Abstract
Using the convex integration technique for the three-dimensional Navier–Stokes equations introduced by Buckmaster and Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier–Stokes equations with fractional hyperviscosity \((-\Delta )^{\theta }\), whenever the exponent \(\theta \) is less than Lions’ exponent 5/4, i.e., when \(\theta < 5/4\).
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In this paper we consider the question of non-uniquness of weak solutions to the 3D Navier–Stokes equations with fractional viscosity (FVNSE) on \(\mathbb {T}^3\)
where \(\theta \in \mathbb {R}\) is a fixed constant, and for \(u \in C^{\infty }(\mathbb {T}^3)\) with \(\int _{\mathbb {T}^3} u(x) dx =0\), the fractional Laplacian is defined via the Fourier transform as
Definition
(weak solutions) A vector field \(v \in C^0_{weak}(\mathbb {R};L^2(\mathbb {T}^3))\) is called a weak solution to the FVNSE if it solves (1) in the sense of distribution.
When \(\theta = 1\), FVNSE (1) is the standard Navier–Stokes equations. Lions first considered FVNSE (1) in [20], and showed the existence and uniqueness of weak solutions to the initial value problem, which also satisfied the energy equality, for \(\theta \in [5/4,\infty )\) in [21]. Moreover, an analogue of the Caffarelli–Kohn–Nirenberg [6] result was established in [18] for the FVNSE system (1), showing that the Hausdorff dimension of the singular set, in space and time, is bounded by \(5 - 4\theta \) for \(\theta \in (1,5/4)\). The existence, uniqueness, regularity and stability of solutions to the FVNSE have been studied in [17, 26, 28, 29] and references therein. Very recently, using the method of convex integration introduced in [12], Colombo et al. [8] showed the non-uniquenss of Leray weak solutions to FVNSE (1) for \(\theta \in (0,1/5)\) and for \(\theta \in (0,1/3)\) in [13].
In the recent breakthrough work [5], Buckmaster and Vicol obtained non-uniqueness of weak solutions to the three-dimensional Navier–Stokes equations. They developed a new convex integration scheme in Sobolev spaces using intermittent Beltrami flows which combined concentrations and oscillations. Later, the idea of using intermittent flows was used to study non-uniqueness for transport equations in [23,24,25] employing scaled Mikado waves, and for stationary Navier–Stokes equations in [7, 22] employing viscous eddies.
The schemes in [5, 24] are based on the convex integration framework in Hölder spaces for the Euler equations, introduced by De Lellis and Székelyhidi [12], subsequently refined in [2, 3, 10, 15], and culminated in the proof of the second half of the Onsager conjecture by Isett in [16]; also see [4] for a shorter proof. For the first half of the Onsager conjecture, see, e.g., [1, 9], and the references therein.
The main contribution of this note is to show that the results in Buckmaster–Vicol’s paper hold for FVNSE (1) for \(\theta < 5/4\):
Theorem 1
Assume that \(\theta \in [1, 5/4)\). Suppose u is a smooth divergence-free vector field, define on \(\mathbb {R}_+ \times \mathbb {T}^3\), with compact support in time and satisfies the condition
Then for any given \(\varepsilon _0 > 0\), there exists a weak solution v to the FVNSE (1), with compact support in time, satisfying
As a consequence there are infinitely many weak solutions of the FVNSE (1) which are compactly supported in time; in particular, there are infinitely many weak solutions with initial values zero.
Remark 1
In the above theorem we assume that \(\theta \in [1, 5/4)\). However, using the constructions in [5] with a slightly different choice of parameters, one can actually show that Theorem 1.2 and Theorem 1.3 in [5] hold for the 3D FVNSE, i.e., there exist non-unique weak solutions \(v \in C_t^0 W_x^{\beta ,2}\), with a different \(\beta > 0\), depending on \(\theta \). However, in this paper we choose to prove a weaker result, Theorem 1, in order to simplify the presentation while retaining the main idea.
Remark 2
For the case \(\theta \in (-\infty ,1)\), the same construction also yields weak solutions \(v \in C^0_t L^2_x \cap C^0_t W^{1,1}_x\) with a suitable choice of parameters.
We now make some comments on the analysis in this paper. Using the technique in [5], we adapt a convex integration scheme with intermittent Beltrami flows as the building blocks. The main difficulty in a convex integration scheme for (FVNSE), is the error induced by the frictional viscosity \(\nu (- \Delta )^{\theta } v\), which is greater for a larger exponent \(\theta \). This error is controlled by making full use of the concentration effect of intermittent flows introduced in [5]. As it is shown in the crucial estimate (36), the error is controllable only for \(\theta < 5/4\). Compared with [5], since our goal is to construct weak solutions \(v \in C^0_t L^2_{x,weak} \cap L^{\infty }_t W^{2\theta - 1,1}_x\), we adapt a slightly simpler cut-off function and prove only estimates that are sufficient for this purpose.
2 Outline
2.1 Iteration lemma
Following [5], we consider the approximate system
where R is a symmetric \(3 \times 3\) matrix.
Lemma 1
(Iteration Lemma for \(L^2\) weak solutions) Let \(\theta \in (-\infty , 5/4)\). Assume \((v_q, R_q)\) is a smooth solution to (2) with
for some \(\delta _{q+1} > 0\). Then for any given \(\delta _{q+2} > 0\), there exists a smooth solution \((v_{q+1}, R_{q+1})\) of (2) with
Here for a given set \(A \subset \mathbb {R}\), the \(\delta \)-neighborhood of A is denoted by
Furthermore, the increment \(w_{q+1} = v_{q+1} - v_q\) satisfies the estimates
where the positive constant C depends only on \(\theta \).
Proof of Theorem 1
Assume Lemma 1 is valid. Let \(v_0 = u\). Then
Let
where \(\mathcal {R}\) is the symmetric anti-divergence operator established in Lemma 5, below. Clearly \((v_0,R_0)\) solves (2). Set
Apply Lemma 1 iteratively to obtain smooth solution \((v_q, R_q)\) to (2). It follows from (6) that
Thus \(v_q\) converge strongly to some \(v \in C^0_t L^2_x\). Since \(\Vert R_{q+1}\Vert _{L^{\infty }_t L^1_x} \rightarrow 0\), as \(q \rightarrow \infty \), v is a weak solution to the FVNSE (1). Estimate (7) leads to
Furthermore, it follows from (5) that
Now we show the existence of infinitely many weak solutions with initial values zero. Let \(u(t,x) = \varphi (t) \sum _{|k| \le N} a_k e^{ik \cdot x}\) with \(a_k \ne 0, a_k \cdot k = 0, a_{-k} = a_k^*\) for all \(|k| \le N\), and \(\varphi \in C_c^{\infty }(\mathbb {R}_+)\). Thus \(\nabla \cdot u = 0\) satisfies the conditions of the theorem. Hence there exists a weak solution v to (1) close enough to u so that \(v \not\equiv 0\). \(\square \)
3 Iteration scheme
3.1 Notations and parameters
For a complex number \(\zeta \in \mathbb {C}\), we denote by \(\zeta ^*\) its complex conjugate. Let us normalize the volume
For smooth functions \(u \in C^{\infty }(\mathbb {T}^3)\) with \(\int _{\mathbb {T}^3} u(x) dx =0\) and \(s \in \mathbb {R}\), we define
For \(M, N \in [0,+\infty ]\), denote the Fourier projection of u by
We also denote \(\mathbb {P}_{\le k} = \mathbb {P}_{[0,k)}\) and \(\mathbb {P}_{\ge k} = \mathbb {P}_{[k,+\infty )}\) for \(k > 0\).
Following the notation in [5], we introduce here several parameters \(\sigma , r, \lambda \), with
where \(\lambda = \lambda _{q+1} \in 5\mathbb {N}\) is the ‘frequency’ parameter; \(\sigma \) with \(1/\sigma \in \mathbb {N}\) is a small parameter such that \(\lambda \sigma \in \mathbb {N}\) parameterizes the spacing between frequencies; \(r \in \mathbb {N}\) denotes the number of frequencies along edges of a cube; \(\mu \) measures the amount of temporal oscillation.
Later \(\sigma , r, \mu \) will be chosen to be suitable powers of \(\lambda _{q+1}\). We also fix a constant \(p > 1\) which will be chosen later to be close to 1. The constants implicitly in the notation ‘\(\lesssim \)’ may depend on p but are independent of the parameters \(\sigma , r, \lambda \).
3.2 Intermittent Beltrami flows
We use intermittent Beltrami flows introduced in [5] as the building blocks. Recall some basic facts of Beltrami waves.
Proposition 1
[5, Proposition 3.1] Given \(\overline{\xi } \in \mathbb {S}^2 \cap \mathbb {Q}^3\), let \(A_{\overline{\xi }} \in \mathbb {S}^2 \cap \mathbb {Q}^3\) be such that
Let \(\Lambda \) be a given finite subset of \(\mathbb {S}^2\) such that \(- \Lambda = \Lambda \), and \(\lambda \in \mathbb {Z}\) be such that \(\lambda \Lambda \subset \mathbb {Z}^3\). Then for any choice of coefficients \(a_{\overline{\xi }} \in \mathbb {C}\) with \(a_{\overline{\xi }}^* = a_{-\overline{\xi }}\) the vector field
is real-valued, divergence-free and satisfies
Furthermore,
Let \(\Lambda , \Lambda ^+, \Lambda ^- \subset \mathbb {S}^2 \cap \mathbb {Q}^3\) be defined by
Clearly we have
Also it is direct to check that
In fact, representations of this form exist for symmetric matrices close to the identity. We have the following simple variant of [5, Proposition 3.2].
Proposition 2
Let \(B_{\varepsilon }(\mathrm {Id})\) denote the ball of symmetric matrices, centered at the identity, of radius \(\varepsilon \). Then there exist a constant \(\varepsilon _{\gamma } > 0\) and smooth positive functions \(\gamma _{(\overline{\xi })} \in C^{\infty }(B_{\varepsilon _{\gamma }}(\mathrm {Id}))\), such that
- 1.
\(\gamma _{(\overline{\xi })} = \gamma _{(-\overline{\xi })}\);
- 2.
for each \(R \in B_{\varepsilon _{\gamma }}(\mathrm {Id})\) we have the identity
$$\begin{aligned} R = \frac{1}{2}\sum _{\overline{\xi } \in \Lambda } \left( \gamma _{(\overline{\xi })}(R)\right) ^2(\mathrm {Id} - \overline{\xi } \otimes \overline{\xi }). \end{aligned}$$
Define the Dirichlet kernel
It has the property that, for \(1 < p \le \infty \),
Following [5], for \(\overline{\xi } \in \Lambda ^+\), define a directed and rescaled Dirichlet kernel by
and for \(\overline{\xi } \in \Lambda ^-\), define
Note the important identity
Since the map \(x \mapsto \lambda \sigma (\overline{\xi } \cdot x + \mu t, A_{\overline{\xi }} \cdot x, (\overline{\xi } \times A_{\overline{\xi }}) \cdot x)\) is the composition of a rotation by a rational orthogonal matrix mapping \(\{e_1, e_2, e_3\}\) to \(\{ \overline{\xi }, A_{\overline{\xi }}, \overline{\xi } \times A_{\overline{\xi }} \}\), a translation, and a rescaling by integers, for \(1 < p \le \infty \), we have
Let \(W_{(\overline{\xi })}\) be the Beltrami plane wave at frequency \(\lambda \),
Define the intermittent Beltrami wave \(\mathbb {W}_{(\overline{\xi })}\) as
It follows from the definitions and (9) that
The following properties are immediate from the definitions.
Proposition 3
[5, Proposition 3.4] Let \(a_{\overline{\xi }} \in \mathbb {C}\) be constants with \(a_{\overline{\xi }}^* = a_{-\overline{\xi }}\). Let
Then W(x) is real valued. Moreover, for each \(R \in B_{\varepsilon _{\gamma }}(\mathrm {Id})\) we have
Proposition 4
[5, Proposition 3.5] For any \(1 < p \le \infty , N \ge 0, K \ge 0\):
3.3 Perturbations
Let \(\psi (t)\) be a smooth cut-off function such that
Take a smooth increasing function \(\chi \) such that
and set
where \(\varepsilon _{\gamma }\) is the constant in Proposition 2. Then clearly
It follows from the above definition that
Therefore, the amplitude functions
are well-defined and smooth. Define the velocity perturbation to be \(w = w_{q+1}\):
where \(\mathbb {P}_{LH} = \mathrm {Id} - \nabla \Delta ^{-1} {\text {div}}\) is the Leray-Helmholtz projection into divergence-free vector field, and \(\mathbb {P}_{\ne 0} f = f - \fint _{ \mathbb {T}^3} f dx\). It is well-known that \(\mathbb {P}_{LH}\) is bounded on \(L^p, 1< p < \infty \) (see, e.g., [14]). It follows from Proposition 3 that
3.4 Estimates for perturbations
Lemma 2
The following bounds hold:
Proof
It follows from (3) that
It is direct to verify (21) and (23), while (22) and (24) follow from (17) and (21). \(\square \)
Now we can estimate the time support of \(w_{q+1}\):
We need the following Lemma, which is a variant of [5, Lemma 3.6].
Lemma 3
([24, Lemma 2.1]) Let \(f,g \in C^{\infty }(\mathbb {T}^3)\), and g is \((\mathbb {T}/N)^3\) periodic, \(N \in \mathbb {N}\). Then for \(1 \le p \le \infty \),
Let us denote
to be some polynomials depending on \(\sup _{\overline{\xi } \in \Lambda }\Vert a_{(\overline{\xi })}\Vert _{C^N_{t,x}}\).
Lemma 4
Suppose the parameters satisfy (8) and
Then the following estimates for the perturbations hold:
for \(1< p < \infty , N \ge 1\).
Proof
Since \(\mathbb {W}_{(\overline{\xi })}\) is \((\mathbb {T}/\lambda \sigma )^3\) periodic, it follows from (15), (23), and Lemma 3 that
In view of (8), (15) and (16) yield that
where the boundedness of \(\mathbb {P}_{LH}\) and \(\mathbb {P}_{\ne 0}\) on \(L^p\), for \(1< p < \infty \), is used in the first inequality of the estimate for \(\Vert w_{q+1}^{(t)}\Vert _{L^{\infty }_t L^p_x}\). In the same way, we can estimate
For \(N \ge 1\), using (15) and (16), we obtain that
where we use (8) and (27). \(\square \)
3.5 Estimates for the stress
Let us recall the following operator in [12].
Lemma 5
(symmetric anti-divergence) There exists a linear operator \(\mathcal {R}\), of order \(-1\), mapping vector fields to symmetric matrices such that
with standard Calderon–Zygmund estimates, for \(1< p < \infty \),
Proof
Suppose \(u \in C^{\infty }(\mathbb {T}^3, \mathbb {R}^3)\) is a smooth vector field. Define
where \(v \in C^{\infty }(\mathbb {T}^3, \mathbb {R}^3)\) is the unique solution to \(\Delta v = u - \fint _{ \mathbb {T}^3} u\) with \(\fint _{ \mathbb {T}^3} v = 0\).
It is direct to verify that \(\mathcal {R}(u)\) is a symmetric matrix field depending linearly on u and satisfies (33). Note that \(\mathcal {R}\) is a constant coefficient ellitpic operator of order \(-1\). We refer to [14] for the Calderon-Zygmund estimates \(\Vert \mathcal {R} \Vert _{L^p \rightarrow W^{1,p}} \lesssim 1\) and \(\Vert \mathcal {R} \mathbb {P}_{\ne 0} u \Vert _{L^p} \lesssim \Vert |\nabla |^{-1}\mathbb {P}_{\ne 0} u\Vert _{L^p}\). Combining these with Sobolev embeddings, we have \(\Vert \mathcal {R} u\Vert _{C^{\alpha }} \lesssim \Vert \mathcal {R} u\Vert _{W^{1,4}} \lesssim \Vert u\Vert _{L^4} \lesssim \Vert u\Vert _{C^0}\), with \(\alpha = 1/4\). \(\square \)
We have the following variant of [5, Lemma B.1] in [5].
Lemma 6
Let \(a\in C^2(\mathbb {T}^3)\). For \(1< p < \infty \), and any smooth function \(f \in L^p(\mathbb {T}^3)\), we have
Proof of Lemma 6
We follow the proof in [5]. Note that
As direct consequences of the Littlewood–Paley decomposition and Schauder estimates we have the bounds for \(1< p < \infty \) (see, for example, [14])
Combining these bounds with Hölder’s inequality and the embedding \(W^{1,4}(\mathbb {T}^3) \subset L^{\infty }(\mathbb {T}^3)\), we obtain
\(\square \)
It follows from the definition of \(w_{q+1}\) that
Hence \(\int _{\mathbb {T}^3} \nu (-\Delta )^{\theta } w_{q+1}dx = 0\) and \(\dfrac{d}{dt} \int _{\mathbb {T}^3} w_{q+1} dx = 0\). We obtain \(R_{q+1}\) by plugging \(v_{q+1} = v_q + w_{q+1}\) in (2), using (33) and the assumption that \((v_q,R_q)\) solves (2):
It follows from Lemma 4 that
Noting that \(\nabla \times \dfrac{w_{q+1}^{(p)}}{\lambda _{q+1}} = w_{q+1}^{(p)} + w_{q+1}^{(c)}\), Lemma 4 and (34) yield that
This is the crucial estimate to control the fractional viscosity. If we assume that \(p \sim 1, r \sim \lambda _{q+1}^{-1}\), we must have \(\theta < 5/4\) in order that the second term in (36) is small for \(\lambda _{q+1}\) sufficiently large.
It remains to estimate \(\widetilde{R}_{oscillation}\), which can be handled in the same way as in [5]. It follows from (19) that
Since \(E_{(\overline{\xi }, \overline{\xi }')}\) has zero mean, we can split it as
Using (15), (34) and (35), we obtain
Recall the vector identity \(A \cdot \nabla B + B \cdot \nabla A = \nabla (A \cdot B) - A \times (\nabla \times B) - B \times (\nabla \times A)\). For \(\overline{\xi }, \overline{\xi }' \in \Lambda \), using the anti-symmetry of the cross product, we can write
For the term \(E_{(\overline{\xi }, \overline{\xi }',2)}\), first consider the case \(\overline{\xi } + \overline{\xi '} \ne 0\). It follows from the above identity and (14) that
where the second term is a pressure, the third can be estimated analogously to \(E_{(\overline{\xi }, \overline{\xi }', 1)}\). Also note that the first and fourth term can estimated analogously. Using (16), (34) and (35), we obtain
Now consider \(E_{(\overline{\xi }, -\overline{\xi },2)}\). We can write
where we use (11) and the fact that \(\{ \overline{\xi }, A_{\overline{\xi }}, \overline{\xi } \times A_{\overline{\xi }} \}\) forms an orthonormal basis of \(\mathbb {R}^3\). Therefore, we can write
Using the identity \(\mathrm {Id} - \mathbb {P}_{LH} = \nabla \Delta ^{-1} {\text {div}}\) , we obtain
where the first and second terms are pressure terms. Using (16), (34) and (35), we obtain
It follows from (16) and (34) that
Let us now give the explicit definition of \(\widetilde{R}_{oscillation}\):
Finally, we estimate the time support of \(R_{q+1}\). Using (25) we obtain
Now we choose the parameters \(r, \sigma , \mu \). Fix \(\alpha \) so that
which is possible since \(\theta \in (-\infty ,5/4)\). Fix
Clearly (27) is satisfied. Choose \(p > 1\) sufficiently close to 1 so that
Note that \(\mathcal {C}_N\) is independent of \(\lambda _{q+1}\), due to (24). Combining the above estimates with Lemma 4, it is easy to check that, by taking \(\lambda _{q+1}\) sufficiently large, we arrive at (4), (6) and (7). This completes the proof of Lemma 1.
References
Bardos, C., Titi, E.S.: Onsager’s conjecture for the incompressible Euler equations in bounded domains. Arch. Ration. Mech. Anal. 228, 197–207 (2018)
Buckmaster, T., De Lellis, C., Isett, P., Székelyhidi Jr., L.: Anomalous dissipation for \(1/5\)-Hölder Euler flows. Ann. Math. 182(1), 127–172 (2015)
Buckmaster, T., De Lellis, C., Székelyhidi, L.: Dissipative Euler flows with Onsager-critical spatial regularity. Commun. Pure Appl. Math. 69(9), 1613–1670 (2016)
Buckmaster, T., De Lellis, C., Székelyhidi, L., Vicol, V.: Onsager’s conjecture for admissible weak solutions. Commun. Pure Appl. Math. 72(2), 229–274 (2019)
Buckmaster, T., Vicol, V.: Nonuniqueness of weak solutions to the Navier–Stokes equation. Ann. Math. 189(1), 101–144 (2019)
Caffarelli, L., Kohn, R., Nirenberg, L.: Partial regularity of suitable weak solutions to the Navier–Stokes equations. Commun. Pure Appl. Math. 35, 771–831 (1982)
Cheskidov, A., Luo, X.: Stationary and discontinuous weak solutions of the Navier–Stokes equations (2019). arXiv:1901.07485
Colombo, M., De Lellis, C., De Rosa, L.: Ill-posedness of leray solutions for the hypodissipative Navier–Stokes equations. Commun. Math. Phys. 362(2), 659–688 (2018)
Constantin, P., Titi, E.S., Weinan, F.: Onsager’s conjecture on the energy conservation for solutions of Euler’s equation. Commun. Math. Phys. 165(1), 207–209 (1994)
Daneri, S., Székelyhidi Jr., L.: Non-uniqueness and h-principle for Hölder-continuous weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 224(2), 471–514 (2017)
De Lellis, C., Székelyhidi Jr., L.: The Euler equations as a differential inclusion. Ann. Math. 170(3), 1417–1436 (2009)
De Lellis, C., Székelyhidi Jr., L.: Dissipative continuous Euler rows. Invent. Math. 193(2), 377–407 (2013)
De Rosa, L.: Infinitely many Leray–Hopf solutions for the fractional Navier–Stokes equations, Commun. Partial Differ. Equations 44(4) (2019)
Grafakos, L.: Classical Fourier analysis, Grad. Texts in Math., vol. 249, 2nd edn. Springer, New York (2008)
Isett, P.: Holder continuous Euler flows with compact support in time. Thesis (Ph.D.)–Princeton University (2013)
Isett, P.: A proof of Onsager’s conjecture. Ann. Math. 188(3), 1–93 (2018). arXiv:1608.08301v1
Jiu, Q., Wang, Y.: On possible time singular points and eventual regularity of weak solutions to the fractional Navier–Stokes equations. Dyn. Partial Differ. Equatinos 11(4), 321–343 (2014)
Katz, N.H., Pavlović, N.A.: A cheap Caffarelli–Kohn–Nirenberg inequality for the Navier–Stokes equation with hyper-dissipation. Geom. Funct. Anal. 12(2), 355–379 (2002)
Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63(1), 193–248 (1934)
Lions, J.L.: Quelques résultats d’existence dans des équations aux dérivées partielles non linéaires. Bull. Soc. Math. Fr. 87, 245–273 (1959)
Lions, J.L.: Quelques Méthodes de Resolution des Problémes aux Limites Non linéaires, vol. 1. Dunod, Paris (1969)
Luo, X.: Stationary solutions and nonuniqueness of weak solutions for the Navier–Stokes equations in high dimensions. Arch. Ration. Mech. Anal. 233, 701–747 (2019)
Modena, S., Sattig, G.: Convex integration solutions to the transport equation with full dimensional concentration (2019). arXiv:1902.08521
Modena, S., Székelyhidi, L.: Non-uniqueness for the transport equation with Sobolev vector fields. Ann. PDE 4, 18 (2018)
Modena, S., Szekelyhidi, L.: Non-renormalized solutions to the continuity equation. Calc. Var. Partial Differ. Equations 58, 208 (2019)
Olson, E., Titi, E.S.: Viscosity versus vorticity stretching: Global well-posedness for a family of Navier–Stokes-alpha-like models. Nonlinear Anal. Theory Methods Appl. 66(11), 2427–2458 (2007)
Scheffer, V.: An inviscid flow with compact support in space-time. J. Geom. Anal. 3(4), 343–401 (1993)
Wu, J.: Generalized MHD equations. J. Differ. Equations 195, 284–312 (2003)
Tao, T.: Global regularity for a logarithmically supercritical hyperdissipative Navier–Stokes equation. Anal. PDE 3, 361–366 (2009)
Acknowledgements
The authors would like to thank H. Ibdah and the anonymous referee for carefully reading the paper and for their constructive suggestions. The authors would also like to thank the “The Institute of Mathematical Sciences”, Chinese University of Hong Kong, for the warm and kind hospitality during which part of this work was completed. The work of T.L. is supported in part by NSFC Grants 11601258. The work of E.S.T. is supported in part by the ONR grant N00014-15-1-2333, the Einstein Stiftung/Foundation - Berlin, through the Einstein Visiting Fellow Program, and by the John Simon Guggenheim Memorial Foundation.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by C. De Lellis.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Luo, T., Titi, E.S. Non-uniqueness of weak solutions to hyperviscous Navier–Stokes equations: on sharpness of J.-L. Lions exponent. Calc. Var. 59, 92 (2020). https://doi.org/10.1007/s00526-020-01742-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-020-01742-4