PDE: Navier–Stokes¶
Job: develop the Navier–Stokes regularity theory at the level of the Clay Millennium Problem — Leray–Hopf weak solutions and the energy inequality, local smooth well-posedness (Fujita–Kato), the Caffarelli–Kohn–Nirenberg partial regularity, the Koch–Tataru \(BMO^{-1}\) critical well-posedness, and the Escauriaza–Seregin–Šverák blow-up criterion — with the precise open problem and what is known.
Core¶
The equations¶
The incompressible Navier–Stokes equations on \(\mathbb{R}^3\) (with viscosity \(\nu>0\) and forcing \(f\)):
The energy equality for smooth solutions: multiplying by \(u\) and integrating,
where the convective term \((u\cdot\nabla)u\) vanishes after integration by parts (skew-symmetry: \(\int u\cdot(u\cdot\nabla)u=0\) for divergence-free \(u\)) and the pressure term vanishes (since \(\nabla\cdot u=0\)). For the unforced equation \(f=0\), the total energy \(\frac12\|u\|_{L^2}^2+\nu\int_0^t\|\nabla u\|_{L^2}^2=\frac12\|u_0\|_{L^2}^2\) is conserved.
Leray–Hopf weak solutions¶
Theorem (Leray 1934, Hopf 1951). For any \(u_0\in L^2(\mathbb{R}^3)\) with \(\nabla\cdot u_0=0\), there exists a Leray–Hopf weak solution \(u\in L^\infty_t L^2_x\cap L^2_t H^1_x\) on \([0,\infty)\times\mathbb{R}^3\) satisfying the equation in the distributional sense and the energy inequality $$ \frac12|u(t)|{L^2}^2+\nu\int_0^t|\nabla u(s)|t\ge 0, $$ with the inequality (not equality) holding. The construction is by Galerkin approximation + Aubin–Lions compactness.}^2\,ds\;\le\;\frac12|u_0|_{L^2}^2\quad\text{for all
The inequality (vs equality for smooth solutions) is the key subtlety: the convective term passes to the weak limit only as an inequality — there may be an energy defect (dissipation in singular regions not captured by \(\nu\|\nabla u\|_{L^2}^2\)). Proving equality for all weak solutions would imply regularity (no energy defect \(\Rightarrow\) smooth); the equality vs inequality distinction is the regularity question.
Local smooth well-posedness¶
Theorem (Fujita–Kato 1964). For \(u_0\in H^s(\mathbb{R}^3)\) with \(s\ge 1/2\) (and \(s>1/2\) for uniqueness), there exists \(T^\ast>0\) and a unique smooth solution \(u\in C([0,T^\ast);H^s)\cap L^2_{\mathrm{loc}}((0,T^\ast);H^{s+1})\) with the blow-up criterion \(\|u(t)\|_{H^s}\to\infty\) as \(t\to T^\ast\) if \(T^\ast<\infty\). The threshold \(s=1/2\) is the critical Sobolev index (scaling-critical for Navier–Stokes).
The local smooth solution exists in \(H^s\) for \(s\ge 1/2\); the global weak Leray–Hopf solution exists for all \(L^2\) data; the gap is whether the smooth solution can be extended globally (for smooth compactly supported data) — the Millennium problem.
Partial regularity (Caffarelli–Kohn–Nirenberg)¶
Theorem (Caffarelli–Kohn–Nirenberg 1982). For a suitable weak solution (a weak solution satisfying a local energy inequality + a local integrability condition), the singular set $$ \Sigma={(x,t)\in\mathbb{R}^3\times(0,\infty)\mid u\text{ is not }C^\infty\text{ in a neighborhood of }(x,t)} $$ has parabolic 1-dimensional Hausdorff measure zero: \(\dim_{\mathcal{P},\mathcal{H}}(\Sigma)\le 1\) (in space-time). The \(\varepsilon\)-regularity criterion: there exists \(\varepsilon_0>0\) such that if the local energy in a parabolic cylinder \(Q_r(x_0,t_0)\) is small, \(\frac{1}{r}\int_{Q_r}|u|^2\,dx\,dt<\varepsilon_0\), then \(u\) is smooth in a smaller cylinder \(Q_{r/2}\).
The CKN theorem bounds the size of the singular set (codimension \(\ge 2\) in space-time, hence 1-dimensional Hausdorff measure zero); it does not rule out singularities. Combined with the Koch–Tataru / ESS blow-up criteria (below), it gives the best known partial regularity.
Critical-space well-posedness and blow-up criteria¶
Theorem (Koch–Tataru 2001). The Navier–Stokes equations are locally well-posed in the critical space \(\mathrm{BMO}^{-1}(\mathbb{R}^3)\) — the largest critical space for which local well-posedness holds — extending the Fujita–Kato \(H^{1/2}\) theory. The blow-up criterion: a smooth solution on \([0,T^\ast)\) with \(T^\ast<\infty\) must satisfy \(\|u(t)\|_{L^\infty_t L^3_x}\to\infty\) as \(t\to T^\ast\).
Theorem (Escauriaza–Seregin–Šverák 2003). The \(L^\infty_t L^3_x\) blow-up criterion is sharp: if a smooth solution satisfies \(\sup_{t<T^\ast}\|u(t)\|_{L^3}<\infty\), then \(T^\ast=\infty\) (no blow-up). Hence \(L^3\) is the critical Lebesgue regularity threshold: blow-up requires \(\|u\|_{L^3_x}\) to diverge.
| Result | Statement |
|---|---|
| Leray–Hopf (1934/1951) | global weak solution + energy inequality for all \(u_0\in L^2\) |
| Fujita–Kato (1964) | local smooth well-posedness in \(H^s\), \(s\ge 1/2\); threshold |
| Koch–Tataru (2001) | local LWP in \(\mathrm{BMO}^{-1}\) (largest critical space) |
| Escauriaza–Seregin–Šverák (2003) | \(L^\infty_t L^3_x\) blow-up criterion (sharp) |
| Caffarelli–Kohn–Nirenberg (1982) | partial regularity: \(\dim_{\mathcal{P},\mathcal{H}}(\Sigma)\le 1\) for suitable weak solutions |
| Open (Millennium) | global smooth well-posedness for \(u_0\in C_c^\infty(\mathbb{R}^3)\) |
The Millennium problem¶
Clay Millennium Problem (Navier–Stokes). Prove (or disprove): for every \(u_0\in C_c^\infty(\mathbb{R}^3)\) with \(\nabla\cdot u_0=0\), there exists a unique smooth solution \(u\in C^\infty([0,\infty)\times\mathbb{R}^3)\) of the Navier–Stokes equations with \(u(0)=u_0\) and bounded energy at all times.
The known: Leray–Hopf global weak solutions exist (with energy inequality); Fujita–Kato local smooth solutions exist (for \(H^{1/2}\) data); CKN partial regularity bounds the singular set; ESS gives the \(L^3\) blow-up criterion. The unknown: whether the smooth solution extends globally (no blow-up), or whether blow-up occurs for some smooth initial data. The gap is between the weak (global, possibly non-smooth) and the strong (smooth, possibly non-global) — the "strong global regularity" question.
See it¶
The Leray–Hopf construction: Galerkin approximation, the energy bound, Aubin–Lions compactness, and where the energy equality is lost. Given \(u_0\in L^2\), project to a finite-dimensional subspace \(V_N=\mathrm{span}\{e_1,\dots,e_N\}\) of divergence-free fields (the Galerkin basis, e.g. the eigenfunctions of the Stokes operator). The Galerkin approximation \(u^N(t)\in V_N\) solves the projected ODE
with \(\mathbb{P}\) the Leray projector (onto divergence-free fields). The energy bound for \(u^N\): multiply by \(u^N\) and use \(\langle\mathbb{P}((u^N\cdot\nabla)u^N),u^N\rangle=\langle (u^N\cdot\nabla)u^N,u^N\rangle=0\) (skew-symmetry, exact on \(V_N\)), giving
The bound is uniform in \(N\). Aubin–Lions compactness (the bound \(u^N\in L^\infty_t L^2_x\cap L^2_t H^1_x\) and the time-derivative bound \(\partial_t u^N\in L^2_t H^{-1}_x\) from the equation gives relative compactness in \(L^2_{\mathrm{loc}}((0,\infty)\times\mathbb{R}^3)\)) extracts a subsequence \(u^N\to u\) in \(L^2_{\mathrm{loc}}\) and weakly in \(L^\infty_t L^2_x\cap L^2_t H^1_x\). The limit \(u\) satisfies the equation in the distributional sense. Where the energy equality is lost: the convective term \((u^N\cdot\nabla)u^N\) converges only weakly, and \(\langle (u^N\cdot\nabla)u^N,u^N\rangle=0\) becomes, in the limit, \(\int_0^t\langle (u\cdot\nabla)u,u\rangle\,ds\le\liminf\int_0^t\langle (u^N\cdot\nabla)u^N,u^N\rangle\,ds=0\) — an inequality (weak lower-semicontinuity gives \(\le\) not \(=\)). The energy defect \(\frac12\|u_0\|_{L^2}^2-\bigl(\frac12\|u(t)\|_{L^2}^2+\nu\int_0^t\|\nabla u\|_{L^2}^2\bigr)\ge 0\) is the possible dissipation in singular regions; for smooth solutions it is zero, for weak solutions it is unknown. This is the operational content of the energy inequality: the Galerkin convergence preserves the bound only as an inequality, and the gap between inequality and equality is the regularity question.
Check¶
State precisely why proving the energy equality for all Leray–Hopf weak solutions would resolve the Millennium regularity problem, and identify the regularity gain it would give.
The energy equality \(\frac12\|u(t)\|_{L^2}^2+\nu\int_0^t\|\nabla u(s)\|_{L^2}^2\,ds=\frac12\|u_0\|_{L^2}^2\) for all weak solutions would give the equality case of the Galerkin lower bound, implying the convective term passes to the limit with no defect: \(\int_0^t\langle (u\cdot\nabla)u,u\rangle=0\) (not just \(\le 0\)). This in turn gives the equality in the weak formulation of the nonlinear term, which is the coercivity needed to pass the regularity bootstrap: the energy equality implies the solution is in \(L^4_t L^3_x\) (via the \(L^2\)-\(H^1\) interpolation and the equality), which is the threshold above the ESS \(L^\infty_t L^3_x\) blow-up criterion — implying no blow-up, hence global smoothness. So the equality \(\Rightarrow L^\infty_t L^3_x\) bound \(\Rightarrow\) no blow-up (ESS) \(\Rightarrow\) global smoothness. Hence the energy equality for all weak solutions would resolve the Millennium problem. (The gap is precisely that the equality is not known for all weak solutions — it holds for smooth solutions and for some weak solutions satisfying extra hypotheses, but not in general.)
Explain why CKN bounds the singular set to dimension \(\le 1\) but does not rule out singularities, and identify the geometric meaning of the \(\varepsilon\)-regularity criterion.
CKN gives an \(\varepsilon\)-regularity criterion: if the local energy \(\frac{1}{r}\int_{Q_r}|u|^2\) in a parabolic cylinder \(Q_r\) is small (below a threshold \(\varepsilon_0\)), then \(u\) is smooth in \(Q_{r/2}\). The criterion is a local smallness condition — singularities can occur only where the local energy is large. The covering argument (Vitali / Besicovitch) bounds the singular set by covering it with cylinders where the local energy is large, and the energy inequality bounds the total measure, giving \(\dim_{\mathcal{P},\mathcal{H}}(\Sigma)\le 1\). The bound \(\le 1\) is a codimension \(\ge 2\) in space-time bound (3 spatial + 1 time = 4 dimensions; singular set has \(\le 1\) dimension, so codimension \(\ge 3\) in space and \(\ge 1\) in time). It does not rule out singularities: a singular set of dimension \(1\) (e.g. a curve of singular points in space-time) is consistent with CKN; whether the singular set is empty (smoothness) or non-empty is the open Millennium question. The geometric meaning of the \(\varepsilon\)-regularity: the singularity, if it exists, must carry a non-trivial local energy density (the "energy concentration" is the obstruction to smoothness), and the smallness rules out singularity.
Pitfalls¶
- The Navier–Stokes regularity problem is open: do not claim smoothness of Leray–Hopf solutions, nor global smooth well-posedness. The known results are: global weak (Leray–Hopf), local smooth (Fujita–Kato), partial regularity (CKN), blow-up criterion (ESS). The strong global regularity is the open Millennium problem.
- The energy inequality (not equality) is the key subtlety: the convective term passes to the weak limit only as \(\le\), not \(=\); the energy defect is the possible singular dissipation. Confusing inequality and equality is the standard error.
- Suitable weak solutions are a stricter class: the CKN partial regularity holds for suitable weak solutions (a local energy inequality + integrability), not all Leray–Hopf solutions; the extension of CKN to all Leray–Hopf solutions is itself partly open.
- The Koch–Tataru \(BMO^{-1}\) LWP is local: it does not give global well-posedness; the global smooth solution requires the additional ESS / energy argument, which is the open part.
- \(L^3\) is the critical Lebesgue threshold: the ESS blow-up criterion is \(L^\infty_t L^3_x\); below \(L^3\) the criterion does not apply. The Fujita–Kato \(H^{1/2}\) and the Koch–Tataru \(BMO^{-1}\) are the Sobolev / Besov analogs.
- CKN is dimension \(\le 1\) in space-time (parabolic Hausdorff): not \(\le 1\) in space; the parabolic metric weights time as \(\sqrt{\cdot}\), so the bound is on the 4-dimensional space-time singular set.
Refs¶
- Leray, Sur le mouvement d'un liquide visqueux (Acta Math. 1934); Hopf (1951)
- Fujita–Kato, On the Navier–Stokes initial value problem I (Arch. 1964)
- Caffarelli–Kohn–Nirenberg, Partial regularity of suitable weak solutions (Acta Math. 1982)
- Koch–Tataru, Well-posedness for the Navier–Stokes equations (Adv. Math. 2001)
- Escauriaza–Seregin–Šverák, Backward uniqueness for parabolic equations (\(L^3\) blow-up, 2003); Seregin, Lecture Notes on Regularity Theory
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