Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations

Dongho Chae

REVIEW

Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations

Dongho Chae

Analysis of PDEs · Submitted 11 Apr 2006 · 35 references

Pith's verdict: The Navier-Stokes half is a genuinely new and clean localized result; the Euler half leans on an unstated theorem from the author's own preprint, and that gap should be fixed before publication.

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CitationDongho Chae. "Nonexistence of asymptotically self-similar singularities in the Euler and the Navier-Stokes equations." arXiv:math/0604234 (2006). https://pith.science/paper/A45C5DL7

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Editor's note

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I read Chae's paper on asymptotically self-similar blow-up. The punchline: the Navier-Stokes part is solid and genuinely useful, while the Euler part has a load-bearing dependency that the manuscript never actually lays out.

What is new and good: Theorem 1.5 is a real improvement over Hou–Li. Allowing convergence in shrinking balls for q in [2,3) (provided it holds for all radii) is a meaningful localization, and the proof goes through the Gustafson–Kang–Tsai regularity criterion in a natural way. The alternative proof of the Hou–Li result via Leray's blow-up rate is clean and much shorter. The continuation principles, Theorem 1.1 and 1.3, are also nice standalone tools; the Besov-space version for Euler is a bit more than BKM and worth having in the literature. Structurally, the similarity-variable passage, weak limit, and continuation step are all standard and handled carefully.

The soft spot is exactly where the reader's stress-test lands. In the proof of Theorem 1.2, after deriving the stationary Euler profile equations (2.9)–(2.10), the paper says the profile satisfies the hypotheses of Theorem 1.1 of [3] and concludes the profile vanishes. But [3] is a preprint, its theorem is not stated, and the manuscript never verifies that the limit profile meets whatever regularity or growth conditions that theorem requires. From (1.3) you get \bar V in C^1 and \bar\Omega in L^q for q in (0,p1); that may not be enough for the cited Besov-based rigidity result. If [3] fails or its conditions are not met, the Euler nonexistence conclusion simply does not follow. The abstract also has a one-line typo, writing \dot B^0_{1,\infty} instead of \dot B^0_{\infty,1}; minor, but symptomatic of a rushed submission.

I should be clear: I don't think the Euler claim is false, and the overall strategy is sound. But an editor or referee cannot check the central step without access to an unpublished theorem whose hypotheses are never spelled out. That is a serious, fixable problem: include the statement of the rigidity result in [3] and show that the weak-limit profile satisfies it, or prove a self-contained version.

My verdict: publish the NS parts, require the Euler part to be self-contained or explicitly conditional. Yes, I would send it to a serious referee. I'd cite the localized NS result in my own work.

Referee report

Significance

If the gaps identified below are repaired, these results are significant: they extend earlier nonexistence results for exact self-similar singularities to the more flexible notion of asymptotic self-similar convergence, and Theorem 1.5 contains a genuinely localized Navier-Stokes statement. The paper also contains useful continuation criteria and a simple alternative proof of a known Navier-Stokes result. The Navier-Stokes half is largely self-contained modulo classical rigidity results and the Gustafson-Kang-Tsai regularity criterion. The Euler half, however, is not self-contained: its decisive rigidity step is imported from the author's own unpublished preprint [3], whose hypotheses are neither stated nor verified, and the limit passage to the stationary Euler profile is not fully justified. The Euler conclusions are therefore conditional as written.

Major objections (2)

[Section 2, proof of Theorem 1.2 (the step after Eq. (2.10))] The deduction \bar\Omega=0, which is the decisive step for the Euler result, is not proved in this manuscript. The proof states that \bar\Omega=\mathrm{curl}\,\bar V satisfies the conditions of Theorem 1.1 of [3] "by our hypothesis" and then applies that theorem. Neither the statement nor the hypotheses of Theorem 1.1 of [3] are included, and the paper does not verify them for the particular limit profile obtained from (1.3). This verification is nontrivial: Theorem 1.2 only guarantees \bar V\in C^1 and \bar\Omega\in L^q for q\in(0,p_1) with an arbitrary p_1>0, while the cited theorem may require a different integrability range or additional Besov regularity or growth conditions. Without a statement of [3]'s hypotheses and a demonstration that they hold, Theorem 1.2 and its corollaries are conditional on an external preprint. Please either state and prove the required rigidity theorem as a lemma within this paper, or verify its hypotheses explicitly in the present setting.
[Section 2, proof of Theorem 1.2 (passage from (2.7)-(2.8) to (2.9))] The passage to the limit in the weak formulation of (E1) is not justified by the displayed convergence. Equation (2.8) gives only \|\nabla V(s)-\nabla\bar V\|_{L^\infty}\to 0. The weak form contains the term \alpha V and the nonlinear term (V\cdot\nabla)V; passing these to the limit requires local convergence of V(s) to \bar V, or at least control of the spatial mean of V(s), not merely convergence of derivatives. The text says "using (2.8)" and the vanishing of \int \xi_s, but this does not explain how the constant (and polynomial) ambiguity inherent in \dot B^0_{\infty,1} convergence of the curl is resolved. Please add the missing compactness or normalization argument before (2.9) is derived; as written, the derivation of the stationary profile equation is incomplete.

Minor objections (5)

[Abstract and Section 1] The abstract mentions the critical Besov space \dot B^0_{1,\infty}(\mathbb R^3), while the body of the paper consistently uses \dot B^0_{\infty,1}(\mathbb R^3); these are different spaces and the notation should be corrected.
[Corollary 1.1] The smallness hypothesis is written as \|\bar\Omega\|_{\dot B^0_{\infty,0}}<\eta, but the surrounding text and the proof use \dot B^0_{\infty,1}; the index 0 appears to be a typo.
[Proofs of Theorem 1.2 and Theorem 1.4] The formulas for t_1 are sign-inconsistent with the definition of s. In the Euler proof, s=(\alpha+1)^{-1}\log(T/(T-t)) gives t=T(1-e^{-(\alpha+1)s}), not T(1-e^{(\alpha+1)s}); similarly, in the Navier-Stokes proof, t=T(1-e^{-2s}), not T(1-e^{2s}). As written, the displayed formulas for t_1 are negative for s_1>0, so the statement "(T-t_1)\|\omega(t_1)\|<\eta" is not meaningful; the sign in the exponents should be corrected.
[Section 2, inequalities (2.2)-(2.3)] Some Besov indices in the displayed estimates appear inconsistent: the second inequality in (2.2) has an integral of \|\omega(\tau)\|_{\dot B^0_{\infty,1}} if the embedding \dot B^0_{\infty,1}\hookrightarrow L^\infty is being used, but the displayed subscript reads \dot B^1_{\infty,1}; please check and harmonize the indices.
[Theorem 1.5 and its proof] In the statement of Theorem 1.5, the notation "either one of the followings hold" and the reuse of t both as the outer limit variable and as the time in the supremum in (1.6) are slightly confusing; renaming one of these variables would improve readability.

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The paper

The paper claims

Self-similar blow-up ruled out in 3D Euler and Navier-Stokes

The reading

This paper argues that a specific route to finite-time singularity—whereby a smooth solution approaches a self-similar profile as the singularity time is approached—cannot occur for the 3D incompressible Euler or Navier-Stokes equations. If the Euler vorticity converges to a self-similar vorticity profile in the critical Besov space $\dot B^0_{\infty,1}$, then the profile must be zero, and the solution extends smoothly past the suspected time. If the Navier-Stokes velocity converges to a self-similar velocity profile in $L^q$ on balls shrinking like $\sqrt{T-t}$, the profile must be zero and the solution is Hölder continuous near the point. Because self-similar profiles are the standard ansatz in singularity searches, ruling them out redirects the blow-up question to non-self-similar or weaker-convergence scenarios.

What carries the argument

The argument runs through similarity variables, $y=x/(T-t)^{1/(\alpha+1)}$ and $s=\frac{1}{\alpha+1}\log(T/(T-t))$ for Euler, and the analogous $\sqrt{T-t}$ variables for Navier-Stokes. In these variables convergence to a profile becomes convergence as $s\to\infty$, and a weak-limit passage using divergence-free test functions shows the limit profile solves the stationary Leray system (for Navier-Stokes, $\bar V+(y\cdot\nabla)\bar V+2(\bar V\cdot\nabla)\bar V=-\nabla\bar P+2\Delta\bar V$) or the stationary Euler profile equation (2.9)-(2.10). Known rigidity theorems for these stationary problems give $\bar V=0$ (or $\bar\Omega=0$), and a continuation principle in the same critical spaces pulls the conclusion back to the original variables.

Extended reading notes

Core claim

The central claim is that asymptotically self-similar singularities are impossible under the paper's convergence hypotheses. For Euler (Theorem 1.2), if a solution $v\in C([0,T);B^1_{\infty,1}(\mathbb{R}^3))$ satisfies $\lim_{t\nearrow T}(T-t)\|\omega(\cdot,t)-\frac{1}{T-t}\bar\Omega\big(\frac{\cdot}{(T-t)^{1/(\alpha+1)}}\big)\|_{\dot B^0_{\infty,1}}=0$ for some $\alpha>-1$ and $\bar\Omega$ in the stated $L^q$ class, then $\bar\Omega=0$ and the solution continues to $C([0,T+\delta);B^1_{\infty,1}(\mathbb{R}^3))$ for some $\delta>0$. For Navier-Stokes (Theorems 1.4 and 1.5), if the velocity converges to $\frac{1}{\sqrt{T-t}}\bar V(\cdot/\sqrt{T-t})$ either globally in $L^p$, $p\in[3,\infty)$, or locally on shrinking balls in $L^q$, $q\in[2,\infty)$, then $\bar V=0$; in the local case $v$ is Hölder continuous near $(z,T)$.

Load-bearing premise

For the Euler result, everything hangs on an earlier theorem, not proved here, that the only stationary self-similar Euler profile with vorticity in the relevant Besov and $L^q$ classes is the zero profile; for the Navier-Stokes side, the corresponding load-bearing premise is the known vanishing of Leray self-similar profiles in $L^p$, $p\ge 3$.

Editorial extensions

If this is right

For 3D Navier-Stokes, no singularity can form at a point where the velocity converges, on parabolically shrinking balls, to a nonzero self-similar profile in $L^q$ with $q\in[2,\infty)$.
For 3D Euler, if the vorticity converges in the critical Besov space $\dot B^0_{\infty,1}$ to a self-similar profile, that profile must be zero and the solution extends smoothly beyond the suspected singularity time.
The local version covers convergence with $q\in[2,3)$, which was outside earlier global $L^q$ results; the conclusion is Hölder continuity near the point, not merely extension.
Under any of these convergence assumptions, finite-time blow-up cannot be asymptotically self-similar; possible singularities would have to fail the convergence assumptions or take a non-self-similar form.
The continuation principles also reproduce the standard lower blow-up rate bounds: $\|\omega(t)\|_{\dot B^0_{\infty,1}}\gtrsim (T-t)^{-1}$ for Euler and $\|v(t)\|_{L^p}\gtrsim (T-t)^{-(p-3)/(2p)}$ for Navier-Stokes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A numerical blow-up candidate that converges to a self-similar profile only pointwise or in $L^p$ with $p<2$ would not be covered by these theorems; one testable reading is that any genuine counterexample must converge in a weaker topology than the critical Besov or local $L^q$ norms used here.
The same weak-limit-to-stationary-profile strategy should transfer to other scale-invariant PDEs whose stationary solutions are known to be rigid, giving a general recipe for excluding asymptotically self-similar blow-up.
The local $L^q$ result for $q\in[2,3)$ may combine with partial regularity theory to constrain the singular set: at any actual singular point, the local convergence used in (1.6) must fail in a quantitative way.

Original abstract

In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as $t$ goes to the possible time of singularity $T$. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, $\dot{B}^0_{1, \infty}(\Bbb R^3)$. For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in $L^q(B(z,r))$ for some $q\in [2, \infty)$, where the ball of radius $r$ is shrinking toward a possible singularity point $z$ at the order of $\sqrt{T-t}$ as $t$ approaches to $T$. In the $L^q (\Bbb R^3)$ convergence case with $q\in [3, \infty)$ we present a simple alternative proof of the similar result in \cite{hou}.

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arXiv listing abstract v8. Results and the main claim are underlined; show every role.

In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as $t$ goes to the possible time of singularity $T$. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, $\dot{B}^0_{1, \infty}(\Bbb R^3)$. For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in $L^q(B(z,r))$ for some $q\in [2, \infty)$, where the ball of radius $r$ is shrinking toward a possible singularity point $z$ at the order of $\sqrt{T-t}$ as $t$ approaches to $T$. In the $L^q (\Bbb R^3)$ convergence case with $q\in [3, \infty)$ we present a simple alternative proof of the similar result in \cite{hou}.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations.	result 0.91	0.95	0.98	asserted 1.00
s1	The notion means that the local in time classical solutions of the equations develop self-similar profiles as $t$ goes to the possible time of singularity $T$.	background 0.89	0.09	0.03	asserted 0.94
s2	For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, $\dot{B}^0_{1, \infty}(\Bbb R^3)$.	method 0.56	0.78	0.04	asserted 0.92
s3	For the Navier-Stokes equations the convergence of the velocity to the self-similar singularity is in $L^q(B(z,r))$ for some $q\in [2, \infty)$, where the ball of radius $r$ is shrinking toward a possible singularity point $z$ at the order of $\sqrt{T-t}$ as $t$ approaches to $T$.	method 0.54	0.78	0.04	asserted 0.95
s4	In the $L^q (\Bbb R^3)$ convergence case with $q\in [3, \infty)$ we present a simple alternative proof of the similar result in \cite{hou}.	contribution 0.82	0.87	0.03	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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References

Works cited (35)

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D. Chae, Nonexistence of self-similar singularities for the 3D inco mpress- ible Euler equations, arXiv-preprint, math.AP/0601060

View on Pith arXiv
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J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations , Comm. Math. Phys., 94, (1984), pp. 61-66

View on Pith 1984
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L. Caﬀarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, (1982), pp.771-831

View on Pith 1982
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D. Chae, A note on ‘Nonexistence of self-similar singularities for t he 3D incompressible Euler equations’, arXiv-preprint, math.AP/0601661. 15

View on Pith arXiv
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D. Chae, Local Existence and Blow-up Criterion for the Euler Equatio ns in the Besov Spaces , Asymp. Anal., 38, no. 3-4, (2004), pp. 339-358

View on Pith 2004
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D. Chae, On the Euler Equations in the Critical Triebel-Lizorkin Spa ces, Arch. Rat. Mech. Anal., 170, no. 3, (2003), pp.185-210

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D. Chae, Remarks on the blow-up of the Euler equations and the related equations, Comm. Math. Phys., 245, no. 3, (2003), 539-550

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J. Y. Chemin, Perfect incompressible ﬂuids, Clarendon Press, Oxford, (1998)

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P. Constantin, C. Feﬀerman and A. Majda, Geometric constraints on po- tential singularity formulation in the 3-D Euler equations , Comm. P.D.E, 21, (3-4), (1996), pp. 559-571

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P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics Series, Univ. Chicago Press (1988)

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L. Escauriaza, G. Seregin and V. Sverak, L3, ∞-solutions of Navier-Stokes equations and backward uniqueness, Rus. Math. Surveys, 58, (2003), pp. 211-250

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Y. Giga and R. V. Kohn, Asymptotically Self-Similar Blow-up of Semi- linear Heat Equations, Comm. Pure Appl. Math., 38, (1985), pp. 297-319

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J. M. Greene and R. B. Pelz, Stability of postulated, self-similar, hydro- dynamic blowup solutions , Phys. Rev. E, 62, no. 6, pp. 7982-7986

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T. Y. Hou and R. Li, Nonexistence of Local Self-Similar Blow- up for the 3D Incompressible Navier-Stokes Equations, arXiv-preprint, math.AP/0603126

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T. Kato, Strong Lp solutions of the Navier-Stokes equations in Rm with applications to weak solutions, Math. Zeit., 187, (1984), pp. 471-480. 16

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How the literature uses it

Papers whose main result relates to this one's

How each label was checked · JSON

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States a conflicting result 1 paper

We provide evidence of a stable finite-time singularity in the 3D Euler equations on the unbounded domain.

Self-Similar Singularity of the Euler Equations on $\mathbb{R}^3$ · 0.97

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [3]

D. Chae, Nonexistence of self-similar singularities for the 3D inco mpress- ible Euler equations, arXiv-preprint, math.AP/0601060

View on Pith arXiv

[2] [1]

J. T. Beale, T. Kato and A. Majda, Remarks on the breakdown of smooth solutions for the 3-D Euler equations , Comm. Math. Phys., 94, (1984), pp. 61-66

View on Pith 1984

[3] [2]

L. Caﬀarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math., 35, (1982), pp.771-831

View on Pith 1982

[4] [4]

D. Chae, A note on ‘Nonexistence of self-similar singularities for t he 3D incompressible Euler equations’, arXiv-preprint, math.AP/0601661. 15

View on Pith arXiv

[5] [5]

D. Chae, Local Existence and Blow-up Criterion for the Euler Equatio ns in the Besov Spaces , Asymp. Anal., 38, no. 3-4, (2004), pp. 339-358

View on Pith 2004

[6] [6]

D. Chae, On the Euler Equations in the Critical Triebel-Lizorkin Spa ces, Arch. Rat. Mech. Anal., 170, no. 3, (2003), pp.185-210

View on Pith 2003

[7] [7]

D. Chae, Remarks on the blow-up of the Euler equations and the related equations, Comm. Math. Phys., 245, no. 3, (2003), 539-550

View on Pith 2003

[8] [8]

J. Y. Chemin, Perfect incompressible ﬂuids, Clarendon Press, Oxford, (1998)

View on Pith 1998

[9] [9]

P. Constantin, Geometric Statistics in Turbulence, SIAM Rev.,36, (1994), pp. 73-98

View on Pith 1994

[10] [10]

P. Constantin, A few results and open problems regarding incompressible ﬂuids, Notices Amer. Math. Soc. 42, no. 6, (1995), pp. 658-663

View on Pith 1995

[11] [11]

P. Constantin, C. Feﬀerman and A. Majda, Geometric constraints on po- tential singularity formulation in the 3-D Euler equations , Comm. P.D.E, 21, (3-4), (1996), pp. 559-571

View on Pith 1996

[12] [12]

P. Constantin and C. Foias, Navier-Stokes Equations, Chicago Lectures in Mathematics Series, Univ. Chicago Press (1988)

View on Pith 1988

[13] [13]

L. Escauriaza, G. Seregin and V. Sverak, L3, ∞-solutions of Navier-Stokes equations and backward uniqueness, Rus. Math. Surveys, 58, (2003), pp. 211-250

View on Pith 2003

[14] [14]

Y. Giga and R. V. Kohn, Asymptotically Self-Similar Blow-up of Semi- linear Heat Equations, Comm. Pure Appl. Math., 38, (1985), pp. 297-319

View on Pith 1985

[15] [15]

J. M. Greene and R. B. Pelz, Stability of postulated, self-similar, hydro- dynamic blowup solutions , Phys. Rev. E, 62, no. 6, pp. 7982-7986

View on Pith

[16] [16]

T. Y. Hou and R. Li, Nonexistence of Local Self-Similar Blow- up for the 3D Incompressible Navier-Stokes Equations, arXiv-preprint, math.AP/0603126

View on Pith arXiv

[17] [17]

T. Kato, Strong Lp solutions of the Navier-Stokes equations in Rm with applications to weak solutions, Math. Zeit., 187, (1984), pp. 471-480. 16

View on Pith 1984

[18] [18]

R. Kerr, Evidence for a singularity of the 3-dimensional, incompres sible Euler equations, Phys. Fluids A, 5, (1993), pp. 1725-1746

View on Pith 1993

[19] [19]

O. A. Ladyzenskaya, The mathematical theory of viscous incompressible ﬂow, Gordon and Breach, (1969)

View on Pith 1969

[20] [20]

J. Leray, Essai sur le mouvement d’un ﬂuide visqueux emplissant l’espace,Acta Math. 63 (1934), pp. 193-248

View on Pith 1934

[21] [21]

F. Lin, A new proof of the Caﬀarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math. 51, no. 3, (1998), pp. 241-257

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