Abstract
The paper is concerned with the IBVP of the Navier-Stokes equations. The result of the paper is in the wake of analogous results obtained by the authors in previous articles Crispo et al. (Ricerche Mat 70:235–249, 2021). The goal is to estimate the possible gap between the energy equality and the energy inequality deduced for a weak solution.
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1 Introduction
This note concerns the 3D-Navier–Stokes initial boundary value problem:
In system (1) \(\Omega \subseteq {\mathbb {R}}^{3}\) is assumed bounded or exterior, and its boundary is assumed smooth.
In the two recent papers [5, 6] the authors look for an energy equality for suitable weak solutions. Here, the term suitable is meant in the sense that a new solution is exhibited and not that an improvement is obtained to the one given in [3]. Actually, the crucial result of papers [5, 6] is the strong convergence in \(L^p(0,T;W^{1,2}(\Omega ))\), for all \(T>0\) and \(p\in [1,2)\), of a sequence \(\{v^m\}\) of smooth solutions to the “Leray’s approximating Navier–Stokes Cauchy problem” (see (4) below), [11].
Since the strong convergence is not in \(L^2(0,T;W^{1,2}(\Omega ))\), the authors attempt to obtain the energy equality employing the (differential and integral) energy equality of the approximating solutions and some auxiliary functions. Actually, the approaches used so far allow to prove an energy equality which involves other quantities. Here it is proved that a suitable weak solution exists and satisfies the following relation
where, thanks to the result of strong convergence in \(L^p(0,T;W^{1,2}(\Omega ))\), \(p\in [1,2)\) (see Lemma 1),
is of full measure in (0, T) for all \(T>0\), and
where \(J^m(\alpha )\) is the union of, at most, a countable sequence (\({\mathbb {N}}(\alpha ,m)\)) of disjoint intervals \((s_h,t_h)\subset (s,t)\) and the following holds:
Instead in the case of \(s=0\), one obtains
where
Roughly speaking the above intervals seem to contain the possible singular points S of the weak solution that, as is known, has \({\mathcal {H}}^\frac{1}{2}(S)=0\) (\({\mathcal {H}}^a\) Hausdorff’s measure), [16]. Of course, independently of the meaning of the conjecture for the intervals, from a physical view point the energy relation (2) would add a dissipative quantity which is not justifiable. If this is a necessary consequence of an initial datum only in \(L^2\), then from a physical point of view it is a right reason to reject the \(L^2\)-class as a class of existence.
Also in [15] the author considers the possibility to add a further dissipative term to the right hand side of the classical energy inequality, but, as already stressed in [5], our result is different, since we obtain the equality (2) with M(s, t) expressed only in terms of energy quantities (“kinetic or dissipated”). We think that this difference is of a special interest.
The proof of our result is based on a new existence theorem, where our weak solution is the limit of the sequence \(\{v^m\}\) of solutions to problem (4). In addition to the usual weak convergences of \(\{v^m\}\), there is the peculiarity that our weak solution is strong limit in \(L^p(0,T;W^{1,2}(\Omega ))\), for all \(T>0\) and \(p\in [1,2)\). This result, proved for the first time in [5] (as far as we know it is also the unique known proof), is obtained under the minimal assumption of \(v_0\in L^2(\Omega )\) and divergence free. As already said, it is important in order to obtain that \(\lim _m\Vert \nabla v^m(t)-\nabla v(t)\Vert _2=0\) almost every where in \(t>0\). This is a main difference with other results of existence of weak solutions, classical or more recent, as the ones furnished in [8] and in [9], obtained with stronger assumptions on the initial datum \(v_0\).
By making the minimal requirement on \(v_0\), from one hand we match the resultFootnote 1 obtained in [13], and from another hand we better match the questions of counterexamples, as we remark below.
The validity of an energy equality, without requiring extra conditions, is interesting to better delimit the case of validity of possible counterexamples.
Actually, in the papers [2] and [1] two examples of non-uniqueness are furnished.
The former works for very-weak solutions, which are continuous in \(L^2\)-norm, but do not verify an energy inequality of the kind given by Leray-Hopf, in other words neglecting the term M(s, t) with \(\ge 0\). Further, in the case of Leray-Hopf weak solutions their counterexample does not work.
The latter works with a homogeneous initial datum. Actually, the non-uniqueness is exhibited for solutions corresponding to a suitable data force, that, among other things, allows an energy equality.
The plan of the paper is the following. In Sect. 2 some preliminary lemmas are recalled and some new results of strong convergence are furnished. In Sect. 3 the statement and the proof of the chief result are performed.
2 Preliminary results
We set \(J^{1,2}(\Omega )\):=completion of \({\mathscr {C}}_0(\Omega )\) in \(W^{1,2}\)-norm, where \({\mathscr {C}}_0(\Omega )\) is the set of the test functions of the hydrodynamics.
Definition 1
For weak solution to the IBVP (1) we mean a field \(v:(0,\infty )\times \Omega \rightarrow {{\mathbb {R}}^{3}}\) such that for all \(T>0\)
-
1.
\(v\in L^\infty (0,T;L^2(\Omega ))\cap L^2(0,T;J^{1,2}(\Omega )) ,\)
-
2.
the field v solves the integral equation
\( \int \limits _{{s}}^{{t}}\Big [(v,\varphi _\tau )-(\nabla v,\nabla \varphi )+(v\cdot \nabla \varphi ,v)+(\pi _v,\nabla \cdot \varphi )\Big ]d\tau +(v(s),\varphi (s))=(v(t),\varphi (t)),\)
for all \(\varphi \in C^1_0([0,T)\times \Omega ),\)
-
3.
\(\displaystyle \lim _{t\rightarrow 0}\Vert v(t)-v_0\Vert _2=0\,.\)
For our goals we consider a mollified Navier–Stokes system. Hence problem (1) becomes
where \(f\in L^2(0,T,L^2(\Omega ))\), \(\{v_0^m\}\subset J^{1,2}(\Omega )\) converges to \(v_0\) in \(J^2(\Omega )\) and \(J_m[\cdot ]\equiv {{\widetilde{J}}}_{\frac{1}{m}}[\cdot ]\) where \({{\widetilde{J}}}_{\frac{1}{m}}[\cdot ]\) is Friedrichs’ (spatial) mollifier and we suppose that \(v^m\) is extended to zero in \({\mathbb {R}}^3-\Omega\).
Lemma 1
For all \(m\in {\mathbb {N}}\) there exists a unique solution to problem (4) such that for all \(T>0\)
Moreover, the sequence \(\{v^m\}\) is strong convergent to a limit v in \(L^p(0,T;W^{1,2}(\Omega ))\cap L^2(0,T;L^2(\Omega ))\), for all \(p\in [1,2)\), and the limit v is a weak solution to problem (1) with \((v(t),\varphi )\in C([0,T))\), for all \(\varphi \in J^2(\Omega )\).
Proof
This lemma for data force \(f=0\) is Theorem 6.1.1 proved in [5]. It is not difficult to image that the proof can be modified without difficulty assuming \(f\ne 0\). So that we consider as achieved the proof of the lemma.\(\square\)
Lemma 2
Let \(\Omega \subseteq {\mathbb {R}}^n\) and let \(u\in W^{2,2}(\Omega )\cap J^{1,2}(\Omega )\). Then there exists a constant c independent of u such that
provided that \(a\in [0,1)\).
Proof
The following lemma furnishes an integrability property of derivatives with respect to t of the sequence \(\{\Vert \nabla v^m\Vert _2\}\). This is made following the approach given in paper [5]. However, there are similar results directly concerning weak solutions. For the sake of completeness, we give the following references [4, 7, 17]. In any case, our proof is different from those given in the quoted papers.
Lemma 3
For any \(T>0\), there exists a constant \(M>0\), not depending on m, such that
where \(v^m\) is the solution of problem (4) stated in Lemma 1.
Proof
By virtue of the regularity of \((v^m,\pi ^m)\) stated in (5), we multiply Eq. (4)\(_1\) by \(P\Delta v^m-v^m_t\). Integrating by parts on \(\Omega\), and applying the Hölder inequality, we get
Applying inequality (6) with \(r=\infty\) and \(q=6\), by virtue of the Sobolev inequality, we obtain
By inequalities (7) and (8), we get
for all \(m\in {\mathbb {N}}\) and a.e. in \(t>0\,\). Substituting in inequality (9) the identity
and dividing by \((1+\Vert \nabla v^m(t)\Vert _2^2)^2\), we get the following estimate
where we set \(\rho _m(t):=\Vert \nabla v^m(t)\Vert _2^2\). Integrating on (0, T) we have
It follows that
Using the identity (10) we get
Using once again identity (10) we get
\(\square\)
Lemma 4
Let \(\{h_m(t)\}\) be a sequence of non-negative functions bounded in \(L^1(0,T)\). Also, assume that \(h_m(t)\rightarrow h(t)\) a.e. in \(t\in (0,T)\) with \(h(t)\in L^1(0,T)\). Let be \(g:(0,\alpha _0)\longrightarrow {\mathbb {R}}\) a continuous and strictly increasing function such that \(\lim \limits _{\alpha \rightarrow \alpha _0}g(\alpha )=+\infty\) and \(p:[0,1)\times [0,\infty )\longrightarrow [0,1]\) a continuous function such that \(p(\alpha ,\rho )=1\) if \(0\le \rho \le g(\alpha )\), \(p(\alpha ,\cdot )\) is weakly decreasing and \(\lim \limits _{\rho \rightarrow +\infty }p(\alpha ,\rho )=0\) for any \(\alpha \in (0,\alpha _0)\).
Then we get
Proof
We have
We fix \(\alpha \in (0,\alpha _0)\) and we consider the first integral. For any \(\varepsilon \in (0,\alpha _0-\alpha )\) we set
Hence we have
By (12) we get
hence, by the dominated convergence theorem, we have
Since \(p(\alpha ,\cdot )\) is decreasing, we get
Using the boundedness of the sequence \(\{h_m\}\) in \(L^1\) we obtain that
Since \(\lim \limits _{\varepsilon \rightarrow 0}p(\alpha ,g(\alpha _0-\varepsilon ))=0\) we have that
Now we consider the integral \(I_2(\alpha ,m)\). Since \(\left| p(\alpha ,h_m(t))h(t)\right| \le 1\) and \(\lim \limits _m h_m(t)=h(t)\) a.e. in \(t\in (0,T)\), by the dominated convergence theorem, we get
Finally, since \(\lim \limits _{\alpha \rightarrow \alpha _0}p(\alpha ,h(t))=1\) we have that
and this completes the proof. \(\square\)
3 The chief result
We recall the definition
where \(\{v^m\}\) is the sequence of solutions to problem (4). By virtue of the strong convergence stated in Lemma 1, the set \({\mathcal {T}}\) is certainly not empty and, as matter of fact, it is of full measure in (0, T) for all \(T>0\).
Theorem 1
Let v be the weak solution and \(\{v^m\}\) the related approximating sequence stated in Lemma 1. Then, for all \(t,s\in {\mathcal {T}}\), v satisfies the relation
with
where, for a suitable positive \(\alpha _0\) depending on (s, t), for all \(\alpha \in (\alpha _0,1)\), \(J^m(\alpha )\equiv {\underset{i\in {\mathbb {N}}(\alpha ,m)}{\cup }}(s_i,t_i)\) with \({\mathbb {N}}(\alpha ,m)\) which is, at most, a sequence of integers, and for all \(i\in {\mathbb {N}}(\alpha ,m)\) \((s_i,t_i)\subset (s,t)\) with \((s_i,t_i)\cap (s_j,t_j)=\emptyset\) for any \(i\not =j\), and
Moreover, if \(s=0\), the relation (16) holds with \(M(0,t)=\lim \limits _k M(s_k,t)\) where \(\{s_k\}\) is any sequence in \({\mathcal {T}}\) converging to 0.
Proof
We consider the sequence \(\{v^m\}\) of solutions to problem (4) whose existence is ensured by Lemma 1. For all \(m\in {\mathbb {N}}\) the Reynolds-Orr equation holds:
We set \(\rho _m(t):=\Vert \nabla v^m(t)\Vert _2^2\) , and we consider
Fix \(s,t\in {\mathcal {T}}\), with \(s<t\) , \({\mathcal {T}}\) given in (15) . Let \(\alpha _1\) be such that
Hence, by virtue of the pointwise convergence, we claim the existence of \(m_0\) such that
We set \(\displaystyle A^m:=\max _{[s,t]}\rho _m(t)\). We denote by
If \(A_m\le \tan \alpha \frac{\pi }{2}\), then \(J^m(\alpha )\) is an empty set. If \(A_m>\tan \alpha \frac{\pi }{2}\) holds, since \(\rho _m(s)<\tan \alpha \frac{\pi }{2}\), there exists the minimum \(\overline{s}>s\) such that \(\rho _m(\overline{ s})=\tan \alpha \frac{\pi }{2}\) , as well, being \(\rho _m(t)<\tan \alpha \frac{\pi }{2}\), there exists the maximum \(\overline{ t}<t\) such that \(\rho _m(\overline{ t})=\tan \alpha \frac{\pi }{2}\). Thus, if \(J^m(\alpha )\) is a non-empty set, by the regularity of \(\rho _m(t)\), we get that \(J^m(\alpha )\) is at most the union of a sequence of open interval \((s_h,t_h)\) such that \(\rho _m(s_h)=\rho _m(t_h)=\tan \alpha \frac{\pi }{2}\). We justify the claim.
The set \(J^m(\alpha )\) is an open set, hence it is at most the countable union of maximal intervals \((s_h,t_h)\). We set \(E^m:=(s, t)-\overline{{\underset{h\in {\mathbb {N}}}{\cup }}(s_h,t_h)}\).
For all \(\tau \in E^m\) we have \(\rho _m(\tau )\le \tan \alpha \frac{\pi }{2}\), thus, by continuity of \(\rho _m\), we get \(\rho _m(s_h) =\tan \alpha \frac{\pi }{2}=\rho _m(t_h)\) for all \(h\in {\mathbb {N}}\). For the measure of \(J^m(\alpha )\) we get
where we took the energy relation (18) into account and the strong convergence of the right-hand side too. Estimate (21) leads to (17). Recalling the definition of \(p(\alpha ,\rho _m(t))\), we have
where we took into account that, for all \(\alpha \in (0,1)\), function p is a Lipschitz’s function in \(\rho _m\), and \(\rho _m(t)\) is a regular function in t. Hence, we get \(p(\alpha ,\rho _m(t))\) is a Lipschitz’s function with respect to t. We multiply Eq. (18) for \(p(\alpha ,\rho _m(\tau ))\), with \(\alpha >\alpha _1\), and we integrate by parts on (s, t):
where we set
where we took (20) and definition of p into account. Letting \(m\rightarrow \infty\) and \(\alpha \rightarrow 1\), by virtue of the pointwise convergence in s and in t, and Lemma 4, we arrive at
where we set
Recalling the properties of \(J^m(\alpha )\), for all \(\alpha\) and m, integrating by parts, we get
Hence, we arrive at
We estimate the last integral. Let be
It results that
Hence, if \(\tau \in {{\widetilde{J}}}(\alpha )\cap {\mathcal {T}}\) we get that
On the complement of the set \({\mathcal {T}}\) we can set \(\rho =0\), since the value on a null measure set does not change the estimates. Since \(0\le \chi _{J^m(\alpha )}\frac{\rho _m^2}{1+\rho _m^2}\le 1\), by Fatou’s lemma, it follows that
Since \(\rho \in L^1\) and, by (26),
the last integral vanishes as \(\alpha\) tends to \(1^-\). Moreover
hence
Concerning the force term we have
It follows that
Using algebraic manipulation we obtain the following relation:
Substituting the above relation in Eq. (24) we get
At last we estimate the integral
where the last inequality follows by Lemma 3. Hence, by (28), we get
Multiplying Eq. (31) by \(\frac{2}{(1-\alpha )\pi }\) and passing to the limit using (32), (30) and (29), we get
By Eq. (18) we get
Let us consider the last integral. Since \(\left| \left( f(\tau ),v^m(\tau )\right) \right| \le \Vert f(\tau )\Vert _2\Vert v^m(\tau )\Vert _2\le c\Vert f(\tau )\Vert _2\) we can apply the Fatou’s lemma to get
with \({{\widetilde{J}}}(\alpha )\) defined in (25). Since \(\Vert f(\tau )\Vert _2\) is summable, considering (27), we get
and this completes the proof in the case of \(s,t\in {\mathcal {T}}\). In order to complete the proof of the theorem, we limit ourselves to remark that, letting \(s\rightarrow 0\), the left-hand side tends to values in 0, in particular on any sequence \(\{s_k\}\subset {\mathcal {T}}\) letting to 0, and as a consequence the limit on \(\{s_k\}\) of the right hand side is well posed. \(\square\)
Notes
In this connection in paper [13], the so called Prodi-Serrin condition for the energy equality for a weak solution is not required on the whole interval of existence, but just on \((\varepsilon ,T)\), that is \(L^4(\varepsilon ,T;L^4(\Omega ))\), for all \(\varepsilon >0\). This means that no extra assumption on the initial datum in \(L^2\) is needed for the validity of the energy equality.
In [8], from a different point of view, the extra condition \(L^4(\varepsilon ,T;L^4(\Omega ))\) is deduced for a special weak solution. Consequently, a local energy equality holds too.
Following the approach given in [10], under the same weaker extra assumption, the energy equality holds in the set of very-weak solutions.
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Acknowledgements
The research activity of F.C. and P.M. is performed under the auspices of GNFM-INdAM, and the research activity of C.R.G. is performed under the auspices of GNAMPA-INDAM. The research activity of F.C has been supported by the Program (Vanvitelli per la Ricerca: VALERE) 2019 financed by the University of Campania “L. Vanvitelli”. The research activity of C.R.G. is partially supported by PRIN 2020 “Nonlinear evolution PDEs, fluid dynamics and transport equations: theoretical foundations and applications.” The author express special thanks to the referees for the interesting comments that make the paper more readable.
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Maremonti, P., Crispo, F. & Grisanti, C.R. Navier–Stokes equations: a new estimate of a possible gap related to the energy equality of a suitable weak solution. Meccanica 58, 1141–1149 (2023). https://doi.org/10.1007/s11012-023-01642-9
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DOI: https://doi.org/10.1007/s11012-023-01642-9