining G2.2–G2.4 and dividing by the exact integrating factors proves

These are original (4.16), derived without discarding pressure or an integration constant.

G3. Exact exterior preservation and stability without radial derivative loss

For two profiles write \(\Delta f=f_2-f_1\) and \(\mathbf m_i=(M_i,I_i,J_i,S_i,C_{p,i})\). Their exact moment differences are

where every integral has measure \(dx\) and parameter \(\eta\) fixed. If the profiles agree for \(X\ge X_h>0\), all these integrands vanish there. Thus \(\Delta\mathbf m(X,\eta)=\Delta\mathbf m(X_h,\eta)\). When this latter function is zero for every \(\eta\), all its parameter derivatives vanish too. With the same \(\Pi_{\rm ax}\), G0.7 and G2.5 give equality of \(\Pi,V_0,Q_s,N_s\) on the entire exterior. Equality of the profiles on that interval also gives equality of their radial derivatives, including their common one-sided value at \(X_h\). G1.5 then proves equality of \(p_s,a,b_s,\mathcal T_0\) there. This proves the full preservation statement of Lemma 4.4(i).

The pressure datum is essential in this calculation. If instead its difference is \(g(\eta)\) while the five moments and exterior profiles agree, then

These exact terms identify what preservation requires; there is no inference that the source neglects them.

The kernel of this pressure-to-stress map can also be determined exactly. On \(-1<\eta<1\), equality of the stresses is equivalent to \(dg'=4A\eta g\), since \(X/(L\sqrt{2X})>0\) for \(X>0\). Product differentiation gives \((d^{2A}g)'=d^{2A-1}(dg'-4A\eta g)=0\). Thus every such difference is \(g(\eta)=c\,d^{-2A}\), with one constant \(c\). Its physical pressure is \(q^{-2A}g=c(qd)^{-2A}=c(1-t)^{-2A}\), spatially constant at each \(t<1\), so it changes no pressure gradient. In the source's smooth closed profile class, boundedness at either endpoint and \(A>0\) force \(c=0\). Hence the common axis datum is necessary as well as sufficient for exterior stress preservation within that class. This does not exclude the usual spatially constant pressure freedom in a larger physical class. The complete teaching calculation is Exercise 2 of the stress and moment lesson.

On \(\mathcal R=[X_0,X_1]\times[-1,1]\), with \(0<X_0<X_1\), set \(\|f\|_j=\max_{0\le r\le j}\sup_{\mathcal R}|\partial_\eta^rf|\), using the componentwise maximum for tuples. Assume the source bounds \(E_i\ge e_{\min}>0\) and finite common \(k+1\) norms. The exact difference expressions are

For clarity, let \(A_I=(1-h)I-D\eta I_\eta-dJ_\eta+2(h-D)\eta J\) and \(A_M=D(M-\eta M_\eta)+4h\eta S-dS_\eta\), retaining every summand. Then

This exposes the one differentiated moment in each term. To bound the other factors explicitly, use \(\partial_\eta^r(fg)=\sum_{a=0}^r\binom ra f^{(a)}g^{(r-a)}\), giving \(\|fg\|_r\le2^r\|f\|_r\|g\|_r\). If \(|f|\ge c>0\), its reciprocal derivatives are recursively

Start with \(\|f^{-1}\|_0\le c^{-1}\); this recursion gives a finite polynomial bound in \(c^{-1}\) and \(\|f\|_r\) for every \(r\). Apply it to \(E_i,H_i,L\), whose lower bounds are \(e_{\min},\sqrt{2X_0}e_{\min},1-2h\). Product differentiation of G3.4 therefore proves, with exactly the source dependence,

Here \(C_k\) depends only on \(k,h,X_0,X_1,e_{\min}\), the common input bounds, and \(\|\Pi_{\rm ax}\|_{k+1}\). No radial derivative of a difference occurs. This estimate does not control the changes of \(a,b_s\) without the corresponding radial derivatives; G1.5 retains that dependence in the full stress.

For the original joining radius \(\rho>0\), the exact comparison, not a replacement of the original objects, is

Changing the integration variable in each of G2.1 proves the five factors. Substitution in G2.5 and G1.5 cancels the same factors, so G3.6 holds in a fixed \(x\) rectangle with a constant independent of \(\rho\) whenever the listed hatted bounds are uniform. The shears obey \(a(\rho x)=\widehat a(x)\), \(b_s(\rho x)=\widehat b_s(x)\), whereas the full stress obeys

Both different powers are necessary and are retained.

G4. Invertibility and the approaching-exponent bound

Let distinct real \(\alpha_1,\ldots,\alpha_m\) and nonnegative nonzero smooth \(\beta_j\) have supports in ordered disjoint compact subintervals \(I_1<\cdots<I_m\) of \((0,\infty)\). Define the actual matrix

A nonzero combination of \(n\) distinct powers has at most \(n-1\) distinct positive zeros. Here is the induction: divide by its least power, which does not change positive zeros. If \(n\) zeros exist, Rolle's theorem gives \(n-1\) zeros of the derivative. Unless that derivative is zero identically, it is a combination of at most \(n-1\) distinct powers and so has at most \(n-2\) zeros by induction. If it is zero identically, the divided function is a nonzero constant and has no zeros. The induction starts with one nonzero monomial. Consequently the evaluation determinant \(\det[x_j^{\alpha_i}]\) is nonzero when \(0<x_1<\cdots<x_m\). It has a constant sign on this connected domain. Expanding the determinant by permutations and applying finite Fubini gives

There is no cancellation: the product is positive on a product of open sets of positive measure. This proves the matrix assertion of Lemmas 4.7 and A.1.

For a smooth compact family preserving these hypotheses, the determinant has a positive absolute minimum. The cofactor formula bounds \(B^{-1}\); all fixed parameter derivatives follow recursively from

The assertion requires smooth dependence of the integrals, as in the source's smooth compact family. It supplies no uniform inverse at colliding exponents. For logarithmic coordinates the exact substitution is \(\int e^{\alpha_i y}\gamma_j(y)\,dy=\int x^{\alpha_i}\gamma_j(\log x)x^{-1}\,dx\). The Jacobian is part of the bump and does not change the ordering or sign.

The loss needed later is explicit. For fixed translated logarithmic bumps \(\beta(y-c_1),\beta(y-c_2)\), let \(\Delta=c_2-c_1>0\) and \(b(s)=\int e^{st}\beta(t)\,dt>0\). The original matrix and its determinant are

The matrix displayed in original (A.1) is obtained by multiplying row \(i\) by \(e^{-s_ic_1}\); recovering the original inverse multiplies by this same diagonal factor on the right. For slopes in a fixed compact interval, those factors and the positive \(b(s_i)\) have two-sided bounds. The mean value theorem gives \(|e^{s_2\Delta}-e^{s_1\Delta}|=\Delta e^{\xi\Delta}|s_2-s_1|\). Cofactors are bounded, hence the original inverse is \(O(|s_2-s_1|^{-1})\).

For weights \(1,x^{-\lambda}\) and fixed ordered bumps \(\beta_1,\beta_2\), set \(m_j=\int\beta_j>0\) and \(\mu_j=m_j^{-1}\int\log x\,\beta_j(x)\,dx\). Separation gives \(\mu_1<\mu_2\). Taylor's formula on their compact positive supports, with its bounded second derivative \((\log x)^2x^{-\lambda}\), gives

The remainder constant is finite on a fixed bounded \(\lambda\) interval. For sufficiently small positive \(\lambda\) it is at most half the magnitude of the displayed linear term, proving the required inverse bound \(O(\lambda^{-1})\). This is an analytic bound, with no numerical matrix sampling.

G5. Exact five-moment correction and the nonlinear solve

On a correction patch in \(X=\rho x\), keep the source base \(U_0=u_c(\eta)\), \(E_0=e_*f_*(\eta)x^\alpha\), where \(e_*>0\), \(f_*>0\). Use two fixed ordered axial bumps and three fixed ordered angular bumps, with additive perturbations \(u=\sum_{i=1}^2c_i\beta_i\), \(e=\sum_{j=1}^3c_{j+2}\gamma_j\). Each block has ordered disjoint supports inside the patch. The original five moment increments, integrated over the entire patch, are exactly

In particular \(ue,u^2,e^2\) remain in the actual moment map. Introduce the explicitly invertible row map \(\mathscr L_\eta\) by

Every factor is retained. Its inverse sends \(d=(d_1,\ldots,d_5)\) to

Thus solving the transformed equation is proved equivalent to restoring the original moments; it does not replace them. The linear map is block diagonal with the actual entries

The exact remainder in the same row order is

Polarization supplies a smooth bilinear map with this diagonal. G4 proves invertibility precisely when \(\alpha\notin\{-1/2,1/2,3/2\}\): the two axial exponents and the three angular exponents are then distinct. At an excluded value the indicated block actually has equal rows, so this particular additive Jacobian is singular. No conclusion about other constructions is inferred. If \(u_c=0\) and \(u=0\), then G5.1 gives exactly \(\delta M=\delta J=0\), while the three angular coefficients control \((I,S,C_p)\).

Here is the full quantitative solve of Lemmas 4.7 and A.2. In \(C^j([-1,1];\mathbb R^m)\), use the source maximum norm, let \(\nu_j\) be the multiplication norm of \(B^{-1}\), and let \(q_j\) be the bilinear norm of \(\mathscr Q\), including all product-rule constants. For a prescribed smooth discrepancy \(d\), set \(r_j=2\nu_j\|d\|_{C^j}\). The iteration

maps the closed ball of radius \(r_j\) into itself and has Lipschitz constant at most \(1/2\) whenever \(8\nu_j^2q_j\|d\|_{C^j}\le1\). In fact

The case \(r_j=0\) gives the exact zero solution. Completeness and the geometric bound on successive differences give convergence and uniqueness in the specified ball. Applying this for \(j=0\) and one preselected finite \(j=k\) gives the same limit: the iterates are literally identical and convergence in \(C^k\) implies convergence in \(C^0\). Therefore \(\|c\|_{C^k}\le2\nu_k\|d\|_{C^k}\).

Smoothness requires no simultaneous smallness of every derivative norm. At the obtained solution the exact differential is \(D_cF=B+\mathscr Q(c,\cdot)+\mathscr Q(\cdot,c)\). Its product with \(B^{-1}\) differs from the identity by an operator of norm at most \(2\nu_0q_0r_0\le1/2\); the Neumann series converges, so it is invertible. The pointwise implicit function theorem gives a smooth branch, equal to the continuous fixed-point branch by local uniqueness. One-sided endpoint derivatives follow either by smooth local extension of the finite-dimensional data or from successive differentiation and continuity up to the endpoints. The first derivative is exactly

For order \(r\), differentiate \(Bc+\mathscr Q(c,c)=d\) using the finite binomial and multinomial product rules. The terms with \(c^{(r)}\) are exactly \(D_cF\,c^{(r)}\). Every other term uses derivatives of \(c\) of order at most \(r-1\) and derivatives of the given smooth data through order \(r\). The inverse bound and induction give finite continuous derivatives of every order on the compact parameter interval. This proves the claimed smoothness without a hidden all-orders smallness assumption. When \(\mathscr Q=0\), the exact solution is \(B^{-1}d\) with no smallness requirement.

The actual support-scale derivative bound is also retained. For a fixed family \(\delta V=v_*\sum c_j(\eta)\beta_j(X/\rho)\),

For fixed-width logarithmic bumps, \(X\partial_X=\partial_{\log X}\) gives the analogous formula with the logarithmic bump derivatives. Each later preservation of a strict inequality must compare this bound with its actual positive margin. This general solve does not certify any unexamined application margin.

In the source's two applications the exponents are explicit. At \(\alpha=1/10\), the blocks have powers \((0,3/5)\) and \((1/2,1/10,-9/10)\). At \(\alpha=-1/2-\lambda\), they have powers \((0,-\lambda)\) and \((1/2,-1/2-\lambda,-3/2-\lambda)\). G4.5 gives the possible \(\lambda^{-1}\) loss in the first block; the second stays separated near zero. The source fixes positive \(\lambda\) before selecting the later radial frequency. The actual comparisons are derived in O, H and Q for the outer and joining constructions, and in V4 for the final shear correction, with positive lambda fixed before the radial frequency.

G6. The exact stress-cone test

For \(a>0\), set \(t_s=-b_s/a\), \(v_s=a(1+t_s^2)\), \(P_c=p_{s,1}+t_sp_{s,2}\), \(J_c=p_{s,2}-t_sp_{s,1}\). These are the original coordinates, with no unit-vector factors suppressed. Define

For \(P>2\), the roots in \(v\) of \(2(P-v)^2-(v-2)J^2\) are \(v_\pm=P+J^2/4\pm|J|\sqrt{(P-2)/2+J^2/16}\). To check their location, put \(x=P-2>0\). The inequality \(\sqrt{x/2+J^2/16}\ge|J|/4\) gives \(v_-\le P\); and \((x+J^2/4)^2-J^2(x/2+J^2/16)=x^2>0\) gives \(v_->2\). The quadratic is positive below the smaller root and above the larger root. Since \(v_+\ge P\), on \(2<v<P\) positivity is exactly \(v<v_-\). This also covers \(J=0\), where both roots equal \(P\). Therefore, for \(v_s>2\), the source's admissible condition is exactly

Its map to the physical stress is exact:

Thus, where \(F>0\), G6.2 is precisely the positive stress-cone inequality used in the covariance derivation C1–C9. This proves the coordinate connection to that receiving argument, without asserting that the complete profile meets the inequality.

Finally retain the source's compact-set sufficient condition. For a nonempty compact \(K\subset\{(a,b_s,w):a>0\}\), suppose its listed pointwise quantities satisfy \(a-b_sw>0\) and \(2b_sw+b_s^2/a+(a-2)w^2<2\). Set \(c=1-b_sw/a\), \(j=w+b_s/a\), \(v=a+b_s^2/a\). Expansion gives the exact identity

Hence choose \(c_K,\gamma_K>0\), \(B_K\ge1\) with \(c\ge c_K\), \(G:=2c^2-(v-2)j^2\ge\gamma_K\), \(v\le B_K\) and \(cv\le B_K\) on \(K\). Let

For \(p_{s,1}=p\ge P_K\), \(p_{s,2}=wp\), one has \(P_c=pc\ge B_K+2>\max(2,v)\), \(J_c=pj\), and

For \(v>2\) the equivalence already proved gives admissibility. For \(v\le2\), the inequality \(\mathcal U(P_c,J_c)>2\) proves the relaxed condition directly. This completes Lemma 4.5, including its original threshold.

G7. Exact scope of this component

G0–G6 prove the local formulas and finite correction mechanisms in Lemmas 4.1, 4.3–4.5 and 4.7, Proposition 4.2, Lemmas A.1–A.2 and Corollary A.3. The five original moments, axis pressure, radial shears, scale factors, viscosity factors, endpoint interpretation and quadratic correction terms are explicit. No source error is established by these calculations.

The actual power-law reference profile and its analytic pressure datum are derived in O0–O8, covering Proposition A.4 and Lemma A.5. The actual heat replacement, three compensating moments, backward stress and terminal endpoint estimates are derived in H0–H7, using the heat factor of L6. All four total moments and the axis pressure are retained. The backward formula is proved for the temporary reference axis powers themselves; no regular-axis construction is assumed. The actual analytic inner solution and its endpoint inequality are derived in J0–J7. Its actual continuation and exact five-moment match are derived in Q0–Q9. The actual shear modification and all six parts of the complete leading-profile Theorem 4.6 are derived in V0–V7. The positive-order base and full stress summation are derived in Z0–Z8. The complete theorem and its proof dependencies are stated in the theorem construction.