Abstract
We consider the Navier–Stokes equations in three spatial dimensions and present a new proof of the Caffarelli–Kohn–Nirenberg theorem, based on a generalized notion of a local suitable weak solution, involving the local pressure. By estimating the integrals involving the pressure in terms of velocity, the pressure term is cancelled in the local decay estimates. In particular, our proof shows that the Caffarelli–Kohn–Nirenberg theorem holds for any open set \(\Omega \) without any restriction on the size and the regularity of the boundary. In addition, the method forms a basis for proving partial regularity results to other fluid models such as non-Newtonian models or models with heat conduction.
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1 Introduction
Let \(\Omega \subset \mathbb {R}^3\) be any domain and let \(0<T<+\infty \). Set \(Q= \Omega \times (0,T)\). We consider the following generalized Navier–Stokes equations (g-NSE)
where
The system (g-NSE) will be completed with the following boundary and initial conditions
where \(\varvec{u}_0\) is a given initial velocity distribution.
1.1 Models for the constitutive law \(\varvec{S}\)
Due to friction the deviatoric stress \(\varvec{S}\) depends on \(\varvec{D}(\varvec{u})\), where
In addition, in case of heat conducting fluids \(\varvec{S}\) can depend also on the temperature \(\theta \) of the fluid which is due to heat transfer. We present various models which are well-known models of fluid motions.
-
(i)
Newtonian fluid with constant viscosity: Here \(\varvec{S}\) is proportional to \(\varvec{D}(\varvec{u})\), i.e. there exists a constant \(\nu >0\) which is called the viscosity of the fluid such that
$$\begin{aligned} \varvec{S}= 2\nu \varvec{D}(\varvec{u}). \end{aligned}$$(1.3)Owing to \(\mathrm{div}\varvec{u}=0\) we have \(\mathrm{div}\varvec{S}= 2\nu \mathrm{div}\varvec{D}(\varvec{u}) = \nu \Delta \varvec{u}\). Thus, (g-NSE) turns into the usual NSE
$$\begin{aligned} \hbox { (NSE) }\qquad \left\{ \begin{array}{rl} \mathrm{div}\varvec{u}&{}\,=\, 0\quad \hbox {in}\quad Q,\\ \partial _t \varvec{u}+ (\varvec{u}\cdot \nabla ) \varvec{u}- \nu \Delta \varvec{u}+\nabla p &{}\,=\,-\mathrm{div}\varvec{f}\quad \hbox {in}\quad Q. \end{array}\right. \end{aligned}$$ -
(ii)
Newtonian fluid with non-constant viscosity: There exists a bounded measurable function \(\nu : Q \rightarrow \mathbb {R}\), such that
$$\begin{aligned} \varvec{S}= \nu \varvec{D}(\varvec{u}),\qquad 0< \nu _0 \le \nu (x,t) \le \nu _1 < +\infty \quad \hbox {for a. e.} (x,t)\in Q. \end{aligned}$$(1.4)where \(\nu _0, \nu _1= \mathrm{const }>0\).
-
(iii)
Non-Newtonian fluids with shear dependent viscosity: There exists a positive function \(\mu : \mathbb {R}_+ \rightarrow \mathbb {R}_+ \) such that
$$\begin{aligned} \varvec{S}= \mu (|\varvec{D}(u)|) \varvec{D}(\varvec{u}). \end{aligned}$$(1.5)In the engineering practice one often makes use of the so called power law model, where
$$\begin{aligned} \mu (s) = (1+ s^2)^{ \frac{q-2}{2} } \quad \hbox {or}\quad \mu (s)= s^{q-2},\quad 1<q<\infty . \end{aligned}$$(1.6)Here we distinguish between the following three cases
$$\begin{aligned} \begin{array}{llll} &{}\mathbf {First \ case}: &{}1<q <2 &{}\quad shear\;thinning,\\ &{}\mathbf {Second \ case}: &{} q=2 &{}\quad Newtonian,\\ &{}\mathbf {Third \ case}: &{}2< q< +\infty &{}\quad shear\;thickening. \end{array} \end{aligned}$$ -
(iv)
Heat conducting fluids: Due to heat conduction, the viscosity may depend on the temperature \(\theta \), such that
$$\begin{aligned} \varvec{S}= \nu (\theta ) \varvec{D}(\varvec{u}). \end{aligned}$$(1.7)and the generalized NSE being coupled by the equation of heat transport,
$$\begin{aligned} \partial _t \theta + \varvec{u}\cdot \nabla \theta - \mathrm{div}(\kappa \nabla \theta ) =\nu (\theta ) |\varvec{D}(\varvec{u})|^2, \end{aligned}$$(1.8)where \(\kappa >0\) denotes the heat capacity due to Fourier’s law. The Eq. (1.8) will be completed by appropriate initial and boundary conditions.
For further details on fluid mechanical background see [2, 13].
1.2 Notion of a weak solution
First, let us introduce the function spaces which will be used in what follows. By \(W^{k,\, q}(\Omega ), W^{k,\, q}_0(\Omega )\) \( \, (k\in \mathbb {N}; 1\le q \le +\infty )\) we denote the usual Sobolev spaces (see, e. g. [1]). Spaces of vector valued functions will be denoted by bold letters, i.e. instead of \( W^{k,\, q}(\Omega ; \mathbb {R}^m) , L^q(\Omega ; \mathbb {R}^m),\) etc. we write shorter \(\varvec{W}^{k,\, q}(\Omega ), \varvec{L}^q(\Omega ),\) etc.
Let \(\varvec{C}^\infty _{0, \sigma } (\Omega )\) denote the space of solenoidal smooth functions having compact support in \(\Omega \). We define
In particular, we set
Let \(X\) be a Banach space with norm \(\Vert \cdot \Vert _X\). Then, by \(L^q(a,b; X)\) we denote the space of Bochner measurable functions, such that
(for details see [18]).
Now, we introduce the notion of a weak solution. For the sake of simplicity we only consider the case of NSE with \(\nu =1\) and \(\varvec{f}= \mathbf{0}\).
Definition 1.1
Let \(\varvec{u}_0 \in \varvec{H}\). A function \(\varvec{u}: Q \rightarrow \mathbb {R}^3\) is called a weak solution to the NSE if
-
(i)
\(\varvec{u}\in L^\infty (0,T; \varvec{H}) \cap L^2(0,T ; \varvec{V})\).
-
(ii)
For every \(\varvec{\varphi }\in C^1([0,T); \varvec{C}^\infty _{0, \sigma }(\Omega ) )\):
$$\begin{aligned} \int \limits _{Q} - \varvec{u}\cdot \partial _t\varvec{\varphi }- \varvec{u}\otimes \varvec{u}: \nabla \varvec{\varphi }+ 2\varvec{D}\varvec{u}: \varvec{D}\varvec{\varphi }dxdt = \int \limits _{\Omega } \varvec{u}_0 \varvec{\varphi }(\cdot , 0) dx. \end{aligned}$$(1.9) -
(iii)
In addition, \(\varvec{u}\) is called a Leray–Hopf solution if the following energy inequality is fulfilled:
$$\begin{aligned} \Vert \varvec{u}(t)\Vert _{\varvec{H}}^2 + 2 \int \limits _{0}^{t}\int \limits _{\Omega } |\nabla \varvec{u}|^2 dx d s \le \Vert \varvec{u}_0\Vert ^2_{\varvec{H}} \quad \hbox {for a.e.} \quad t\in (0,T). \end{aligned}$$(1.10)
The existence of a Leray–Hopf solution is well-known and can be found in Leray [14] for the case \(\Omega = \mathbb {R}^3\) and in Hopf [11] for general bounded domains. However, the notion of a Leray–Hopf solution is not sufficient for the study of local regularity properties, since in general there is no control of the local energy. For this reason, first Scheffer [17] introduced the notion of a suitable weak solution and then he proves the partial regularity for such solutions. Later, this notion has been also used in the celebrated paper by Caffarelli–Kohn–Nirenberg [4] to obtain the partial regularity, proving that the 1-dimensional parabolic Hausdorff measure of the singular set is zero. For more simplified proofs of this result see [12, 15]. Recently a new proof of the Caffarelli–Kohn–Nirenberg theorem has been given by Vasseur in [21].
Let us now recall the notion of a suitable weak solution due to Scheffer. A pair \((\varvec{u}, p)\) is called suitable weak solution to (NSE) if \(\varvec{u}\) is a weak solution to (NSE), \(p \in L^{\frac{3}{2}}(Q)\) and the following local energy inequality is fulfilled for all nonnegative \(\phi \in C^\infty _0(Q)\) and for almost all \(t\in (0,T)\):
Clearly, in order to establish such suitable weak solutions to (NSE) one has to define a pressure function \(p\), which is only available if \(\Omega \) is a uniform \(C^2\) domain (cf. [5]). Therefore, we cannot expect to obtain such solutions for general domains. The same difficulties occur if one considers other models like heat conducting fluids or non-Newtonian fluids. The purpose of this paper is to provide a new method in constructing suitable weak solutions for general domains and general models. As we will see below, we may introduce a notion of a local suitable weak solution to the NSE instead defining a global pressure, we introduce a local pressure decomposition \(p = \partial _t p_h + p_0\). However, this notion requires some study on the steady Stokes equation, which will be the subject of Sect. 2.
2 Local estimate of the pressure
2.1 The definition of the spatial pressure
The aim of this section is the construction of a special operator which maps every \( \varvec{F}\in W^{-1,\, q}(\Omega )\) into a class \([p] \in L^q_\mathrm{loc}(\Omega )/ \mathbb {R}\). To begin with, we recall a well-known result due to Galdi et al. [7] (for a similar result on Lipschitz domains see [3]), which is the following
Lemma 2.1
(Galdi–Simader–Sohr) Let \(G \subset \mathbb {R}^n\) be a bounded domain with \(\partial G\in C^1\). Then, for every \(\varvec{F}\in \varvec{W}^{ -1,\, q}(G)\, (1<q<\infty )\) there exists exactly one pair \((\varvec{v}, p)\in \varvec{W}^{ 1,\, q}_{0,\sigma } (G)\times L^q_0(G)\) Footnote 1 such that
In addition, there holds
As an immediate consequence of Lemma 2.1 we have the
Lemma 2.2
For every \(1<q<\infty \) there holds
where
Lemma 2.2 enables us to define the surjective bounded operator \(\fancyscript{P}_{q, G}: \varvec{W}^{- 1,\, q}(G) \rightarrow L^q_0(G)\) by setting \(\fancyscript{P}_{q, G}\varvec{F}= p\), where \(\varvec{F}- \nabla p\in \varvec{W}^{- 1,\, q}_\mathrm{div} (G)\). Clearly, \(\fancyscript{P}_{q, G}\nabla p = p\) for all \(p\in L^q_0(G)\). Consequently, \(\nabla \fancyscript{P}_{q, G} \) defines a projection of \( \varvec{W}^{- 1,\, q}(G)\) onto \( \varvec{W}^{- 1,\, q}_\mathrm{grad} (G)\).
Notice, due to uniqueness of the weak solution we have \(\fancyscript{P}_{q, G} \varvec{F}= \fancyscript{P}_{s, G} \varvec{F}\) for all \(\varvec{F}\in \varvec{W}^{- 1,\, q}(G)\cap \varvec{W}^{- 1,\, s}(G)\) (\(1<s,q<+\infty \)). Thus, in what follows \(\fancyscript{P}_{G}\) stands for \(\fancyscript{P}_{q, G} \) for some \(1<q<+\infty \).
By means of elliptic regularity we have the following regularity property of \(\fancyscript{P}_{G}\):
Lemma 2.3
Let \(G\subset \mathbb {R}^n\) be a bounded \(C^2\) domain. Let \(\varvec{f}\in \varvec{L}^q(G) \hookrightarrow \varvec{W}^{ -1,\, q}(G)\). Then \(\fancyscript{P}_{G} \varvec{f}\in W^{1,\, q}(G)\) and there holds the estimate
where \(c=\mathrm{const }>0\) depending on \(q,n\) and the geometry of \(G\) only.
2.2 The time dependent case
Next, let \(\varvec{F}\in L^s(a,b, \varvec{W}^{ -1,\, q}(G))\). With help of Pettis’ theorem (e.g. see [24]; Chap. V.4.) we may define \(\fancyscript{P}_{G}: L^s(a,b,\varvec{W}^{- 1,\, q}(G)) \rightarrow L^s(a,b,L^q_0(G)) \) according to
Furthermore, for every \(\varvec{F}\in L^s(a,b,\varvec{W}^{- 1,\, q}(G))\) we define
Using the above definition of the projections \(\fancyscript{P}_{q, G}\) we have the following
Lemma 2.4
Let \(\varvec{u}\in L^2(0,T; \varvec{V}) \cap L^\infty (0,T; \varvec{H})\) be a weak solution to the (NSE) with \(\varvec{f}=\mathbf{0}\). Then for every bounded subdomain \(G\subset \Omega \) with \(\partial G\in C^2\) there holds
for all \(\varvec{\varphi }\in \varvec{C}^\infty _0(G\times (0,T))\), whereFootnote 2
In addition, we have the estimates,
for almost all \(t\in (0,T)\). Here \(c=\mathrm{const }>0\) depends on the geometry of \(G\) and in (2.7) on \(m\) only. In particular, if \(G\) is the ball \(B_R(x_0)\) then \(c\) in (2.7) depends only on \(m\), while in (2.8) and (2.9) \(c\) is an absolute constant.
Proof
According to Lemma 2.1 there exists \(\varvec{v}_{h}, \varvec{v}_2 \in L^2(0,T; \varvec{W}^{ 1,\, 2}_{0, \sigma } (G)), \varvec{v}_{1} \in L^{\frac{3}{2}}(0,T; \varvec{W}^{ 1,\, \frac{3}{2}}_{0, \sigma } (G))\) such that
Since \(\varvec{u}\) is a weak solution to (NSE) we see that
for all \(\varvec{\varphi }\in C^1_0(0,T; \varvec{C}^\infty _{0,\sigma } (G))\). Introducing the Steklov mean of \(f\in L^1(Q ) \) by
the above identity leads to
for all \(0<\lambda <T\).
Let \(0<\lambda <T\) be arbitrarily chosen. Fix \(\eta \in C^1_0(0,T-\lambda )\) and define
Inserting \(\varvec{\varphi }(x,t)= \varvec{\psi }(x) \eta (t)\) for \(\varvec{\psi }\in \varvec{C}^\infty _{0, \sigma } (G)\) into (2.13) and using Fubini’s theorem, it follows that
By the definition of \(\varvec{v}_h, \varvec{v}_1\) and \(\varvec{v}_2\) we see that \( \varvec{w}\in \varvec{W}_{0, \sigma } ^{1,\, \frac{3}{2}}(G)\). Clearly, \(\varvec{w}\) solves the Stokes system (2.1), (2.2) with \(\varvec{F}=0\). Thus, thanks to Lemma 2.1 we have \(\varvec{w}=0\). Recalling the definition of \(\varvec{w}\) this shows that
Whence, \( \partial _t (\varvec{v}_h)_{\lambda } + (\varvec{v}_1)_\lambda + (\varvec{v}_2)_\lambda =0\) a. e. in \(G\times (0,T-\lambda )\). Now, observing
applying integration by parts and passing to the limit \(\lambda \rightarrow 0\) we arrive at
Whence, the assertion follows from (2.10)–(2.12) using (2.14).
In order to verify (2.7) we may chose \(N\subset (0,T)\) with Lebesgue measure zero such that \(\varvec{u}(t)\in \varvec{L}^6(G)\) for all \(t\in (0,T)\setminus N\). Then (2.7) is an immediate consequence of (2.5) (cf. Lemma 2.3). Similarly, (2.8) and (2.9) follows by using (2.3) (cf. Lemma 2.1). \(\square \)
Remark 2.5
-
1.
Owing to \(\mathrm{div}\varvec{u}= 0\) we see that both \(p_{h,G} (t)\) and \(p_{2,G} (t)\) defined above are harmonic for a. e. \(t\in (0,T)\).
-
2.
As it is readily seen the statement of Lemma 2.4 remains true if one replaces the convective term \(\mathrm{div}\varvec{u}\otimes \varvec{u}\) by \(\mathrm{div}\varvec{H}\) for a general matrix \(\varvec{H}\in \varvec{L}^{\frac{3}{2}}(G\times (0,T)) \) and replacing \(p_{1,G} \) by \(- \fancyscript{P}_{G} \mathrm{div}\varvec{H}\).
3 Local suitable weak solutions to the Navier–Stokes equations
Based on the local projections introduced in Sect. 2 we are in a position to introduce a notion of local suitable weak solutions, which reads as follows
Definition 3.1
A weak solution \(\varvec{u}\) to (NSE) is called a local suitable weak solution, if for every ball \(B\subset \subset \Omega \) there holds
for all nonnegative \(\phi \in C^\infty _0(B\times (0,T))\) and for almost all \(t\in (0,T)\), where \(\varvec{v}_B = \varvec{u}+ \nabla p_{h,B} \) and
For the definition of the operator \(\fancyscript{P}_{q, B}\) see Sect. 2.
Now, we obtain the following result on the existence of a local suitable weak solution
Theorem 3.2
For every \(\varvec{u}_0\in \varvec{H}\) there exists a local suitable weak solution to (NSE).
Proof
Let \(\eta \in C^\infty (\mathbb {R}) \) with \(\eta \equiv 1\) on \((-\infty ,1)\) and \(\eta \equiv 0\) in \((2, +\infty )\). Set \(\eta _\varepsilon (\tau )= \eta (\varepsilon \tau )\, (\tau \in \mathbb {R}; \varepsilon >0) \). Clearly, there exists a unique weak solution \(\varvec{u}_\varepsilon \in L^2(0,T; \varvec{V})\cap L^\infty (0,T; \varvec{H})\) to the approximate system
Integration by parts gives
Furthermore, by using Hölder’s inequality and Sobolev’s embedding theorem we infer that
Thus, by means of reflexivity and Banach–Alaoglu’s theoremFootnote 3 one finds a sequence \((\varepsilon _k)\) with \(\varepsilon _k \rightarrow 0\) and a function \(\varvec{u}\in L^2(0,T; \varvec{V})\cap L^\infty (0,T; \varvec{H})\) such that
In addition, we have for a. e. \(t\in (0,T)\)
Next, fix a ball \(B\subset \subset \Omega \). Define
Owing to (2.5) and (3.2) we have
Thus, eventually passing to a subsequence there exists \(p_h \in L^2(0,T; W^{ 1,\, 2}(B))\) such that
Observing (3.4) by virtue of the boundedness of \(\fancyscript{P}_{2, B}\) there holds \(p_h = p_{h, B} = \fancyscript{P}_{2, B} \varvec{u}\). In addition, thanks to (3.6) we find
for almost all \(t\in (0,T)\). Let \(t\in (0,T)\) such that (3.7) is true. Let \(x\in B\) and \(B_r(x) \subset B\). Since \(p_{h, B}^{k}(t)\) and \(p_{h,B} (t)\) are harmonic functions by using the mean value property we have
This shows that
On the other hand, by (3.2) for every \(0< \tau <1\) and every multi-index \(\alpha \in \mathbb {N}_0^3 \) we have
Then, by virtue of (3.8) taking into account that \(p_{h, B}^k\) are harmonic, we deduce that
Define, \(\varvec{v}_B^k := \varvec{u}_{\varepsilon _k} + \nabla p_{h,B}^k\) a. e. in \(B\times (0,T)\). By Lemma 2.4 we see that \(\varvec{v}_B^k \in L^\infty (0,T; \varvec{L}^2(B))\cap L^2(0,T; \varvec{W}^{1,\, 2}(B))\) solving the equation
in the sense of distributions. According to (3.2) and the boundedness of \(\fancyscript{P}_{\frac{3}{2}, B} \) we infer that
Hence, the sequence \((\varvec{v}_{B}^k )\) is relatively compact in \(L^1(B\times (0,T))\). Taking into account (3.2), (3.3), (3.5) and (3.9) we deduce that
Once more appealing to (3.9) we get
Consequently, there holds
This shows that \(\varvec{u}\) is a weak solution to (NSE). In addition, recalling the definition of \(p^k_{1, B} \) and \(p^k_{2,B} \) from (3.5) and (3.11) we deduce that
Now, it only remains to verify the validity of the local energy inequality (3.1). To this aim we fix \(t\in (0,T)\) such that (3.6) and (3.7) are fulfilled. Then, let \(\phi \in C^\infty _0(B\times (0,T))\) be a nonnegative function. In (NSE)\(_{ \varepsilon _k } \) testing with \(\varvec{v}_{B}^k \phi \) and using integration by parts we are led toFootnote 4
where
By means of (3.4), (3.5), (3.7) and (3.9) using Banach–Steinhaus’ theorem we see that
On the other hand, with the help of (3.9), (3.10), (3.11), (3.12) and (3.13) we verify
Thus, from (3.14), (3.15) and (3.16) it follows that
This completes the proof of (3.1). \(\square \)
Remark 3.3
-
1.
Besides the local energy inequality (3.1) we may write an alternative one. In fact, the last integral on the right of (3.1) can be rewritten as follows. Using integration by parts together with the identity \(2(\varvec{u}\cdot \nabla ) \varvec{u}=2 \mathrm{curl }\varvec{u}\times \varvec{u}+ \nabla |\varvec{u}|^2\) we get
$$\begin{aligned}&\int \limits _{0}^{t} \int \limits _{B} 2\varvec{u}\otimes \varvec{u}:\nabla (\nabla p_{h, B} \phi ) - |\varvec{u}|^2 \nabla p_{h, B}\cdot \nabla \phi dxds\nonumber \\&\quad = -2\int \limits _{0}^{t} \int \limits _{B} (\mathrm{curl }\varvec{u}\times \varvec{u}) \cdot \nabla p_{h,B} \phi dxdt.\qquad \end{aligned}$$(3.17) -
2.
Observing that
$$\begin{aligned} 2\fancyscript{P}_{q, B} (\mathrm{curl }\varvec{u}(t) \times \varvec{u}(t) )&= 2\fancyscript{P}_{q, B} \mathrm{div}(\varvec{u}(t) \otimes \varvec{u}(t) ) - \fancyscript{P}_{q, B} \nabla |\varvec{u}(t)|^2\\&= - 2p_{1,B}(t) - |\varvec{u}(t)|^2+ (|\varvec{u}(t)|^2)_B \quad \forall \, 1<q \le 3, \end{aligned}$$using Lemma 2.3 and Sobolev’s embedding theorem we obtain the estimate
$$\begin{aligned} {\left\{ \begin{array}{ll} \Big \Vert 2p_{1,B} (t)+ |\varvec{u}(t)|^2- (|\varvec{u}(t)|^2)_B \Big \Vert _{L^{q}(B) } \le c \Vert \mathrm{curl }\varvec{u}(t) \times \varvec{u}(t) \Vert _{\varvec{L}^{\frac{3q}{3+q}}(B) }\\ \hbox {for a.e.} \, t\in (0,T),\quad \forall \, \frac{3}{2}< q \le 3. \end{array}\right. } \end{aligned}$$(3.18)
4 Caccioppoli-type inequalities in terms of \(\varvec{u}\) only
In this section we will derive two Caccioppoli-type inequalities, which play the main role in the proof of the partial regularity. First let us introduce the notations which will be used below. For \(x_0\in \mathbb {R}^3\) and \(0<R<+\infty \) we denote by \(B_R=B_R(x_0)\) the usual ball with center \(x_0\) and radius \(R\). In addition, for \(X_0=(x_0, t_0)\in \mathbb {R}^3\times \mathbb {R}\) we introduce the parabolic cylinder
For the proof of partial regularity of suitable weak solutions it will be important to obtain decay estimates for quantities like \(R^{-\alpha } \Vert \varvec{u}\Vert _{L^q(Q_R)} \) which are invariant under the natural scaling of the Navier–Stokes equations. The main quantities we are going to work with are the following,
Lemma 4.1
Let \(\varvec{u}\in L^2(0,T; \varvec{V})\cap L^\infty (0,T; \varvec{H}) \) be a local suitable weak solution to (NSE). Then for every \(Q_{2R} =Q_{2R} (X_0)\subset Q\) we have the following two Caccioppoli-type inequalities
where \(c>0\) denotes an absolute constant.
Proof
\(1^\circ \) Proof of (4.1). Let \(R\le r < \rho \le 2R\) be arbitrarily chosen. Set \(\sigma := \frac{r+\rho }{2}\). Let \(\phi \in C^\infty _0 (Q)\) such that \(0\le \phi \le 1\) in \(Q\), \( \phi \equiv 0\) in \(Q\setminus B_\sigma \times (t_0 - \sigma ^2, t_0 + \sigma ^2)\), \(\phi \equiv 1\) on \(Q_r\) and \( |\partial _t\phi |+ |\Delta \phi | + |\nabla \phi |^2 \le c (\rho -r)^{-2}\) in \(Q\). Then, from (3.1) with \(B= B_\rho \) replacing \(\phi \) by \(\phi ^2\) therein we get
for almost all \(t\in (t_0-\rho ^2, t_0)\).
-
(i)
Since \(p_{h, B} = -\fancyscript{P}_{2, B}\varvec{u}\) , according to Lemma 2.3 we have
$$\begin{aligned} I_1 \le c(\rho -r)^{-2} \int \limits _{Q_\rho } |\varvec{u}|^2 + |\nabla p_{h,B}|^2 dx ds \le c(\rho -r)^{-2} \Vert \varvec{u}\Vert ^2_{\varvec{L}^2(B_\rho )}. \end{aligned}$$ -
(ii)
Recalling that \(p_{0,B} = p_{1,B} + p_{2,B} = - \fancyscript{P}_{2, B}\mathrm{div}(\varvec{u}\otimes \varvec{u}) + \fancyscript{P}_{\frac{3}{2} , B} \Delta \varvec{u}\) by virtue of Lemma 2.1 we find
$$\begin{aligned} \Vert p_{1,B}\Vert _{\varvec{L}^{3/2} (Q_\rho )}&\le c \Vert \varvec{u}\Vert _{\varvec{L}^3(Q_\rho )}^2,\\ \Vert p_{2,B}\Vert _{\varvec{L}^{2} (Q_\rho )}&\le c \Vert \nabla \varvec{u}\Vert _{\varvec{L}^2(Q_\rho )}. \end{aligned}$$Thus, with help of Hölder’s inequality together with the estimates above we easily deduce
$$\begin{aligned} I_2 \le c(\rho -r)^{-1} \Vert \varvec{u}\Vert _{\varvec{L}^3(Q_\rho )}^2 \Vert \varvec{v}_B\Vert _{\varvec{L}^{3} (Q_\rho )} + c (\rho -r)^{-1} \Vert \nabla \varvec{u}\Vert _{\varvec{L}^2(Q_\rho )} \Vert \varvec{v}_B \Vert _{\varvec{L}^{2} (Q_\rho )}. \end{aligned}$$On the other hand, by using Lemma 2.3 having \( \Vert \varvec{v}_B\Vert _{\varvec{L}^{3} (Q_\rho )} \le c\Vert \varvec{u}\Vert _{\varvec{L}^{3} (Q_\rho )} \) and \( \Vert \varvec{v}_B\Vert _{\varvec{L}^{2} (Q_\rho )} \le c\Vert \varvec{u}\Vert _{\varvec{L}^{2} (Q_\rho )} \), we apply Young’s inequality to arrive at
$$\begin{aligned} I_2 \le c(\rho -r)^{-1} \Vert \varvec{u}\Vert _{\varvec{L}^3(Q_\rho )}^3 + c (\rho -r)^{-2} \Vert \varvec{u}\Vert ^2_{\varvec{L}^{2} (Q_\rho )} + \frac{1}{4}\Vert \nabla \varvec{u}\Vert ^2_{\varvec{L}^2(Q_\rho )}. \end{aligned}$$ -
(iii)
Again using Hölder’s inequality recalling that \(\Delta p_{h,B}=0\) in \(B\times (0,T)\) and arguing as above we infer
$$\begin{aligned}&I_3 \le c(\rho -r)^{-1} \Vert \varvec{u}\Vert _{\varvec{L}^3(Q_\rho )}^3 + c \Vert \varvec{u}\Vert _{\varvec{L}^3(Q_\rho )}^2 \left( \int \limits _{Q_\sigma } |\nabla ^2 p_{h,B} |^3 dxds\right) ^{1/3}\\&\quad \le c(\rho -r)^{-1} \Vert \varvec{u}\Vert _{\varvec{L}^3(Q_\rho )}^3. \end{aligned}$$
Inserting estimates of \(I_1-I_3\) into (4.3) and applying Hölder’s inequality we are let to
By means of Sobolev’s embedding theorem using Hölder’s inequality we get
Recalling \(\Delta p_{h,B} =0\) and applying Lemma 2.3 we see that
Combining the last two inequalities we deduce that
Similarly, one proves
By means of (4.5) and (4.6) we obtain
with an absolute constant \(c>0\). By a well-known algebraic iteration argument (see, e. g. [8] ) we get
Multiplying both sides by \(R^{-1} \) we complete the proof of the first inequality (4.1).
\(2^\circ \) Proof of (4.2). For given \(R\le r< \rho \le 2R\) let \(\phi \in C^\infty _0(Q) \) denote the same cut-off function as in \(1^\circ \). Appealing to Remark 3.3/1. besides (4.3) we have the inequalityFootnote 5
-
(i)
Clearly, as in \(1^\circ \) we see that
$$\begin{aligned} I_1 \le \rho (\rho -r)^{-2} \Vert \varvec{u}\Vert ^2_{L^3(t_0-\rho ^2, t_0; \varvec{L}^{9/4} (B_\rho ))}. \end{aligned}$$ -
(ii)
Making use of (3.18) with \(q=2\) with help of Hölder’s inequality arguing similar as in \(1^\circ \) we find
$$\begin{aligned} I_2&\le c (\rho -r)^{-1} \Vert \mathrm{curl }\varvec{u}\times \varvec{u}\Vert _{\varvec{L}^{6/5}(Q_\rho )} \Vert \phi \varvec{v}_B\Vert _{L^6(t_0-\rho ^2, t_0; \varvec{L}^2(B_\rho ))}\\&\qquad \qquad + c (\rho -r)^{-1}\Vert \nabla \varvec{u}\Vert _{\varvec{L}^2(Q_\rho )} \Vert \varvec{v}_B\Vert _{\varvec{L}^2(Q_\rho )}\\&\le c \rho ^{\frac{2}{3}} (\rho -r)^{-2} \Vert \mathrm{curl }\varvec{u}\Vert ^2_{\varvec{L}^{2}(Q_\rho )} \Vert \varvec{u}\Vert ^2_{\varvec{L}^3(Q_\rho )} \!+\! c \rho (\rho \!-\!r)^{-2} \Vert \varvec{u}\Vert ^2_{L^3(t_0\!-\!\rho ^2, t_0; \varvec{L}^{9/4} (B_\rho ))}\\&\qquad \qquad + \frac{1}{4} \Vert \nabla \varvec{u}\Vert ^2_{\varvec{L}^2(Q_\rho )} + \frac{1}{4}\Vert \phi \varvec{v}_B\Vert ^2_{L^\infty (t_0-\rho ^2, t_0; \varvec{L}^2(B_\rho ))}. \end{aligned}$$ -
(iii)
Similarly as above using Lemma 2.2 we estimate
$$\begin{aligned} I_3' \le c \rho ^{\frac{2}{3}} (\rho -r)^{-2} \Vert \mathrm{curl }\varvec{u}\Vert ^2_{\varvec{L}^{2}(Q_\rho )} \Vert \varvec{u}\Vert ^2_{\varvec{L}^3(Q_\rho )} + \frac{1}{4}\Vert \phi \varvec{v}_B\Vert ^2_{L^\infty (t_0-\rho ^2, t_0; \varvec{L}^2(B_\rho ))}. \end{aligned}$$Inserting the estimates of \(I_1, I_2\) and \(I_3'\) into (4.7) we are led to
$$\begin{aligned}&\Vert \phi \varvec{v}_B \Vert ^2_{L^\infty (t_0-\rho ^2; \varvec{L}^2(B_\rho ))} + \Vert \phi \nabla \varvec{v}_B \Vert ^2_{\varvec{L}^2(Q_\rho )}\nonumber \\&\quad \le c \rho (\rho -r)^{-2} \Vert \varvec{u}\Vert ^2_{L^3(t_0-\rho ^2; \varvec{L}^{9/4} (B_\rho ))} \!+\! c \rho ^{\frac{2}{3}} (\rho -r)^{-2} \Vert \mathrm{curl }\varvec{u}\Vert ^2_{\varvec{L}^{2}(Q_\rho )} \Vert \varvec{u}\Vert ^2_{\varvec{L}^3(Q_\rho )} \nonumber \\&\qquad \qquad \qquad + \frac{1}{4}\Vert \nabla \varvec{u}\Vert ^2_{\varvec{L}^2(Q_\rho )}. \end{aligned}$$(4.8)Using the same reasoning as in \(1^\circ \) from (4.8) we deduce that
$$\begin{aligned}&\Vert \varvec{u}\Vert ^2_{L^3(t_0-r ^2; \varvec{L}^{18/5} (B_r))} +\Vert \nabla \varvec{u}\Vert ^2_{\varvec{L}^2(Q_r)}\\ \,&\quad \le \, c R (\rho -r)^{-2} \Big (\Vert \varvec{u}\Vert ^2_{L^3(t_0-4R ^2; \varvec{L}^{9/4} (B_{2R} ))} \!+\! c R ^{-\frac{1}{3}} \Vert \mathrm{curl }\varvec{u}\Vert ^2_{\varvec{L}^{2}(Q_{2R} )} \Vert \varvec{u}\Vert ^2_{\varvec{L}^3(Q_{2R})} \Big )\\&\qquad \qquad \qquad + \frac{1}{2}\Vert \nabla \varvec{u}\Vert ^2_{\varvec{L}^2(Q_\rho )}, \end{aligned}$$where \(c>0\) denotes an absolute constant. As in \(1^\circ \) from the last estimate we obtain the second inequality (4.2). \(\square \)
5 Partial regularity
On the basis of the two Caccioppoli-type inequalities (4.1) and (4.2), we are now in a position to prove the following local regularity result
Theorem 5.1
Let \(\varvec{u}\) be a local suitable weak solution to (NSE).
-
1.
There exists \(\varepsilon _1>0\) such that for any \(Q_R(X_0)\subset Q\) there holds
$$\begin{aligned} R^{-\frac{4}{3}} \left( \,\,\int \limits _{Q_R(X_0)} |\varvec{u}|^3dxdt \right) ^{\frac{2}{3}} \le \varepsilon _1\quad \Longrightarrow \quad \varvec{u}|_{Q_{R/2}} \in \varvec{C}^0(\overline{Q_{R/2}(X_0)} ). \end{aligned}$$(5.1) -
2.
There exists \(\varepsilon _2>0\), such that for every \( X_0\subset Q\) there holds
$$\begin{aligned}&\limsup _{R \rightarrow 0}R^{-1} \int \limits _{Q_R(X_0)} |\mathrm{curl }\varvec{u}|^2 dx dt \le \varepsilon _2 \nonumber \\&\quad \Longrightarrow \quad \exists \rho >0:\quad \varvec{u}|_{Q_{\rho }(X_0)} \in \varvec{C}^0(\overline{Q_{\rho }(X_0)} ). \end{aligned}$$(5.2)In particular, if \(S(\varvec{u})\) is the set of possible singularities then
$$\begin{aligned} \mathcal{P}_1(S(\varvec{u})) = 0, \end{aligned}$$(5.3)where \(\mathcal{P}_1\) stands for the one-dimensional parabolic Hausdorff measure.
Proof
\(1^\circ \) Let \(Q_R = Q_R(X_0)\subset Q\) be fixed. Let \(\zeta \in C^\infty _0(B_{R/2} )\) be a cut-off function, with \(\zeta \equiv 1 \) on \(B_{R/4} \). Noticing, that
by the \(L^p-L^q\) theory of the heat equation (cf. [10]) there exists a unique weak solution
to the system of the heat equations
Then, applying divergence to both sides of the above system we se that the vector function \(\varvec{w}= \mathrm{div}\varvec{W}\) is a weak solution to the system
By the \(L^p-L^q\) theory of the heat equation (cf. [10]) making use of Gagliardo–Nirenberg’s inequality we infer
In addition, verifying that
it follows that \(\varvec{w}\in L^\infty (t_0-R^2/4, t_0; \varvec{L}^{3/2} )\).
Next, applying Hölder’s inequality together with Sobolev’s inequality, making use of Lemma 2.1 and (4.1), from the inequality above we infer
Next, set \( \varvec{U}:= \varvec{u}- \varvec{w}\) a. e. in \(Q_{R/2}\). Clearly, according to Lemma 2.2 \(\varvec{U}\) satisfies the following equation
in the sense of distributions. Define,
where
Recalling that both \( p_{h, B_{R/2} }\) and \(p_{2, B_{R/2} }\) are harmonic we see that \(\varvec{V}\) is a caloric function in \(Q_{R/4} \), i.e.
Thus, we get for \(0 < \tau < \frac{1}{8}\)
On the other hand, using the mean value property of harmonic functions, from the definition of \(\varvec{V}\) we get for \(0< \tau < \frac{1}{8}\)
By means of Lemmas 2.1 and 2.3, using (4.1) we infer
with an absolute constant \(c>0\), where
Alternatively, using (4.2) we have
Applying (4.1) and then using (5.6) and (5.4) we estimate
Thus, there exists an absolute constant \(c_1\), such that for every \(0< \tau < 1\) and for every \(Q_R = Q_R(X_0) \subset Q\),
If \( E(R) \le \frac{1}{4 (2c_1)^{16} } \) then with \( \tau := \frac{1}{(2 c_1)^3}\) we get
In fact, from (5.8) and the definition of \(\tau \) it follows that
Define,
Let \(Q_{R}(X_0) \subset Q\), such that \(E(R, X_0) \le \varepsilon _1\). Then, observing \(Q_{R/2}(Y)\subset Q_R(X_0) \) for every \(Y \in Q_{R/2}(X_0) \) we have
Hence, (5.8) gives
Thus, by using a standard iteration argument the above inequality yields
with an absolute constant \(C>0\). By a similar reasoning as in [22] and [23] from (5.9) we get \(\varvec{u}+\nabla p_{h, B_R}\in \varvec{C}^{0,\alpha } (\overline{Q_{R/2}(X_0)} )\) for some \(0<\alpha < 1 \) and \(\nabla p_{h, B_R} \in \varvec{C}^{0 } (Q_{R}(X_0) )\). Whence, the claim.
\(2^{\circ } \) Given \(Q_R = Q_R(X_0)\subset Q\) let \(\varvec{w}\in L^{\frac{3}{2}}(t_0- R^2/4, t_0; \varvec{W}^{1,\frac{9}{5}} ) \cap L^{\infty }(t_0- R^2/4, t_0; \varvec{L}^{\frac{3}{2}} )\) denote the unique solution to the heat equation
Once more using the \(L^p-L^q\) theory of the heat equation (cf. [10]) we see that
By the aid of Lemma 2.3 and Hölder’s inequality taking into account (4.2) we are led to
Next, setting \(\varvec{U}:= \varvec{u}- \varvec{w}\) we see that \(\varvec{U}\) verifies (5.5). Accordingly, as in \(1^\circ \) with help of (5.7), (5.10) and (4.2) we estimate
Thus, there exists an absolute constant \(c_2>0\), such that for every \(0<\tau <1\) there holds
Define \(\tau := \frac{1}{(2 c_2)^3} \) and \(\varepsilon _2 := \frac{1}{4 (2c_2)^{16} } \). Let \(X_0 \in Q\), such that \( \limsup _{R \rightarrow 0} \widetilde{B }(R,X_0) \le \varepsilon _2\). Arguing as above we see that there exists \(R_0 >0 \) such that
which shows that \(\lim _{R \rightarrow 0} E(R, X_0) = 0\). According to the first statement of the theorem \(\varvec{u}\) is continuous in a neighbourhood of \(X_0\).
Finally, the set of singular points \(S(\varvec{u})\) is containted in the set of all \(X_0 \in Q_T\) for which \(\limsup _{R \rightarrow 0^+} R^{-1} \int \limits \nolimits _{Q_R(X_0)} |\nabla \varvec{u}|^2 dxdt >0 \). Thus, as in [4] one proves \(\mathcal{P}_1 (S(\varvec{u}))=0\). This completes the proof of the theorem. \(\square \)
Notes
Here \(L^q_0(G)\) means the space of all \(f\in L^q(G)\) with \(\int \limits \nolimits _{G} f dx=0.\)
Here \(\varvec{I}\) stands for the identity matrix in \(\mathbb {R}^{3\times 3} \).
Note, \(L^\infty (0,T; \varvec{H})\) can be identified with \((L^1(0,T; \varvec{H}))^*\).
Here we argue as in the proof of Lemma 2.4 replacing \(\varvec{u}\otimes \varvec{u}\) by \(\eta _{\varepsilon _k} (|\varvec{u}_{\varepsilon _k} |^2) \varvec{u}_{\varepsilon _k} \otimes \varvec{u}_{\varepsilon _k} \) (cf. also Remark 2.5, 2.).
Here we have used the fact that \((\mathrm{curl }\varvec{u}\times \nabla p_{h,B} )\cdot \nabla p_{h,B}= 0\) a. e. in \(Q_\rho \).
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Wolf , J. On the local regularity of suitable weak solutions to the generalized Navier–Stokes equations . Ann Univ Ferrara 61, 149–171 (2015). https://doi.org/10.1007/s11565-014-0203-6
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DOI: https://doi.org/10.1007/s11565-014-0203-6