On the uniqueness of a suitable weak solution to the Navier–Stokes Cauchy problem

Abstract

The paper is concerned with the Navier–Stokes Cauchy problem. We investigate on some results of regularity and uniqueness related to suitable weak solutions corresponding to a special set of initial data. The suitable weak solution notion is meant in the sense introduced by Caffarelli–Kohn–Nirenberg. As further result we discuss the uniqueness of a set of suitable weak solutions (wider than the previous one) enjoying a “Prodi–Serrin” condition which is “relaxed” in space.

1 Introduction

We deal with the Navier–Stokes Cauchy problem

$$\begin{aligned}&u_t+u\cdot \nabla u+\nabla \pi _u=\varDelta u,\;\nabla \cdot u=0, \text{ in } (0,T)\times \mathbb {R}^3, \nonumber \\&u=u_0(x) \text{ on } \{0\}\times \mathbb {R}^3. \end{aligned}$$

(1)

In system (1) u is the kinetic field, $\pi _u$ is the pressure field, $u_t:= \frac{\partial }{\partial t}u$ and $u\cdot \nabla u:= u_k\frac{\partial }{\partial x_k}u$. The aim of the note is to investigate on some questions of regularity and uniqueness of weak solutions to the Cauchy problem (1). We believe that for this topic two different approaches are possible. One looks for establishing results by comparing a weak solution in the sense of Leray–Hopf with another weak solution enjoying extra conditions. We believe that this is the major question investigated in a very wide literature originated by Prodi and Serrin papers [15, 16], whose results employ for the uniqueness the technique of the famous Leray’s paper [12]. However the goal is slightly different in [12], because the uniqueness result has a precise role in the proof of the structure theorem. This approach conceals a possible compatibility between the extra conditions assumed for the weak solution and the initial datum. Hence, as consequence of the further hypothesis burdening the initial datum, the simple character of Leray–Hopf weak solution is lost. In this regard, some authors look for a characterization between the initial datum and the Prodi–Serrin conditions, see the recent contribution given by Kozono et al. [8] and see [5] for a review of results on this topic. In their approach there is no discrepancy between the extra conditions fit to obtain the uniqueness and the ones related to the regularity of solutions. Actually, the assumptions fit to ensure the uniqueness imply the regularity of a weak solution. However in [13] relaxed Prodi–Serrin conditions are considered, valid on any time interval properly contained in the one of existence, implying the regularity of the solutions for $t>0$. Since no equivalence between the Prodi–Serrin conditions and the uniqueness is known, as well as no example of non-uniqueness is known for weak solutions corresponding to an initial datum only in $L^2$, it is an open question if a weak solution satisfying a relaxed Prodi–Serrin condition is also unique.

Another possible approach looks for the coincidence between a weak solution and a solution in the existence class related to the initial datum. This second topic, which is more classical and posed by Leray in [12], is matter of study of a more recent literature. We refer to the interesting and clarifying paper [11]. As a consequence of the weaker assumptions on the initial datum with respect to the $J^{1,2}$ assumption considered by Leray, we believe that they represent the real generalization of Leray’s result [12], that is, “Comparison d’une solution réguliére et d’une solution turbulent”.

In this note we follow this task. In particular, our existence class is in some sense close to the one of weak solutions of the $L^2$-theory. So that the results could achieve a further interest.

In order to better discuss the results of this note, we need some definitions and notations.

Definition 1

Let $u_0\in J^2(\mathbb {R}^3)$. A pair $(u,\pi _u)$, such that $u:(0,\infty )\times \mathbb {R}^3 \rightarrow \mathbb {R}^3$ and $\pi _u:(0,\infty )\times \mathbb {R}^3 \rightarrow \mathbb {R}$, is said a weak solution to problem (1) if

• (i) for all $T>0$, $u\in L^2(0,T; J^{1,2} (\mathbb {R}^3 ))$ and $\pi _u\in L^\frac{5}{3}((0,T)\times \mathbb {R}^3)$, for all $\psi \in {J}^{2} (\mathbb {R}^3), {(u(t), {\psi }) \in C((0, T))}$

• (ii) $\lim \limits _{t\rightarrow 0}(u(t), {\psi }) = ({u_0}, \psi ), \; {\text {for all }} {\psi \in \mathscr {C}_ {0} ({\mathbb {R}^3)}}$

• (iii) for all $t,s\in (0,T)$, the pair $(u,\pi _u)$ satisfies the equation:

  $$\begin{aligned}&\int \limits _{s}^{t}\Big [(u,\varphi _\tau )-(\nabla u,\nabla \varphi )+(u\cdot \nabla \varphi ,u)+(\pi _u,\nabla \cdot \varphi )\Big ]d\tau +(u(s),\varphi (s)), \\&\quad =(u(t),\varphi (t)) \hbox { for all }\varphi \in C^1_0([0,T)\times \mathbb {R}^3 ). \end{aligned}$$

In [1] an energy relation having also a local character is introduced to investigate on the regularity of a weak solution:

Definition 2

A pair $(u,\pi _u)$ is said a suitable weak solution if it is a weak solution in the sense of the Definition 1 and, moreover,

$$\begin{aligned}&\int \limits _{\mathbb {R}^3}|u(t)|^2\phi (t)dx+2\int \limits _{\sigma }^{t}\int \limits _{\mathbb {R}^3}|\nabla u|^2\phi \, dxd\tau \le \int \limits _{\mathbb {R}^3}|u(\sigma )|^2\phi (\sigma )dx \nonumber \\&\quad +\int \limits _{\sigma }^{t}\int \limits _{\mathbb {R}^3}|u|^2(\phi _\tau +\varDelta \phi )dxd\tau +\int \limits _{\sigma }^{t}\int \limits _{\mathbb {R}^3}(|u|^2+2\pi _u) u\cdot \nabla \phi dxd\tau , \end{aligned}$$

(2)

for all $t\ge \sigma$, for $\sigma =0$ and a.e. in $\sigma \ge 0$, and for all nonnegative $\phi \in C_0^\infty (\mathbb {R}\times \mathbb {R}^3)$.

We introduce the acronimus sws to mean a suitable weak solution in the sense of Definition 2 . In [1], the following existence result is proved:

Theorem 1

For all $u_0\in J^2(\mathbb {R}^3)$ there exists a sws.

As a consequence of inequality (2) and of the existence theorem one gets

Corollary 1

A sws enjoys the strong energy inequality:

$$\begin{aligned}&||u(t)||_2^2+2\int \limits _{s}^{t} ||\nabla u(\tau )||_2^2d\tau \le ||u(s)||_2^2,\nonumber \\&\quad \text{ for } \text{ all } t\ge s, \text{ for } s=0 \text{ and } \text{ a.e. } \text{ in } s> 0\,. \end{aligned}$$

(3)

Moreover for all s such that (3) holds we get

$$\begin{aligned} \lim _{t\rightarrow s^+}||u(t)-u(s)||_2=0\,. \end{aligned}$$

(4)

Following [1], in the sequel by the symbol $Q_r(t,x)$ we mean the parabolic cylinder

$$\begin{aligned} Q_r(t,x):=\{(\tau ,y):t-r^2<\tau<t \text{ and } |y-x|<r\}. \end{aligned}$$

In paper [4], as natural continuation of the results proved in [3] (actually some hints are already given in [3]) we proved two results of partial regularity that we resume in the following statement:

Theorem 2

Let u be a sws. Then there exists a set ${\mathbb {E}}\subseteq \mathbb {R}^3$, with $\mathbb {R}^3-{\mathbb {E}}$ having zero $H^a$-Hausdorff measure, with $a=\frac{3}{2}+\varepsilon$ and arbitrary $\varepsilon >0$, enjoying the property: for all $x\in {\mathbb {E}}$ there exist $\delta \in [0,1)$ and $t>0$ such that

$$\begin{aligned} u\in L^\infty \Bigg (Q_{( \frac{(1-\delta )s}{4})^\frac{1}{2}}\Bigg ( {\frac{7}{6}}s,x\Bigg )\Bigg ), \text{ for } \text{ all } s\in (0,t). \end{aligned}$$

(5)

In particular, if $(\tau ,y)$ is a Lebesgue’s point in $Q_{\big (\frac{(1-\delta )s}{4}\big )^\frac{1}{2}}({\frac{7}{6}}s,x)$, then we get

$$\begin{aligned} |u(\tau ,y)|\le \overline{c}\tau ^{-\frac{1}{2}} \,. \end{aligned}$$

(6)

Moreover, for all $R>0$ and $\sigma >0$ there exists a closed set ${\mathbb {F}}_\sigma \subset {\mathbb {E}}\cap B_R$ such that $|B_R-{\mathbb {F}}_\sigma | <\sigma$ and (5) holds for all $(s,x)\in (0,T_{{\mathbb {F}}_\sigma })\times {\mathbb {F}}_\sigma$ and (6) holds for the Lebesgue’s point $(\tau ,y)$ belonging to $\in Q_{\big (\frac{(1-\delta )s}{4}\big )^\frac{1}{2}}( {\frac{7}{6}}s,x)$, provided that $x\in {\mathbb {F}}_\sigma$.

Remark 1

• The set ${\mathbb {E}}$ depends on the initial datum $u_0$ . Actually, the set ${\mathbb {E}}$ is the set of those x, ensured by the Hardy–Littlewood–Sobolev inequality, for which

  $$\begin{aligned} \int \limits _{\mathbb {R}^3}\frac{\,|u_0(y)|^2 }{|x-y| }\,dy \,<\infty \,. \end{aligned}$$

  (7)

  From (7) the claim on the Hausdorff measure of $\mathbb {R}^3-{\mathbb {E}}$ is immediate.

• The set ${\mathbb {E}}$ is independent of the sws corresponding to the initial datum $u_0$. In particular, if we assume the existence of two sws corresponding to $u_0$, ignoring their possible coincidence, we can select not only the same set ${\mathbb {E}}$, but also the same set ${\mathbb {F}}_\sigma$. Different is the claim related to the existence of $\delta (x)$ and t(x). They can depend on the solution.

The aim of this note is to prove some uniqueness results. For this task we have to improve the results given in Theorem 2. In particular we cannot limit ourselves to prove (5) and (6). More precisely, by means of the initial datum, we have to give a upper bound which is useful for the uniqueness.

We introduce also the notion of $J^{1,2}$-regular solution:

Definition 3

A pair $(v,\pi )$ is said a $J^{1,2}$-regular solution if $(v,\pi _v)$, for some $T\le \infty$, solves problem (1) almost everywhere in $(t,x)\in (0,T)\times \mathbb {R}^3$, and

$$\begin{aligned} v\in C([0,T);J^{1,2}(\mathbb {R}^3))\cap L^2(0,T;W^{2,2}(\mathbb {R}^3))\,,\;v_t,\nabla \pi \in L^2(0,T;L^2(\mathbb {R}^3))\,. \end{aligned}$$

(8)

It is well known that for all $v_0\in J^{1,2}(\mathbb {R}^3)$ there exists on some interval $(0,T(v_0))$ a unique $J^{1,2}$-regular solution (cf. e.g. [12]).

The following theorem is an improvement of Theorem 2. To obtain the improvement we consider a suitable subset of $J^2(\mathbb {R}^3)$ as set of the initial data for problem (1).

Theorem 3

Let u(t, x) be a sws. Assume the existence of $v_0\in J^{1,2}(\mathbb {R}^3)$ such that

$$\begin{aligned} \sup _{\mathbb {R}^3}\int \limits _{\mathbb {R}^3}\frac{|u_0(y)-v_0(y)|^2}{|x-y| }\,dy<\frac{1}{(4c)^2 }\,\,, \end{aligned}$$

(9)

where c is an absolute constant. Denote by v the kinetic field of the $J^{1,2}$-regular solution corresponding to $v_0$, and by $(0,T_0)$ the related existence interval. For all $x\in \mathbb {R}^3$, there exists a $\delta \in [0,1)$ such that

$$\begin{aligned} u\in L^\infty \Bigg (Q_{(\frac{(1-\delta )s}{4})^\frac{1}{2}}(s,x)\Bigg )\,, \text{ for } \text{ all } s\in (0,T_0)\,, \end{aligned}$$

(10)

and for all Lebesgue’s points $(\tau ,y) \in Q_{\big (\frac{(1-\delta )s}{4}\big )^\frac{1}{2}}( s,x)$

$$\begin{aligned} |u(\tau ,y)|\le c[(1-\delta )\tau ]^{-\frac{1}{2}}+|v(\tau ,y)|\,. \end{aligned}$$

(11)

If $v_0\equiv 0$, then we get $T_0=\infty$ .

Remark 2

The assumption (9), which detects a subset of $J^2(\mathbb {R}^3)$, allow us to consider ${\mathbb {E}}\equiv \mathbb {R}^3$ and $\inf _{x\in \mathbb {R}^3}t(x)\ge T_0$. This is the meaning of assumption (9) and the improvement with respect Theorem 2. Assumption (9) with $v_0\equiv 0$ and, as a consequence, $T_0=\infty$, corresponds to a suitable small data. Actually, Theorem 3 is either a new existence theorem of regular solutions (local for large data and global for small data) and a theorem of regularity for a sws. Although the local character of regularity, we are not able to prove the uniqueness for the sws of Theorem 3. This motives the further following results.

In the following theorems the symbols $c_0$ and $\varepsilon _1$ denote two absolute constants whose meaning is the same of the one given in Proposition 1 by Caffarelli-Kohn-Nirenberg in paper [1]. We employ these constants in the proof of Lemma 6 sect. 4.

Theorem 4

Assume that u(t, x) is a sws. There is an absolute constant $c_1$ such that if for some $R_0>0$

$$\begin{aligned} \int \limits _{\mathbb {R}^3}\frac{\,| u_0(y)|^2 }{|x-y|}\,dy<{\frac{\varepsilon _1}{2c_1 }}\,, \text{ for } |x|>R_0\,, \end{aligned}$$

(12)

then

$$\begin{aligned} t^\frac{1}{2}|u(t,x)| <c_0\varepsilon _1^\frac{2}{3}\,, \text{ for } \text{ all } (t,x)\in (0,\infty )\times \{x:|x|>R_0\}\,. \end{aligned}$$

(13)

Moreover, assume $v_0,\,v_0^*\in J^{1,2}(\mathbb {R}^3)$ such that

$$\begin{aligned} \sup _{|x|\le R_0}\int \limits _{\mathbb {R}^3}\frac{\,|u_0(y)-v_0(y)|^2}{|x-y| } dy<{\frac{1}{(4c)^2 }}\,\,,\;\;\sup _{|x|\le R_0}\int \limits _{\mathbb {R}^3}\frac{\,|u_0(y)-v_0^*(y)|^2 }{|x-y|} dy<{\frac{\varepsilon _1}{2c_1 }}\,, \end{aligned}$$

(14)

where the constant c is as in (9). Set v and $v^*$ the kinetic fields of the $J^{1,2}$-regular solutions to problem (1) with initial datum $v_0$ and $v_0^*$, and with existence intervals $(0,T_0)$ and $(0, T_0^*)$, respectively. Then, for some $T^*\le T^*_0$, we get

$$\begin{aligned} t^\frac{1}{2}|u(t,x)| <c_0\varepsilon _1^\frac{2}{3}+ t^\frac{1}{2}|v^*(t,x)|\,, \text{ for } \text{ all } (t,x)\in (0,T^*)\times \mathbb {R}^3\,, \end{aligned}$$

(15)

and, if $T^*<T_0$, the following estimate holds:

$$\begin{aligned} |u(t,x)|\le \overline{c}{T^*}^{-\frac{1}{2}}+|v(t,x)|\,, \text{ for } \text{ all } (t,x)\in [T^*,T_0)\times \mathbb {R}^3\,, \end{aligned}$$

(16)

for some constant $\overline{c}$.

Remark 3

If we denote by $(0,T^*_0)$ the interval of existence of the solution $v^*$, then a priori we have $T^*\le T^*_0$. Since we do not compare the numerical value of $\varepsilon _1/2c_1$ and $1/(4c)^2$, a priori we consider $v_0\ne v_0^*$ and as a consequence $T_0\ne T^*_0$. Actually, the “perturbations $v_0$ and $v_0^*$ to the initial datum $u_0$” fulfill a different role.

The following theorem is a weak version of the above one

Theorem 5

Assume that u(t, x) is a sws. There is an absolute constant $c_1$ such that if for some $R'_0>0$

$$\begin{aligned} \int \limits _{\mathbb {R}^3}\frac{\,| u_0(y)|^2 }{|x-y|}\,dy<{ \frac{\varepsilon _1}{2c_1 }}\,, \text{ for } \text{ all } |x|>R'_0\,, \end{aligned}$$

(17)

then (13) holds on $(0,\infty )\times \{|x|>R_0'\}$. Moreover, for all $\sigma >0$ there exists a closed set ${\mathbb {F}}_\sigma$ such that $|B_{R_0'}-{\mathbb {F}}_\sigma |<\sigma$ and there exist functions $v{^*}{(t, x)}$ and v(t, x) enjoying (8) on (0, T*) and on (0, $T_0$), respectively, independent of u(t, x), such that

$$\begin{aligned} t^\frac{1}{2}|u(t,x)| <c_0{\varepsilon _1}^\frac{2}{3}+ t^\frac{1}{2}|v^*(t,x)|\,, \text{ for } \text{ all } (t,x)\in (0,T^*)\times {\mathbb {F}}_\sigma \,, \end{aligned}$$

(18)

and if $T^*<T_0$, the following estimate holds:

$$\begin{aligned} |u(t,x)|\le \overline{c}{T^*}^{-\frac{1}{2}}+|v(t,x)|\,, \text{ for } \text{ all } (t,x)\in [T^*,T_0)\times {\mathbb {F}}_\sigma \,, \end{aligned}$$

(19)

for some constant $\overline{c}$.

Remark 4

We believe that the assumptions (12) and (17) are of some interest. Actually they ensure the regularity of a sws in the exterior of a ball. This result is hailing from [1] where, being $\nabla u_0\in L^2(|x|>R)$, the assumption on the initial datum is slightly stronger. In the prospective to look for sufficient conditions for the well-posedness of the Navier–Stokes, at least in the case of the Cauchy problem (or more in general for some unbounded domains), they appear useful, because these assumptions ensure regularity in the exterior of the ball for all $t>0$. So that the regularity question of a solution (or equivalently of a possible blow-up) is in the ball. We stress that ${\mathbb {F}}_\sigma$ depends on the initial datum. That is, the set is independent of the particular sws u. Roughly speaking, the theorem represents the spatial counterpart of the structure theorem (in time) give in [12]. This also is the meaning of Theorem 5.

We get the following uniqueness results:

Theorem 6

A sws of Theorem 4 is unique on $(0,T_0)$ provided that $\varepsilon _1^\frac{2}{3}<1/(\sqrt{2}\pi c_0)$.

The second uniqueness result is concerned with sws of Theorem 5 for which a sort of (actually weaker) “Prodi–Serrin” condition is employed [6, 15, 16].

We set ${\mathbb {F}}^c_\sigma :=B_{R^{\prime }_0}-{\mathbb {F}}_\sigma$. For arbitrary $\rho >0$, the set $I_\rho :={\underset{x\in {\mathbb {F}}^c_{\sigma }}{\cup }B_\rho (x)}$ is a neighborhood of ${\mathbb {F}}^c_\sigma \,.$ The symbol $\chi _\rho$ is the characteristic function of $I_\rho$. The symbol $J_\delta [\cdot ]$ denotes the mollifier, and, for $\delta \in (0,\rho )$, we set

$$\begin{aligned} h(x):=J_\delta [\chi _\rho ](x)\in [0,1], \text{ with } h(x)=1 \text{ for } \text{ all } x\in {\mathbb {F}}^c_\sigma \,. \end{aligned}$$

(20)

We denote by ${\mathbb {H}}:= \text{ supp } h$.

Theorem 7

Let u be a sws of Theorem 5. If, for some $\sigma > 0$ and $s > 3$, u belongs to $L^r(0,T_0;L^s({\mathbb {H}}))$, with $\frac{2}{r}+\frac{3}{s}=1$, then, for all $\eta >0$, u is a $J^{1,2}$-regular solution on $(\eta ,T_0)\times \mathbb {R}^3$, and it is the unique corresponding to $u_0$ in the set of solutions detected in Theorem 5, provided that $\varepsilon ^\frac{2}{3}_1<1/(\sqrt{2}\pi c_0)$.

Remark 5

• Theorem 6 states the uniqueness of the solutions furnished by Theorem 4. In the framework of the above settings, we consider Theorem 6 belonging to the second kind of approaches to the uniqueness, and like a generalization of the Leray one (actually by means of Hardy’s inequality all $u_0\in J^{1,2}$ satisfies (12) and (14)).

• An initial datum satisfying (12) and (14) is an element of $BMO^{-1}_T$. In this setting we have the results of existence and uniqueness (both local) proved in [7] (see also [10]) and the result of uniqueness proved in [9]. However, in our local uniqueness theorem the hypotheses require no condition of smallness to the initial datum in $BMO^{-1}_T$, or else the $\lim _{t\rightarrow 0}t^\frac{1}{2}||u(t)||_{\infty }=0$, u weak solution. Further comparison is due for the assumptions (12) and (14) with the ones made for other uniqueness results reproduced in [11] by Lemarié–Rieusset. Here we prefer to refer [11]. In [11] data $u_0$ are assumed in ${L^2} \cap BMO^{-1}_0\cap \dot{B}^{-s}_{q,\infty }$, $q\in (3,\infty )$ and $-s>-1+\frac{2}{q}$. Here

  $$\begin{aligned} \dot{B}^{-s}_{q,\infty } \text{ is } \text{ normed } \text{ with } \sup _{t>0}t^{\frac{s}{2}}||U(t)||_q<\infty , \end{aligned}$$

  (21)

  where U is the heat solution corresponding to $u_0$. The uniqueness result concerns the coincidence of the Leray weak solution with a mild solution both corresponding to $u_0$. This result, that we set in the second kind of uniqueness results, that is uniqueness in the class of existence, are a generalization of the conditions on the initial data in order to obtain uniqueness with respect to the one by Leray [12].

  Our result of uniqueness, that we consider also as belonging to the second kind, has a special character ascribable to two different facts. The first is that no comparison is possible between the space ${L^2} \cap BMO^{-1}_0\cap \dot{B}^{-s}_{q,\infty }$ and the one of the assumptions (12) and (14). Actually, the solution to the heat equation with $u_0\in L^2(\mathbb {R}^3)$ satisfying (12) and (14) does not satisfy (21) for any $q>2$ . In this connection see Theorem A$'$ in [14] for the characterization of convolutions in weighted spaces (in Sect. 2, for the reader convenience we reproduce the statement of Theorem A$'$). The second is the fact that we can consider uniqueness in the case of $\sup _{t\in (0,T_0)}t^\frac{1}{2}||u(t)||_\infty <\infty$, (u weak solution to problem(1)) which is out of the results furnished in [11]. This is due to their approach based on the energy inequality. Instead, thanks the assumption of smallness on the “perturbation” $u_0-v_0$, and the fact that we argue by duality in the way suggested in [6], we can avoid the condition.

• Theorem 7 is stated for $s>3$. This restriction can be removed, hence the result also holds for $s=3$. For the sake of brevity we omit the proof.

• Theorem 7 is meant in the first kind of uniqueness results. The theorem is based on a relaxed space Prodi–Serrin condition furnishing regularity and uniqueness. Actually, it represents a different relaxed condition with regard to the one introduced in [13], that is a relaxed condition just with respect to the time variable. In fact, for arbitrary $\sigma >0$, on set ${\mathbb {F}}^c_\sigma \subset \mathbb {R}^3$, related to the “unknown” behavior of a weak solution, we have a Prodi–Serrin condition in an arbitrary neighborhood ${\mathbb {H}}\supset {\mathbb {F}}^c_\sigma$, and on ${\mathbb {F}}_\sigma$ the solution $u\in L^2_{weak}(0,T;L^\infty ({\mathbb {F}}_\sigma ))$. As far as we are able to recognize, in the light of the regularity expressed in Theorem 5, these extra assumptions appear as the most congenial for a sws.

The plan of the paper is the following. In Sect. 2 we recall some well known results preliminary for the next sections. In Sect. 3 we formulate a weighted energy relation, which is a key tool in order to apply the criterium of regularity established in [1]. In Sect. 4 we extend to the solutions of a “perturbed Navier–Stokes system” the criterium of regularity stated in [1] for the solutions to the ordinary Navier–Stokes system. In Sect. 5 we furnish the proof of Theorems 3–5. Finally, in Sect. 6,7 we give the proof of the uniqueness results.

2 Some preliminary results

We start recalling the following result concerning a (relaxed) Prodi–Serrin condition for regularity:

Theorem 8

Assume that a weak solution v satisfies the condition:

$$\begin{aligned} \text{ for } \text{ all } \eta >0,\;v\in L^\rho (\eta ,T;L^\sigma (\varOmega )) \text{ with } \frac{3}{\sigma }+\frac{2}{\rho }=1,\;\rho <\infty , \end{aligned}$$

(22)

then v is a $J^{1,2}$-regular solution on $(\eta ,T)$ .

Proof

For the proof see [13] Theorem 2.

Lemma 1

Assume that $(u,\pi _u)$ is a sws. Then the pressure field admits the following representation formula

$$\begin{aligned} \pi _u(t,x)=-D_{x_i}D_{x_j}\int \limits _{\mathbb {R}^3}{\mathcal {E}} (x-y)u^i(y)u^j(y)dy\,, \text{ a.e. } \text{ in } (t,x)\in (0,\infty )\times \mathbb {R}^3\,, \end{aligned}$$

(23)

where ${\mathcal {E}}(z)$ denotes the fundamental solution of the Laplace operator, and the following estimates hold:

$$\begin{aligned}&\pi _u(t,x)\in L^\frac{5}{3}(0,T;L^\frac{5}{3}(\mathbb {R}^3))\,, \end{aligned}$$

(24)

$$\begin{aligned}&||\pi _u(1+|x|)^{-\frac{4}{3}}||_\frac{3}{2}\le c||u(1+|x|)^{-\frac{2}{3}}||_3^2\,, \end{aligned}$$

(25)

where c is a constant independent of $(u,\pi _u)$.

Indicate by $(u,\pi _u)$ and $(\overline{u}, \pi _{\overline{u}})$ two weak solutions in the sense of Definition 1, set $w=\overline{u}-u$ and $\pi _w:=\pi _{\overline{u}}-\pi _u$. The pair $(w,\pi _w)$ satisfies the following integral equation:

$$\begin{aligned}&\int \limits _{s}^{t}\Big [(w,\varphi _\tau )-(\nabla w,\nabla \varphi )+(w\cdot \nabla \varphi ,\overline{u})+(u\cdot \nabla \varphi ,w)+(\pi _w,\nabla \cdot \varphi )\Big ]d\tau \nonumber \\&\quad +(w(s),\varphi (s))=(w(t),\varphi (t)), \text{ for } \text{ all } \varphi \in C^1_0([0,T)\times \mathbb {R}^3 ). \end{aligned}$$

(26)

Lemma 2

The pressure field $\pi _w$ in (26) solves the equation

$$\begin{aligned} (\pi _w,\varDelta g)=-(w\otimes \overline{u}+ u\otimes w,\nabla \nabla g) \text{ for } \text{ all } g\in C_0^\infty (\mathbb {R}^3) \end{aligned}$$

(27)

and, for $q\in (1,3]$, $\pi _w$ enjoys the following estimate:

$$\begin{aligned} ||\pi _w||_q\le c||w(|u|+|\overline{u}|)||_q\,, \text{ a.e. } \text{ in } t>0\,, \end{aligned}$$

(28)

and for $q=1$

$$\begin{aligned} ||\pi _w||_{{\mathcal {H}}^1}\le c||w(|u|+|\overline{u}|)||_1\,. \end{aligned}$$

(29)

Moreover, for all smooth function h, the function $h\pi _w$ solves the equation

$$\begin{aligned} (h\pi _w,\varDelta g)= & {} -(\pi _w,g\varDelta h)-2(\pi _w,\nabla h\cdot \nabla g ) -(w\otimes \overline{u}+ u\otimes w,h\nabla \nabla g) \nonumber \\&-(w\otimes \overline{u}+ u\otimes w,g\nabla \nabla h)-(w\otimes \overline{u}+ u\otimes w,\nabla h\otimes \nabla g+\nabla g\otimes \nabla h ), \end{aligned}$$

(30)

provided that $g\in C_0^\infty (\mathbb {R}^3)$. Moreover, if function h and its derivates are bounded functions, we get $h\pi _w=\pi _1+\pi _2$ with

$$\begin{aligned} \pi _1(x):= & {} -\int \limits _{\mathbb {R}^3} {\mathscr {E}}(x)\pi _w \varDelta h-(w\otimes \overline{u}+u\otimes w)\cdot \nabla \nabla hdy\,, \nonumber \\ \pi _2(x):= & {} -\int \limits _{\mathbb {R}^3}2\pi _w\nabla {\mathscr {E}}(x)\cdot \nabla h+[w\otimes \overline{u}+u\otimes w]\cdot D(x) dy \nonumber \\&-\int \limits _{\mathbb {R}^3}\nabla \nabla {\mathscr {E}}(x)\cdot h(w\otimes \overline{u}+u\otimes w)dy \end{aligned}$$

(31)

where ${\mathscr {E}}(x)$ is the fundamental solution of Poisson’s equation and $D(x):=[\nabla {\mathscr {E}}(x)\otimes \nabla h+\nabla h\otimes \nabla {\mathscr {E}}(x)]$. Finally, the following estimates hold:

$$\begin{aligned}&||\pi _2||_2\le c(||h u||_\infty +||h\overline{u}||_\infty )||w||_2 +c||w||_2\big [||u||_3+||\overline{u}||_3\big ]\,, \text{ a.e. } \text{ in } t>0\,, \nonumber \\&||\nabla \pi _1||_2\le c||w||_2\big [||u||_3+||\overline{u}||_3\big ]\,, \text{ a.e. } \text{ in } t>0\,. \end{aligned}$$

(32)

Proof

The results (27)–(31) of the lemma are known. Hence we limit ourselves to prove (32). Actually with obvious meaning of the symbols, we have $\pi _2=\pi _{HLS}+\pi _{CZ}$. Estimating $||\pi _{HLS}||_2$ via the Hardy–Littlewood–Sobolev theorem, we get $||\pi _{HLS}||_2\le c(||\pi _w||_\frac{6}{5}+|||\overline{u}+u|w||_\frac{6}{5})$. Hence, recalling estimate (28), Hölder’s inequality leads to $||\pi _{HLS}||_2\le c||w||_2(||u||_3+||\overline{u}||_3)$. Estimating $||\pi _{CZ}||_2$ via the Calderon–Zigmund theorem, we get $||\pi _{CZ}||_2\le c||( hu+h\overline{u})w||_2)$. Hence Hölder’s inequality leads to estimate $||\pi _{CZ}||_2\le c(||hu||_\infty +||h\overline{u}||_\infty )||w||_2$. In the case of $||\nabla \pi _1||_2$, applying the Hardy–Littlewood–Sobolev theorem, we get $||\nabla \pi _1||\le c (||\pi _w||_\frac{6}{5}+ |||w|(|\overline{u}|+|u|)||_\frac{6}{5})$. Hence, recalling estimate (28) for $||\pi _w||_\frac{6}{5}$, Hölder’s inequality leads to $||\nabla \pi _1||_2\le c||w||_2(||u||_3+\overline{u}||_3)$, which completes the proof.

Lemma 3

Suppose that $|x|^\beta u\in L^2(\mathbb {R}^3)$ and $|x|^\alpha \nabla u\in L^2(\mathbb {R}^3)$. Also

• (i) $r\ge 2$, $\gamma +\frac{3}{r}>0$, $\alpha +\frac{3}{2}>0$, $\beta +\frac{3}{2}>0$, and $a\in [\frac{1}{2},1]$,

• (ii) $\gamma +\frac{3}{r}=a(\alpha +\frac{1}{2})+(1-a)(\beta +\frac{3}{2})$ (dimensional balance),

• (iii) $a(\alpha -1)+(1-a)\beta \le \gamma \le a\alpha +(1-a)\beta$.

Then, with a constant c independent of u, the following inequality holds:

$$\begin{aligned} |||x|^\gamma u||_r\le c|||x|^\alpha \nabla u||_2^a|||x|^\beta u||^{1-a}_2. \end{aligned}$$

(33)

Proof

See [1] Lemma 7.1 .

Let us consider the convolution of functions on $\mathbb {R}^n$:

$$\begin{aligned} K[\,g\,](x):=\int \limits _{\mathbb {R}^n}K(x-y)g(y)dy\,,\;\;x\in \mathbb {R}^n\,. \end{aligned}$$

Set

$$\begin{aligned} p\in [1,\infty ]\,,\;\alpha \in \mathbb {R}\,,\;\,L^p_\alpha (\mathbb {R}^n):=\Big \{g:\Big [\int \limits _{\mathbb {R}^n}|g(x)|^p|x|^{\alpha p}dx\Big ]^\frac{1}{p}<\infty \Big \}\,. \end{aligned}$$

Theorem 9

Suppose $r,q\in (1,\infty )$ and $p\in (1,\infty ]$. Then we get

$$\begin{aligned} ||K[\,g\,]||_{L^r_{-\gamma }}\le c||K||_{L^q_{\beta }}||g||_{L^p_\alpha } \text{ for } \text{ all } g\in L^p_\alpha (\mathbb {R}^n)\,, \end{aligned}$$

(34)

if, and only if,

$$\begin{aligned} \frac{1}{r}= & {} \frac{1}{p}+\frac{1}{q}+\frac{\alpha +\beta +\gamma }{n}-1\, \nonumber \\ \alpha< & {} \frac{n }{p' }\,,\,\;\beta<\frac{n}{q' }\,,\,\;\gamma<\frac{n}{r}\,\; \text{ if } p<\infty \,, \nonumber \\ \alpha \in (0,n),\,\;\beta< & {} \frac{n}{q' }\,,\,\;\gamma <\frac{n}{r}\,\; \text{ if } p=\infty \,, \end{aligned}$$

(35)

and

$$\begin{aligned} \alpha +\gamma \ge 0\,,\;\alpha +\beta \ge 0\,,\;\beta +\gamma \ge 0\,. \end{aligned}$$

(36)

Proof

See [14] Theorem A’.

By virtue of this result, we deduce that if $u_0\in L^2_{-\frac{1}{2}}(\mathbb {R}^3)$, then the corresponding heat solution does not belongs to $L^r(\mathbb {R}^3)$ (that is $L^r_0$) since (36)$_1$ does not hold (that is $-\frac{1}{2}+0\ngeq 0$). It is not difficult to verify that $L^3(\mathbb {R}^3)$ is not included in $L^2_{-\frac{1}{2}}(\mathbb {R}^3)$. As a consequence the $L^2_{-\frac{1}{2}}(\mathbb {R}^3)$ cannot be compared with Besov spaces detected with (21).

3 A weighted energy relation for suitable weak solutions

We reproduce, in a suitable way, Lemma 3.1 contained in [3]. Actually the proof of Lemma 3.1 of [3] consists of 5 steps. The first four steps realize a statement that we consider in the following Lemma 5.

We set $p(x,y,\mu ):=(|x-y|^2+\mu ^2)^{-\frac{1}{2}},\,\mu >0$, and

$$\begin{aligned} {\mathscr {E}}(v,t,x,\mu ):= \int \limits _{\mathbb {R}^3} |v(\tau ,y)|^2p(x,y)dy, \;{\mathscr {D}}(v,\tau ,x,\mu ):= \int \limits _{\mathbb {R}^3} |\nabla v(\tau ,y)|^2p(x,y)dy.\nonumber \\ \end{aligned}$$

(37)

We put before a lemma

Lemma 4

Let $(v,\pi _v)$ be a $J^{1,2}$-regular solution. Then the following weighted energy inequality hold:

$$\begin{aligned}&{\mathscr {E}}(v,t,x,\mu )+\int \limits _{0}^{t}{\mathscr {D}}(v,\tau ,x,\mu )d\tau +3\mu ^2 \int \limits _{0}^{t}\int \limits _{\mathbb {R}^3}\frac{v^2(\tau ,y)}{(|x-y|^2+\mu ^2)}_{\frac{5}{2}}dyd\tau \nonumber \\&\quad \le {\mathscr {E}}(v,0,x,\mu )+c\int \limits _{0}^{t}||v(\tau )||_2||\nabla v(\tau )||_2^5d\tau \,, \end{aligned}$$

(38)

for all $\mu >0$, $x\in \mathbb {R}^3$ and $t\in [0,T)$.

Proof

In Lemma 2.5 of [3] it is proved the following:

$$\begin{aligned}&{\mathscr {E}}(v,t,x,\mu )+\int \limits _{0}^{t}{\mathscr {D}}(v,\tau ,x,\mu )d\tau +3\mu ^2\int \limits _{0}^{t}\int \limits _{\mathbb {R}^3}\frac{v^2(\tau ,y)}{(|x-y|^2+\mu ^2)}_{\frac{5}{2}}dyd\tau \nonumber \\&\quad \le {\mathscr {E}}(0,x,\mu )+c\int \limits _{0}^{t}{\mathscr {E}}(v,\tau ,x,\mu )||\nabla v(\tau )||_2^4d\tau \,. \end{aligned}$$

(39)

Taking the definition of ${\mathscr {E}}(v,t,x,\mu )$ into account, and Lemma 3, then we get ${\mathscr {E}}(v,t,x,\mu )\le c||v(t)||_2||\nabla v(t)||_2$. Hence (38) follows from (39).

By the symbols ${\mathscr {E}}(w,t,x)$ and ${\mathscr {D}}(w,t,x)$ we mean (37) with $\mu =0$.

Lemma 5

Let $(u,\pi _u)$ be a suitable weak solution. Assume that there exists $v_0\in J^{1,2}(\mathbb {R}^3)$ such that for some $\varepsilon _0\le \frac{1}{4c^2 }$ and ${\mathbb {E}}\subseteq \mathbb {R}^3$

$$\begin{aligned} \sup _{x\in {\mathbb {E}}}\int \limits _{\mathbb {R}^3}\frac{|u_0(y)-v_0(y)|^2 }{|x-y| }\,dy<\varepsilon _0\,. \end{aligned}$$

(40)

Let $(v,\pi _{v})$ be the $J^{1,2}$-regular solution corresponding to $v_0$ and set $w:=u-v$. Then there exists a $t^*>0$ such that

$$\begin{aligned} {\mathscr {E}}(w,t,x)+\frac{1}{2}\int \limits _{0}^{t}{\mathscr {D}}(w,\tau ,x)d\tau \le \varepsilon _0, \text{ for } \text{ all } t\in [0,t^*) \text{ and } \text{ for } \text{ all } x\in {\mathbb {E}} \,. \end{aligned}$$

(41)

If assumption (40) holds with $v_0=0$, then we get (41) with $t^*=\infty$.

Proof

The proof of estimate (41) reproduces in a suitable way an idea employed in [2]. This idea follows the Leray arguments employed for the proof of the energy inequality in strong form. The proof is achieved by means of four steps.

Step 1. We start proving that for all $t>0,\,x\in \mathbb {R}^3$ and $\mu >0$

$$\begin{aligned}&{\mathscr {E}}(u,t,x,\mu ) +2\int \limits _{0}^{t}{\mathscr {D}}(u,\tau ,x,\mu )d\tau +\int \limits _{0}^{t} \int \limits _{\mathbb {R}^3} {|u(\tau )|^2}\varDelta _y pdyd\tau \le {\mathscr {E}}(u,0,x,\mu ) \nonumber \\&\quad +\int \limits _{0}^{t}\int \limits _{\mathbb {R}^3}\frac{|u(\tau )|^2\,u\cdot (x-y)}{(|x-y|^2+\mu ^2)^\frac{3}{2}} dyd\tau +2\int \limits _{0}^{t}\int \limits _{\mathbb {R}^3} \frac{\pi _{u}(\tau )u(\tau )\cdot (x-y)}{(|x-y|^2+\mu ^2)^\frac{3}{2}}dyd\tau \,. \end{aligned}$$

(42)

In the energy inequality (2) we set $\phi (\tau ,y):=(|x-y|^2+\mu ^2)^{-\frac{1}{2}}h_R(y)k(\tau )\in C_0^\infty (\mathbb {R}\times \mathbb {R}^3)$, with $h_R$ and k such that

$$\begin{aligned} h_R(y):=\left\{ \begin{array}{ll}1&{}\quad \text{ if } |y|\le R\\ \in (0,1)&{}\quad \text{ if } |y|\in (R,2R)\\ 0&{}\quad \text{ for } |y|\ge 2R\,, \end{array}\right. \text{ and } k(\tau ):=\left\{ \begin{array}{ll}1&{}\quad \text{ if } | \tau |\le t\\ \in (0,1)&{}\quad \text{ if } |\tau |\in (t,2t)\\ 0&{}\quad \text{ for } |\tau |\ge 2t\,. \end{array}\right. \end{aligned}$$

We get

$$\begin{aligned}&\int \limits _{\mathbb {R}^3}|u(t)|^2h_Rpdy +2\int \limits _{0}^{t}\int \limits _{\mathbb {R}^3}|\nabla u(\tau )|^2h_Rpdyd\tau +3\mu ^2\int \limits _{0}^{t} \int \limits _{\mathbb {R}^3} \frac{|u(\tau )|^2h_R}{(|x-y|^2+\mu ^2)}_{\frac{5}{2}}dyd\tau \nonumber \\&\quad \le \int \limits _{\mathbb {R}^3}|u_0|^2h_Rpdy +\int \limits _{0}^{t}\int \limits _{\mathbb {R}^3}\frac{|u(\tau )|^2h_R\,u\cdot (x-y)}{(|x-y|^2+\mu ^2)^\frac{3}{2}}\, dyd\tau \nonumber \\&\qquad +2\int \limits _{0}^{t}\int \limits _{\mathbb {R}^3} \frac{\pi _{u}(\tau )h_Ru(\tau )\cdot (x-y)}{(|x-y|^2+\mu ^2)^\frac{3}{2}}\,dyd\tau +F(t,R) \nonumber \\&\quad :=\int \limits _{\mathbb {R}^3}|u_0|^2h_Rpdy+I_1(t,x)+I_2(t,x)+F(t,R), \end{aligned}$$

(43)

where we set

$$\begin{aligned} F(t,R):= & {} \int \limits _{0}^{t} \int \limits _{\mathbb {R}^3}|u|^2\Big [2\nabla h_R\cdot \nabla p+\varDelta h_Rp+u\cdot \nabla h_Rp\Big ]dyd\tau +\int \limits _{0}^{t} \int \limits _{\mathbb {R}^3}\pi _uu\cdot \nabla h_Rpdyd\tau . \end{aligned}$$

Since $\pi _u,\,u^2\in L^\frac{5}{3}(0,T;L^\frac{5}{3}(\mathbb {R}^3))$, applying Hölder’s inequality and employing the decay of $\nabla h_R,\,\varDelta h_R$, for all $t>0$, we get $F(t,R)=o(R)$. We estimate the terms $I_i,\,i=1,2$. Since $\mu >0$, by virtue of the integrability properties of a suitable weak solution, applying Lemma 3 we get

$$\begin{aligned} |I_1(t,x)|\le \int \limits _{0}^{t}||{u}p^{-\frac{2}{3}}||_3 ^3d\tau \le c\int \limits _{0}^{t} ||{u}p^{-\frac{1}{2}}||_2 ||{(\nabla u)}p^{-\frac{1}{2}}||_2^2d\tau . \end{aligned}$$

For $I_2$, applying Hölder’s inequality and inequality (25), we obtain

$$\begin{aligned} |I_2(t,x)|\le c\int \limits _{0}^{t}||{u}p^{-\frac{2}{3}}||_3 ||{\pi _{u}}p^{-\frac{4}{3}}||_\frac{3}{2}d\tau \le c\int \limits _{0}^{t} ||{u}p^{-\frac{2}{3}}||_3^3d\tau . \end{aligned}$$

Hence, as in the previous case, applying Lemma 3, we get

$$\begin{aligned} |I_2(t,x)|\le c\int \limits _{0}^{t} ||{u}p^{-\frac{1}{2}}||_2 ||({\nabla u})p^{-\frac{1}{2}}||_2^2d\tau . \end{aligned}$$

Recalling that F(R, t) is an o(R), employing the estimates obtained for $I_i,i=1,2$, via the Lebesgue dominated convergence theorem, in the limit as $R\rightarrow \infty$, for all $t>0$ we deduce the inequality (42).

Step 2. In this step we derive a sort of Green’s identity between solutions $(u,\pi _u)$, which is a suitable weak solution, and $(v,\pi _{v})$, where $(v,\pi _{v})$ is the regular solution considered in Lemma 4, corresponding to the initial data $v_0\in J^{1,2}(\mathbb {R}^3)$. In the following (0, T) is the interval of existence of $(v,\pi _{v})$. We also recall that the regular solution $(v,\pi _{v})$ is smooth for $t\in (0,T)$. We denote by $\lambda (\tau )$ a smooth cutoff function such that $\lambda (\tau )=1$ for $\tau \in [s,t]$ and $\lambda (\tau )=0$ for $\tau \in [0,\frac{s}{2}]$.

For all $t,s\in (0,T)$, we consider the weak formulation iii) of Definition 1 written with $\varphi =\lambda vp$:

$$\begin{aligned}&\int \limits _{s}^{t}\Big [(pu,v_\tau )-(p\nabla u,\nabla v)+(pu\cdot \nabla v,u)+(\pi _u,v\cdot \nabla p)\Big ]d\tau +(pu(s),v(s) ) \nonumber \\&\quad =(pu(t),v(t)) + \int \limits _{s}^{t}\Big [ (\nabla u,v\otimes \nabla p)+(u\otimes u, v\otimes \nabla p) \Big ]d\tau . \end{aligned}$$

(44)

We multiply equation (1)$_1$ written for $(v,\pi _v)$ by up. After integrating by parts on $(s,t)\times \mathbb {R}^3$, we get

$$\begin{aligned}&\int \limits _{s}^{t}\Big [(pu,v_\tau )+(p\nabla u,\nabla v)+(pv\cdot \nabla v,u)-(\pi _{v},u\cdot \nabla p)\Big ]d\tau \nonumber \\&\quad = \int \limits _{s}^{t}\Big [(\nabla u ,v\otimes \nabla p)+(u\cdot v,\varDelta p)\Big ]d\tau \, . \end{aligned}$$

(45)

making the difference between formulas (44) and (45) we get

$$\begin{aligned}&\int \limits _{s}^{t}\Big [ -2(p\nabla u,\nabla v)+(pu\cdot \nabla v,u)-(pv\cdot \nabla v,u)+(\pi _u,v\cdot \nabla p)+(\pi _{v},u\cdot \nabla p)\Big ]d\tau \\&\quad =(pu(t),v(t)) -(pu(s),v(s))+ \int \limits _{s}^{t}\Big [ (u\otimes u, v\otimes \nabla p)-(u\cdot v,\varDelta p)\Big ] d\tau \, , \end{aligned}$$

Since in a suitable neighborhood of 0 all the terms of the last integral equation are continuous on the right, letting $s\rightarrow 0^+$, we get

$$\begin{aligned}&\int \limits _{0}^{t}\Big [(pu\cdot \nabla v,u) -2(p\nabla u,\nabla v) -(pv\cdot \nabla v,u)+(\pi _u,v\cdot \nabla p)+(\pi _{v},u\cdot \nabla p)\Big ]d\tau \nonumber \\&\quad =(pu(t),v(t)) -(pu(0),v_0)+ \int \limits _{0}^{t}\Big [ (u\otimes u, v\otimes \nabla p)-(u\cdot v,\varDelta p)\Big ] d\tau , \end{aligned}$$

(46)

which furnishes the wanted Green’s identity.

Step 3. Setting $w:=u-v$ and $\pi _w:=\pi _u-\pi _{v}$, let us derive the following estimate

$$\begin{aligned}&{\mathscr {E}}(w,t,x,\mu )+\int \limits _{0}^{t}{\mathscr {D}}(w,\tau ,x,\mu )d\tau \nonumber \\&\quad \le {\mathscr {E}}(w,0,x,\mu ) +c\int \limits _{0}^{t}{\mathscr {E}}^\frac{1}{2}(w,\tau ,x,\mu ){\mathscr {D}}(w,\tau ,x,\mu )d\tau \nonumber \\&\qquad + H(v,t,x,\mu ), \text{ for } \text{ all } t\in [0,T),x\in \mathbb {R}^3,\mu >0, \end{aligned}$$

(47)

where we set

$$\begin{aligned} H(v,t,x,\mu ):=c\int \limits _{0}^{t} ||\nabla v(\tau )||_2^4d\tau +c\int \limits _{0}^{t}{\mathscr {E}}(v,\tau ,x,\mu ) {\mathscr {D}}(\tau ,v,x,\mu )d\tau \,. \end{aligned}$$

We remark that from the representation formula (23) and the regularity of v we get that

$$\begin{aligned} \pi _w= & {} \pi ^1+\pi ^2\,, \nonumber \\ \pi ^1:= & {} D_{x_j}\int \limits _{\mathbb {R}^3}D_{y_i}{\mathcal {E}}(x-y)w^i(y)w^j(y)dy, \nonumber \\ \pi ^2:= & {} 2 \int \limits _{\mathbb {R}^3}D_{y_j} {\mathcal {E}}(x-y)w(y)\cdot \nabla {v}^j(y)dy. \end{aligned}$$

(48)

We sum estimates (38) and (42), then we add twice formula (46). written for $s=0$. Recalling the definition of $(w,\pi _w)$ and formula (48), after a straightforward computation we get

$$\begin{aligned}&{\mathscr {E}}(w,t,x,\mu )+2\int \limits _{0}^{t}{\mathscr {D}}(w,\tau ,x,\mu )d\tau +3\mu ^2\int \limits _{0}^{t}\int \limits _{\mathbb {R}^3}\frac{w^2(\tau ,x)}{(|x-y|^2+\mu ^2)}_\frac{5}{2}d\tau \nonumber \\&\quad \le {\mathscr {E}}(w,0,x,\mu )+F_1(w,t,x,\mu )+F_2(w,v,t,x,\mu ), \end{aligned}$$

(49)

where

$$\begin{aligned} F_1:= & {} F_1(w,t,x,\mu ):=\int \limits _{0}^{t}(w\otimes w,w\otimes \nabla p)d\tau +2\int \limits _{0}^{t}( \pi _1, w\cdot \nabla p)d\tau \\ F_2:= & {} F_2(w,v,t,x,\mu ):= 2\int \limits _{0}^{t}(\pi _2,w\cdot \nabla p)d\tau -2\int \limits _{0}^{t}(w\cdot \nabla v,wp)d\tau + \int \limits _{0}^{t}(v\cdot \nabla p,w^2) \,. \end{aligned}$$

The term $F_1$ admits the same estimate as $I_1$ and $I_2$ given in Step 1, hence we get

$$\begin{aligned} |F_1|\le c \int \limits _{0}^{t} {\mathscr {E}}^\frac{1}{2}(\tau ,w,x,\mu ){\mathscr {D}}(\tau ,w,x,\mu )d\tau \, \text{ for } \text{ all } t\in (0,T),\,x\in \mathbb {E},\,\mu >0. \end{aligned}$$

For term $F_2$ we estimate the first two terms in a different way from the last. Taking the representation formula of $\pi _2$ into account, we get

$$\begin{aligned} |\int \limits _{0}^{t}(\pi _2,w\cdot \nabla p)d\tau -\int \limits _{0}^{t}(w\cdot \nabla v,wp)d\tau |=|\int \limits _{0}^{t} p\nabla \pi _2\cdot wdyd\tau +\int \limits _{0}^{t}(w\cdot \nabla v,wp)dyd\tau |. \end{aligned}$$

Hence, applying the same arguments employed in Lemma 4 to estimate $J_1$ and $J_2$, for all $t\in [0,T),\,x\in \mathbb {E},\,\mu >0$, we get

$$\begin{aligned}&\left| \int \limits _{0}^{t}(\pi _2,w\cdot \nabla p)d\tau -\int \limits _{0}^{t}(w\cdot \nabla v,wp)d\tau \right| \le \int _0^t||wp^\frac{1}{2}||_4^2||\nabla v||_2d\tau \\&\quad \le \int \limits _{0}^{t} {\mathscr {E}}^\frac{1}{3}(w,\tau ,x,\mu ){\mathscr {D}}(w,\tau ,x,\mu )d\tau +c\int \limits _{0}^{t} ||\nabla v(\tau )||_2^4d\tau \,. \end{aligned}$$

For the last term in $F_2$, applying Hölder’s inequality, we get

$$\begin{aligned} \left| \int \limits _{0}^{t}(v\cdot \nabla p,w^2)d\tau \right| \le \int _0^t||wp^\frac{1}{2}||_4^2||vp||_2^2d\tau \,. \end{aligned}$$

By virtue of estimate (33), applying Young’s inequality with obvious meaning of the symbols we deduce:

$$\begin{aligned} |\int \limits _{0}^{t}(v\cdot \nabla p, w^2)d\tau |\le & {} c\int \limits _{0}^{t}{\mathscr {E}}^\frac{1}{4}(w,\tau ) {\mathscr {D}}^\frac{3}{4}(w,\tau ) {\mathscr {E}}^\frac{1}{4}(v,\tau ){\mathscr {D}}^\frac{1}{4}(v,\tau )d\tau \\\le & {} \int \limits _{0}^{t}{\mathscr {E}}^\frac{1}{3}(w,\tau ){\mathscr {D}}(w,\tau )d\tau +c\int \limits _{0}^{t}{\mathscr {E}}(v,\tau ){\mathscr {D}}(v,\tau )d\tau . \end{aligned}$$

Hence, we obtain

$$\begin{aligned} |F_2|\le & {} 2 \int \limits _{0}^{t}{\mathscr {E}}^\frac{1}{3}(w,\tau ,x,\mu ){\mathscr {D}}(w,\tau ,x,\mu )d\tau +c\int \limits _{0}^{t} ||\nabla v(\tau )||_2^4d\tau \\&+c\int \limits _{0}^{t}{\mathscr {E}}(v,\tau ,x,\mu ){\mathscr {D}}(v,\tau ,x,\mu )d\tau , \text{ for } \text{ all } t\in [0,T),\,x\in \mathbb {E},\,\mu >0. \end{aligned}$$

Finally, applying Young’s inequality with obvious meaning of the symbols we get

$$\begin{aligned} |F_2|\le & {} \int \limits _{0}^{t}{\mathscr {D}}(w,\tau )d\tau + c \int \limits _{0}^{t}{\mathscr {E}}^\frac{1}{2}(w,\tau ){\mathscr {D}}(w,\tau )d\tau +c\int \limits _{0}^{t} ||\nabla v(\tau )||_2^4d\tau \\&\quad +c\int \limits _{0}^{t}{\mathscr {E}}(v,\tau ){\mathscr {D}}(v,\tau )d\tau , \text{ for } \text{ all } t\in [0,T),\,x\in \mathbb {E},\,\mu >0. \end{aligned}$$

Estimates for $F_1,F_2$ and (49) furnish the integral inequality (47).

Step 4. Deduction of estimate (41).

Under our assumptions, a fortiori we get

$$\begin{aligned} {\mathscr {E}}(w,0,x,\mu )<\varepsilon _0,\ \text{ for } \text{ all } \mu >0 \text{ and } x\in \mathbb {E}. \end{aligned}$$

(50)

Moreover, by virtue of the $J^{1,2}$-regularity of the solution $(v,\pi _{v})$ there exists a $t^*$ such that

$$\begin{aligned} H(v, t,x,\mu )<\varepsilon _0-{\mathscr {E}}(w,0,x,\mu )\,, \text{ for } \text{ all } x\in \mathbb {E} \text{ and } \mu >0 \text{ and } t\in (0,t^*). \end{aligned}$$

(51)

Since u and v are right continuous in $L^2$-norm in $t=0$, for all $\mu >0$ and $x\in \mathbb {E}$ the same continuity property holds for ${\mathscr {E}}(w,t,x,\mu )$. Therefore there exists a $\delta =\delta (x,\mu )>0$ such that

$$\begin{aligned} {\mathscr {E}}(w,t,x,\mu )\le \varepsilon _0 \,, \text{ for } \text{ all } t\in [0,\delta ). \end{aligned}$$

(52)

Hence the validity of estimates (47) and (51) yields for any $t \in [0,\delta )$

$$\begin{aligned} {\mathscr {E}}(w,t,x,\mu )+\int \limits _{0}^{t}{\mathscr {D}}(w,\tau ,x,\mu )d\tau <\varepsilon _0+c\int \limits _{0}^{ t}{\mathscr {E}}^\frac{1}{2}(w,\tau ,x,\mu ){\mathscr {D}}(w,\tau ,x,\mu )d\tau , \end{aligned}$$

(53)

that, thanks to (52) and our assumption on $\varepsilon _0$, gives (41) on $[0,\delta )$.

Let us show that estimate (52) holds for $t\in [0,t^*)$. For all $x\in \mathbb {E}$ and $\mu >0$, the function

$$\begin{aligned} f(t,x,\mu ):={\mathscr {E}}(w,0,x,\mu ) +c\int \limits _{0}^{t}{\mathscr {E}}^\frac{1}{2}(w,\tau ,x,\mu ){\mathscr {D}}(w,\tau ,x,\mu )d\tau + H(v,t,x,\mu ) \end{aligned}$$

is uniformly continuous on $[0,t^*]$. Hence there exists $\eta =\eta (x,\mu )>0$ such that

$$\begin{aligned} |t_1-t_2|<\eta \Rightarrow |f(t_1)-f(t_2)|<\varepsilon \varepsilon _0\,. \end{aligned}$$

We claim that estimate (52) and, consequently, estimate (41), also holds for $t\in [\delta ,\delta +\eta )$. Assuming the contrary, there exists $\overline{t}\in [\delta ,\delta +\eta )$ such that for some $\varepsilon >0$ we have

$$\begin{aligned} {\mathscr {E}}(w,\overline{t},x,\mu )\ge (1+\varepsilon )\varepsilon _0\,. \end{aligned}$$

(54)

On the other hand, the validity of (47) and assumption (51) yield

$$\begin{aligned}&{\mathscr {E}}(w,\overline{t},x,\mu )+\int \limits _{0}^{\overline{t}}{\mathscr {D}}(w,\tau ,x,\mu )d\tau \le (f(\overline{t})-f(\delta ))+f(\delta )\\&\quad <(1+\varepsilon )\varepsilon _0+c\int \limits _{0}^{\delta }{\mathscr {E}}^\frac{1}{2}(w,\tau ,x,\mu ){\mathscr {D}}(w,\tau ,x,\mu )d\tau . \end{aligned}$$

Estimate (52) allows to deduce that

$$\begin{aligned} c\int \limits _{0}^{\delta }{\mathscr {E}}^\frac{1}{2}(w,\tau ,x,\mu ){\mathscr {D}}(w,\tau ,x,\mu )d\tau < \frac{1}{2}\int \limits _{0}^{\delta }{\mathscr {D}}(w,\tau ,x,\mu )d\tau . \end{aligned}$$

Hence the last two estimates imply

$$\begin{aligned} {\mathscr {E}}(w,\overline{t},x,\mu )<(1+\varepsilon )\varepsilon _0, \end{aligned}$$

which is in contradiction with (54). Since the arguments are independent of $\delta$, the result holds for any $t\in [0, t^*)$, which, for all $\varepsilon >0$, proves

$$\begin{aligned} {\mathscr {E}}(w,t,x,\mu )<(1+\varepsilon )\varepsilon _0, \text{ for } \text{ all } t\in [0,t^*)\,,\;x\in \mathbb {E} \text{ and } \mu >0\,. \end{aligned}$$

This last and the validity of estimates (47) and assumption (51), for all $\varepsilon >0$, allow us to deduce

$$\begin{aligned} {\mathscr {E}}(w,t,x,\mu )+\frac{1-\varepsilon }{2}\int \limits _{0}^{t}{\mathscr {D}} (w,\tau ,x,\mu )d\tau <\varepsilon _0, \text{ for } \text{ all } t\in [0,t^*),\;x\in \mathbb {E} \text{ and } \mu >0\,. \end{aligned}$$

By virtue of dominate convergence theorem, and then employing the arbitrariness of $\varepsilon >0$, we also get

$$\begin{aligned} {\mathscr {E}}(w,t,x,0)+\frac{1}{2}\int \limits _{0}^{t}{\mathscr {D}}(w,\tau ,x,0)d\tau \le \varepsilon _0, \text{ for } \text{ all } t\in [0,t^*),\;x\in \mathbb {E}\,, \end{aligned}$$

which completes the proof of (41). Assuming $v_0=0$, in the above computations we get $H(v,t,x,\mu )=0$ for all $t>0$. Starting from (53), and being $t^*$ free from constrains, the claim is immediate on $(0,\infty )$.

4 A result of partial regularity for a suitable weak solution to the perturbed Navier–Stokes system

Let us consider the following perturbed Navier–Stokes Cauchy problem:

$$\begin{aligned}&w_t-\varDelta w+\nabla \pi =-w\cdot \nabla w-v\cdot \nabla w-w\cdot \nabla v\,, \nonumber \\&\nabla \cdot w=0\,, \text{ in } (0,T)\times \mathbb {R}^3\,, \nonumber \\&w=w_0\,, \text{ on } \{0\}\times \mathbb {R}^3\,. \end{aligned}$$

(55)

We assume that for all $T>0$

$$\begin{aligned} v\in L^\infty (0,T;J^{1,2}(\mathbb {R}^3))\cap L^2(0,T;W^{2,2}(\mathbb {R}^3))\,. \end{aligned}$$

(56)

For weak solution to the Cauchy problem (55) we mean a pair $(w,\pi )$ which satisfies Definition 1 with the integral equation iii) substituted by the weak formulation of (55).

For any non negative scalar function $\phi (t,x)\in C_0^\infty (\mathbb {R}\times \mathbb {R}^3)$ , we set

$$\begin{aligned} {\mathscr {E}}(t,\phi ):=\int \limits _{\mathbb {R}^3}|w|^2\phi dx \,,\quad {\mathscr {D}}(t,\phi ):=\int \limits _{\mathbb {R}^3}|\nabla w|^2\phi dx\, . \end{aligned}$$

For a suitable weak solution to problem (55) we mean a weak solution $(w,\pi )$ to problem (55) such that for all $t>s$ a.e. in $s >0$ and for $s=0$

$$\begin{aligned} {\mathscr {E}}(t,\phi )+2\int \limits _{s}^{t}{\mathscr {D}}(\tau ,\phi )d\tau\le & {} {\mathscr {E}}(s,\phi )+\int \limits _{s}^{t} (|w|^2,\phi _\tau +\varDelta \phi )d\tau \nonumber \\&+\int \limits _{s}^{t}(|w|^2,(w+v)\cdot \nabla \phi )d\tau -2\int \limits _{s}^{t} (w\cdot \nabla v,w\phi )d\tau \nonumber \\&+2\int \limits _{s}^{t}(\pi ,w\cdot \nabla \phi )d\tau . \end{aligned}$$

(57)

We set

$$\begin{aligned} M(w,t,x,r):=r^{-2}\int \int \limits _{Q_r}(|w|^3+|w||\pi |) +r^{-\frac{13}{4}}\int \limits _{t-r^2}^{t}\big (\int \limits _{|x-y|<r} |\pi |dy\big )^\frac{5}{4}d\tau \,, \end{aligned}$$

(58)

where, we recall, we set

$$\begin{aligned} Q_r=Q_r(t,x):=\{(\tau ,y):t-r^2<\tau<t \text{ and } |y-x|<r\}. \end{aligned}$$

(59)

When the parabolic cylinder $Q_r$ is centered in (0, 0), then the above neighborohood becomes

$$\begin{aligned} Q_r=Q_r(0,0):=\{(\tau ,y):-r^2<\tau<0 \text{ and } |y|<r\}. \end{aligned}$$

(60)

In the computations one can indifferently argument on the parabolic cylinder $Q_r(t,x)$ and $Q_r(0,0)$. Actually, as remarked in [1], in some steps of the proofs the adoption of the cylinder $Q_r(0,0)$ simplifies the notation, as the inductive procedure employed to prove Lemma 6 below. The usual euclidean ball is denoted by $B_r(x)$ and by $B_r$, when no confusion occurs.

Our aim is to prove

Lemma 6

Let $(w,\pi )$ be a suitable weak solution to problem (55) in some parabolic cylinder $Q_r(t,x)$. There exist absolute positive constants $\varepsilon _1,c_0>0$ and $\varepsilon _2>0$ such that, assuming

$$\begin{aligned} M(w,t,x,r)\le \varepsilon _1, \end{aligned}$$

(61)

and

$$\begin{aligned} r{\underset{Q_r}{\int \int }}|v|^6+r^2\int \limits _{t-r^2}^{t}||v(\tau )||_\infty ^4d\tau +r{\underset{Q_r}{\int \int }}|\nabla v|^3<\varepsilon _2\,, \end{aligned}$$

(62)

then

$$\begin{aligned} |w(\tau ,y)|\le c_0\varepsilon ^{\frac{2}{3}}_1r^{-1}, \text{ a.e. } \text{ in } (\tau ,y)\in Q_{\frac{r}{2}}(t,x). \end{aligned}$$

(63)

In particular, a suitable weak solution w is regular in $Q_{\frac{r}{2}}(t,x)$.

Of course the proof of the above proposition consists in an adaption to problem (55) of the well known result by Caffarelli–Kohn–Nirenberg (see [1] Proposition 1 and Corollary 1).

In order to prove Lemma 6 we need some preliminary lemmas that are related to the terms of the “unperturbed” field v.

We assume that $\{\phi _m\}$ is a sequence of cutoff functions such that

$$\begin{aligned} \begin{array}{lll} \frac{1}{c} r_m^{-3}\le &{}\phi _m\le cr^{-3}_m\,,\quad |\nabla \phi _m|\le cr_m^{-4} \text{ on } Q_m\,,&{} 2\le m\,, \\ &{}\phi _m\le cr_k^{-3}\,,\quad |\nabla \phi _m|\le c r_k^{-4} \text{ on } Q_{k-1}-Q_k\,,\, 1< &{}k\le m\,,\end{array} \end{aligned}$$

(64)

with $r_k:=2^{-k}$ and $Q_k$ given by (59) with radius $r_k:=2^{-k}$. Moreover, for some $\varepsilon _1>0$ and for all k, we set

(65)

Lemma 7

Let v and w be two fields enjoying (56) and (65), respectively. Then we get

$$\begin{aligned} \Big |\int \int \limits _{ Q_1}w^2v\cdot \nabla \phi _m \Big |+\Big |\int \int \limits _{ Q_1}w\cdot \nabla v\cdot w\phi _m\Big | \le c\varepsilon _1^\frac{4}{9} \Big [\big (\int \int \limits _{ Q_1}|v|^6\Big )^\frac{1}{6}+\Big (\int \int \limits _{ Q_1}|\nabla v|^3\Big )^\frac{1}{3}\Big ]\,,\quad \quad \end{aligned}$$

(66)

where $\phi _m$ is defined in (64).

Proof

We separate the estimates of the two integrals. Taking into account the properties (64) and employing (65), applying Hölder’s inequality, we get

Analogously, we get

The above inequalities lead to the proof of (66).

Denoted by $\phi$ a smooth nonnegative cutoff function with support in $B_\rho$ and such that $\phi =1$ on $B_{\frac{3}{4}\rho }$, taking into account that from (55) we formally deduce $\varDelta \pi =\nabla w\cdot (\nabla w)^T+2\nabla w\cdot (\nabla v)^T$, then for the pressure field $\pi$ of (55) the following representation formula holds:

$$\begin{aligned} \pi \phi :=\pi _w+\pi _v\,, \end{aligned}$$

where we set

$$\begin{aligned} \pi _w:= & {} {\widetilde{\pi }}_w+\pi _{w_2}+\pi _{w_3}\,,\; \text{ with } \;{\widetilde{\pi }}_w:=-\frac{3}{4\pi }\nabla _{x}\int \limits _{\mathbb {R}^3}\nabla _{y}\frac{1}{|x-y|}\phi w\otimes wdy\,, \\ \pi _{w_2}:= & {} \frac{3}{2\pi }\int \limits _{\mathbb {R}^3}w\otimes w\cdot \nabla \frac{1}{|x-y|}\otimes \nabla \phi dy+\frac{3}{4\pi }\int \limits _{\mathbb {R}^3}w\otimes w\cdot \frac{1}{|x-y|}\cdot \nabla \nabla \phi dy\,, \\ \pi _{w_3}:= & {} \frac{3}{2\pi }\int \limits _{\mathbb {R}^3}\pi \nabla \frac{1}{|x-y|}\cdot \nabla \phi dy+\frac{3}{4\pi }\int \limits _{\mathbb {R}^3}\pi \frac{1}{|x-y|}\varDelta \phi dy\,, \end{aligned}$$

and

$$\begin{aligned} \pi _v:= & {} {\widetilde{\pi }}_v+\pi _{v_2}\,,\; \text{ with } \;{\widetilde{\pi }}_v:=-\frac{3}{2\pi }\nabla _{x}\int \limits _{\mathbb {R}^3}\nabla _{y}\frac{1}{|x-y|}\phi w\otimes vdy\,, \\ \pi _{v_2}:= & {} \frac{3}{\pi }\int \limits _{\mathbb {R}^3}w\otimes v\cdot \nabla \frac{1}{|x-y|}\otimes \nabla \phi dy+\frac{3}{2\pi }\int \limits _{\mathbb {R}^3}\frac{1}{|x-y|}w\otimes v\cdot \nabla \nabla \phi dy\,. \end{aligned}$$

For our aims is important to evaluate the term

$$\begin{aligned} L(r):=r^{-2}\int \int \limits _{ Q_r} |w|\cdot |\pi -\overline{\pi }_r|\,, \end{aligned}$$

(67)

where we set

that we increase by

$$\begin{aligned} L(r)\le r^{-2}\int \int \limits _{ Q_r} |w|\cdot |\pi _w-{\overline{\pi }_w}_r|+r^{-2}\int \int \limits _{ Q_r} |w|\cdot |\pi _v-{\overline{\pi }_v}_r|=: L_w(r)+L_v(r)\,, \end{aligned}$$

(68)

where we set

We introduce the following quantities related to $(w,\pi )$:

$$\begin{aligned} A(r):= & {} \sup _{-r^2<t<0}r^{-1}\int \limits _{B_r}|w(t)|^2\,,\quad \delta (r):=r^{-1}\int \int \limits _{ Q_r}|\nabla w|^2\,,\\ G(r):= & {} r^{-2}\int \int \limits _{ Q_r}|w|^3\,,\quad K(r):=r^{-\frac{13}{4}}\int \limits _{-r^2}^{0}\Big [\int \limits _{B_r}|\pi |\Big ]^\frac{5}{4}d\tau \,. \end{aligned}$$

The definition of suitable weak solution to problem (55) ensures that $\pi \in L^{\frac{5}{3}}(0,T;L^\frac{5}{3}(\mathbb {R}^3))$. Hence we get

$$\begin{aligned} K(r)<\infty \, \text{ for } \text{ all } r>0. \end{aligned}$$

(69)

Analogously, via assumption (62) related to $(v,\pi _v)$, for all $r>0$, the following is well posed

$$\begin{aligned} V(r):=r^{-2}\int \int \limits _{ Q_r}|v|^3\,. \end{aligned}$$

Lemma 8

The following estimate holds:

$$\begin{aligned} G(r)\le cA^\frac{3}{4}(r)\big [A^\frac{3}{4}(r)+\delta ^\frac{3}{4}(r)\big ]\,, \end{aligned}$$

(70)

where c is a constant independent of r, w.

Proof

See [1] Lemma 3.1 .

In [1] the authors prove Lemma 3.2 related to $L_w(r)$:

Lemma 9

Let $r\le \frac{1}{2}\rho$, then

$$\begin{aligned} L_w(r)\le & {} c\Big [\frac{r}{\rho }\Big ]^\frac{7}{5}A^\frac{1}{5}(r)G^\frac{1}{5}(r)K^\frac{4}{5}_w(\rho )+c\big [\frac{r}{\rho }\Big ]^\frac{5}{3}G^\frac{1}{3}(r)G^\frac{2}{3}(\rho ) \nonumber \\&+cG^\frac{1}{3}(r)G^\frac{2}{3}(2r)+cr^3G^\frac{1}{3}(r)\sup _{-r^2<t<0} \int \limits _{2r<|y|<\rho } \frac{|w(t)|^2 }{\,|y|^4} \end{aligned}$$

(71)

where c is a constant independent of $r,w,\pi _w$ and $\rho$.

By the same arguments employed in [1] to obtain Lemma 9 (that is, Lemma 3.2 in [1]), we prove for the term $L_v(r)$ the following

Lemma 10

Let $r\le \frac{1}{2}\rho$, then

$$\begin{aligned} L_v(r)\le & {} c\Big [\frac{r}{\rho }\Big ]^\frac{5}{3}G^\frac{1}{3}(r)G^\frac{1}{3}(\rho )V^\frac{1}{3}(\rho ) +cG^\frac{1}{3}(r)G^\frac{1}{3}(2r)V^\frac{1}{3}(2r) \nonumber \\&+cr^2G^\frac{1}{3}(r)\sup _{-r^2<t<0} \Big [\int \limits _{2r<|y|<\rho }\frac{|w|^2 }{|y|^4 }\,dy\Big ]^\frac{1}{2}\Big [\int \limits _{-r^2}^{0}||v(t)||_{L^\infty (B_\rho )}^4dt\Big ]^\frac{1}{4}, \end{aligned}$$

(72)

where c is a constant independent of $r,v,\pi _v$ and $\rho$.

Proof

We recall that, via the representation formula of $\pi _v={\widetilde{\pi }}_v+\pi _{v_2}$, we are going to discuss the following inequality:

$$\begin{aligned} L_v(r)= & {} r^{-2}{\underset{Q_r}{\int }\int }|w|\cdot |\pi _v-\overline{\pi }_v|\nonumber \\\le & {} r^{-2}{\underset{Q_r}{\int }\int }|w|\cdot |{\widetilde{\pi }}_v- \overline{{\widetilde{\pi }}}_v|+r^{-2}{\underset{Q_r}{\int }\int }|w| \cdot |{\pi _v}_2-{{\overline{\pi }}_v}_2|\,. \end{aligned}$$

(73)

For the latter term on the right hand side of (73), in a sequential way we apply Hölder’s inequality:

$$\begin{aligned} r^{-2}{\underset{Q_r}{\int }\int }|w|\cdot |{\pi _v}_2-{{\overline{\pi }}_v}_2|\le & {} cr\int \limits _{-r^2}^{0}\Big [\int \limits _{B_r}|w(\tau ,y)|^3dy\Big ]^\frac{1}{3} \sup _{B_r}|\nabla {\pi _v}_2(\tau )|d\tau \\\le & {} c\frac{r}{\rho ^4}\int \limits _{-r^2}^{0}\Big [\int \limits _{B_r}|w(\tau ,y)|^3dy\Big ]^\frac{1}{3} \int \limits _{B_\rho }|w(\tau ,y)||v(\tau ,y)|dyd\tau \\\le & {} c\frac{r}{\rho ^3}\,\Big [{\underset{Q_r}{\int }\int }|w|^3 \Big ]^\frac{1}{3}\Big [{\underset{Q_\rho }{\int }\int }|w|^3 \Big ]^\frac{1}{3} \Big [{\underset{Q_\rho }{\int }\int }|v|^3\Big ]^\frac{1}{3}\\\le & {} c\Big [\frac{r}{\rho }\Big ]^\frac{5}{3}G^\frac{1}{3}(r)G^\frac{1}{3}(\rho )V^\frac{1}{3}(\rho )\,, \end{aligned}$$

thus we have obtained the first term on the right hand side of (72). For the former term on the right-hand side of (73), we consider ${\widetilde{\pi }}_v:=\pi '_v+\pi ''_v\,,$ where we set

$$\begin{aligned} \pi '_v:= & {} -\frac{3}{4\pi }\nabla _{x}\int \limits _{|x-y|<2r}\nabla _{y}\frac{1}{|x-y|}\phi w\otimes vdy\\ \pi ''_v:= & {} -\frac{3}{4\pi }\nabla _{x}\int \limits _{|x-y|>2r}\nabla _{y}\frac{1}{|x-y|}\phi w\otimes vdy\,. \end{aligned}$$

The term $\pi '_v$ has the kernel which is singular of Calderon–Zigmund type. Hence one easily deduce

$$\begin{aligned} r^{-2}{\underset{Q_r}{\int \int }}|w||\pi '_v- \overline{\pi }{'}_{v}|\le cr^{-2}\Big [{\underset{Q_r}{\int \int }}|w|^3\Big ]^\frac{1}{3} \Big [{\underset{Q_{2r}}{\int \int }}|w|^\frac{3}{2}|v|^\frac{3}{2}\Big ]^\frac{2}{3}\le c G^\frac{1}{3}(r)G^\frac{1}{3}(2r)V^\frac{1}{3} (2r). \end{aligned}$$

For $\pi ''_v$ we initially stress that

$$\begin{aligned} ||\pi _{v}''-{\overline{\pi }}_{v}''||_{L^\infty (B_r)}\le cr||\nabla \pi _{v}''||_{L^\infty (B_r)}\,, \end{aligned}$$

and

$$\begin{aligned} ||\nabla \pi _{v}''||_{L^\infty (B_r)}\le c\int \limits _{2r<|y|<\rho }\frac{|w||v|}{|y|^4}dy\le c \Big [\int \limits _{2r<|y|<\rho }\frac{|w|^2 }{|y|^4 } \,dy\Big ]^\frac{1}{2}\Big [\int \limits _{2r<|y|<\rho } \frac{|v|^2 }{|y|^4 }\,dy\Big ]^\frac{1}{2} . \end{aligned}$$

Thus

$$\begin{aligned} r^{-2}{\underset{Q_r}{\int }\int }|w||{\pi _v''}-{{\overline{\pi }}_v''}|\le & {} cr\Big [{\underset{Q_r}{\int \int }}|w|^3\Big ]^\frac{1}{3} \Big [\int \limits _{-r^2}^{0}||\nabla \pi ''_v(t)||_{L^\infty (B_r)}^\frac{3}{2}dt\Big ]^\frac{2}{3}\\\le & {} cr\Big [{\underset{Q_r}{\int \int }}|w|^3\Big ]^\frac{1}{3} \Big [\int \limits _{-r^2}^{0}\Big [\int \limits _{2r<|y|<\rho } \frac{|w|^2 }{|y|^4 }\,dy\Big ]^\frac{3}{4} \Big [\int \limits _{2r<|y|<\rho } \frac{|v|^2 }{|y|^4 }\,dy\Big ]^\frac{3}{4}dt\Big ]^\frac{2}{3}\\\le & {} cr^\frac{5}{3}G^\frac{1}{3}(r)\Big [\int \limits _{-r^2}^{0} \Big [\int \limits _{2r<|y|<\rho } \frac{|w|^2 }{|y|^4 }\,dy\Big ]^\frac{3}{4}\Big [\int \limits _{2r<|y|<\rho } \frac{|v|^2 }{|y|^4 }\,dy \Big ]^\frac{3}{4}dt\Big ]^\frac{2}{3}\,. \end{aligned}$$

Applying Hölder’s inequality, we get

$$\begin{aligned} r^{-2}{\underset{Q_r}{\int }\int }|w||{\pi _v''}-{{\overline{\pi }}_v''}|\le & {} cr^\frac{5}{3}G^\frac{1}{3}(r)\sup _{-r^2<t<0}\Big [\int \limits _{2r<|y|<\rho } \frac{|w|^2 }{|y|^4 }dy \Big ]^{\frac{1}{2}}\Big [\int \limits _{-r^2}^{0}\Big [\int \limits _{2r<|y|<\rho } \frac{|v|^2 }{|y|^4 }dy \Big ]^\frac{3}{4}dt\Big ]^{\frac{2}{3}} \\\le & {} cr^\frac{7}{6}G^\frac{1}{3}(r)\sup _{-r^2<t<0} \Big [\int \limits _{2r<|y|<\rho } \frac{|w|^2 }{|y|^4 } \,dy\Big ]^\frac{1}{2}\Big [\int \limits _{-r^2}^{0}||v(t) ||_{L^\infty (B_\rho )}^\frac{3}{2}dt\Big ]^{\frac{2}{3}}\\\le & {} cr^2G^\frac{1}{3}(r)\sup _{-r^2<t<0}\Big [\int \limits _{2r<|y|<\rho } \frac{|w|^2 }{|y|^4 }\,dy \Big ]^\frac{1}{2}\Big [\int \limits _{-r^2}^{0}||v(t)||_{L^\infty (B_\rho )}^4dt \Big ]^{\frac{1}{4}}. \end{aligned}$$

We are in a position to prove Lemma 6. The proof completely follows the arguments employed in [1]. Actually we have only to show that the adjoint of the terms $v\cdot \nabla w,\,w\cdot \nabla v$ and the modified pressure field work in the inductive procedure.

Proof of Lemma 6

For the proof we adopt the parabolic cylinder $Q_1(0,0)$. The aim is to prove that

(74)

for each $(s,a)\in Q_{\frac{1}{2}}(0,0)$ and for all $m\ge 2$, where we set $r_m:=2^{-m}$. Step 1. Setting up the induction.

The assumptions are

$$\begin{aligned} \int \int \limits _{ Q_1}(|w|^3+|w||\pi |)+\int \limits _{-1}^{0} \Big [\int \limits _{B_1}|\pi |dx\Big ]^\frac{5}{4}dt\le \varepsilon _1\,, \end{aligned}$$

(75)

and, for the coefficient v,

$$\begin{aligned} {\underset{Q_1}{\int \int }}|v|^6+\int \limits _{-1}^{0}||v(\tau )||_\infty ^4d\tau +{\underset{Q_1}{\int \int }}|\nabla v|^3<\varepsilon _2\,. \end{aligned}$$

(76)

Considering the generalized energy inequality (57) with function $\phi =1$ on $Q_2$ and support enclosed in $Q_1$, assuming (75) and (76), by virtue of Lemma 7, we obtain

provided that $\varepsilon _1$ and $\varepsilon _2$ are suitable. Since $Q_\frac{1}{2}(s,a)\subset Q_1(0,0)$ for all $(s,a)\in Q_\frac{1}{2}(0,0)$, we get

$$\begin{aligned} \int \int \limits _{ Q_\frac{1}{2}}(|w|^3+|w||\pi |)+\int \limits _{s-\frac{1}{4}}^{s} \Big [\int \limits _{B_\frac{1}{2}}|\pi |dx\Big ]^\frac{5}{4}dt\le \varepsilon _1\,. \end{aligned}$$

(77)

The inductive procedure is the following: for $k\ge 3$,

(78)

and, for $k\ge 2$,

(79)

then we also have

$$\begin{aligned} \begin{array}{ll}&{}(78) \text{ true } \text{ for } 2\le k\le m \;\Rightarrow \quad (79) \text{ for } m\,,\\ \text{ and } &{}\\ &{}(79) \text{ true } \text{ for } 3\le k\le m\;\Rightarrow \quad (78) \text{ for } m+1\,. \end{array} \end{aligned}$$

(80)

Step 2. Implication (80)$_2$ .

We start remarking that (78) regards two terms. As proved in [1], by choosing $c\varepsilon _1\le \frac{1}{2}$, the following chain of inequalities

(81)

is a consequence of Lemma 8 and the inductive assumption (79) for $k=m$. Hence we omit further comments. Now, we consider the latter term of (78) which involves the pressure field. The proof is analogous to the one furnished by Caffarelli–Kohn–Nirenberg, we have just to consider the adjoint of the terms in v. The key tools for the estimate are Lemma 9 and Lemma 10. We premise that, by the definition of the quantities A(r), G(r), and by the inductive hypothesis (79) for $k=m$, easily we obtain

$$\begin{aligned} \begin{array}{c}G(r_{m+1})\le cG(r_m)\le c\varepsilon _1 r_m^3\,, \\ A(r_{m+1})\le cA(r_m)\le c\varepsilon _1^\frac{2}{3}r^2_m\,, \end{array} \end{aligned}$$

(82)

and via (69) we also deduce

$$\begin{aligned} K\left( {\frac{1}{4}}\right) \le c\varepsilon _1\,. \end{aligned}$$

(83)

The estimate of the quoted latter term trivially admits the following

(84)

where we took (67) and (68) into account. By virtue of Lemma 9, following all the arguments employed in [1], one obtain

$$\begin{aligned} L_w(r_{m+1})\le c r^{\frac{12}{5}}_m\varepsilon _1\,. \end{aligned}$$

(85)

Now, reproducing the arguments, we realize the same estimate for $L_v(r_{m+1})$ starting by Lemma 10. We set ${\mathfrak V}:=\Big [\int \limits _{-1}^{0}||v(t)||_\infty ^4dt\Big ]^\frac{1}{4}\,.$ Recalling the definition of V, applying Hölder’s inequality, we have

$$\begin{aligned} V(r_m)\le cr_m^\frac{3}{2}\Big [\int \limits _{s-r_m^2}^{s}||v(t)||_{L^\infty (B_\rho )}^{4}\Big ]^\frac{3}{4}\le cr_m^\frac{3}{2}{\mathfrak V}^3 \end{aligned}$$

(86)

Being possible to set $\rho =\frac{1}{4}$, for the first two terms on the right hand side of (72) we have

$$\begin{aligned} r_{m+1}^\frac{7}{5}A^\frac{1}{5}(r_{m+1})G^\frac{1}{5}(r_{m+1})K^\frac{4}{5}\left( \frac{1}{4}\right)\le & {} cr_m^\frac{12}{5}\varepsilon _1\,, \nonumber \\ \quad r^\frac{5}{3}_{m+1}G^\frac{1}{3}(r_{m+1})G^\frac{1}{3}(\frac{1}{4})V^\frac{1}{3}\left( \frac{1}{4}\right)\le & {} cr_m^\frac{8}{3}\varepsilon _1^\frac{2}{3}\Bigg [\int \limits _{s-\frac{1}{4}}^{s}||v(t)||_{L^3(B_\frac{1}{4})}^3dt\Bigg ]^\frac{1}{3}\,. \end{aligned}$$

(87)

For the third term on the right hand side of (72), employing (86), we get

$$\begin{aligned} G^\frac{1}{3}(r_{m+1})G^\frac{1}{3}(r_m)V^\frac{1}{3}(r_m)\le cr_m^\frac{5}{2}\varepsilon _1^\frac{2}{3}{\mathfrak V} \end{aligned}$$

(88)

and, finally, by virtue of (82), we obtain for the last term of (72)

$$\begin{aligned}&r^2_{m+1}G^\frac{1}{2}(r_{m+1}) \sup _{-r^2_{m+1}<t<0}\Big [\int \limits _{r_m<|y|<\frac{1}{4}} \frac{|w|^2 }{|y|^4 }\,dy\Big ]^\frac{1}{2}{\mathfrak V}\nonumber \\&\le cr_{m}^3\varepsilon _1^\frac{1}{3}\Big [\,{{\underset{k=2}{\overset{m}{\sum }}}}\int \limits _{r_k<|y|<r_{k-1}} \frac{|w|^2 }{|y|^4 }dy\Big ]^{\frac{1}{2}}{\mathfrak V} \le cr_{m}^3\varepsilon _1^\frac{1}{3}\Big [\,{{\underset{k=2}{\overset{m}{\sum }}}}\frac{1}{r_k^3 }A_k\Big ]^\frac{1}{2}{\mathfrak V} \nonumber \\&\le cr_{m}^3\varepsilon _1^\frac{2}{3}\Big [\,{{\underset{k=2}{\overset{m}{\sum }}}} \frac{1}{r_k }\Big ]^\frac{1}{2}{\mathfrak V} \le cr_{m}^\frac{5}{2}\varepsilon _1^\frac{2}{3}\Big [\,{{\underset{p=0}{\overset{\infty }{\sum }}}}\,\frac{1}{2^{^p} }\Big ]^\frac{1}{2} {\mathfrak V}\, \end{aligned}$$

(89)

Considering for the $L_v(r)$ estimate (72), and increasing the right hand side by means of estimates (87) and (89), since $r<1$ and we can consider $\varepsilon _2$ such that $c\varepsilon _2<\varepsilon _1^\frac{1}{3}$, we get

$$\begin{aligned} L_v(r)\le c\varepsilon _1\,. \end{aligned}$$

(90)

Hence, recalling that for all $r<\frac{3}{4}\rho$ we have $L(r)\le L_w(r)+L_v(r)$, via estimates (85) and (90), we complete the proof of step 2.

Step 3. Implication (80)$_1$.

Recalling that the center of the parabolic cylinder is the point (0, 0), in the generalized energy inequality (57) we insert the scalar test function $\phi _m:=\chi \psi _m$ where function $\chi$ is smooth nonnegative cutoff function with support $Q_{\frac{1}{3}}(0,0)$ and $\chi =1$ on $Q_{\frac{1}{4}}(0,0)$, and, for $t<r_m^2$,

$$\begin{aligned} \psi _m:=\frac{1}{(r_m^2-t)^\frac{3}{2} }\,\exp \bigg (-\frac{|x|^2}{4(r_m^2-t)}\bigg )\,. \end{aligned}$$

For all $m\in \mathbb {N}$, function $\psi _m$ is a solution backward in time of the heat equation. The sequence $\{\phi _m\}$ enjoys the property listed in (64) , and also

$$\begin{aligned} {\phi _m}_t+\varDelta \phi _m=0\,,\; \text{ in } Q_\frac{1}{4}(0,0)\,, \text{ and } |{\phi _m}_t+\varDelta \phi _m|\le c\,,\; \text{ everywhere. } \end{aligned}$$

(91)

Substituting $\phi _m$ in (57), by virtue of (64), we realize the inequality

(92)

where we set

$$\begin{aligned} I_1:= & {} {\underset{Q_1}{\int \int }}|w|^2\big |{\phi _m}_t+\varDelta \phi _m\big |\,,\;I_2:={\underset{Q_1}{\int \int }}|w|^3|\nabla \phi _m|\,, \;I_3:=\big |{\underset{Q_1}{\int \int }}\pi w\cdot \nabla \phi _m\big |\,,\;\\&\quad I_4:=\big |{\underset{Q_1}{\int \int }}w\cdot \nabla v\cdot w\phi _m\big |\,,\;I_5:=\big |{\underset{Q_1}{\int \int }}v\cdot \nabla w\cdot w\phi _m\big |\,. \end{aligned}$$

Now we should have to estimate each term from $I_1$ to $I_5$. Since the estimates from $I_1$ to $I_3$ are the same of the ones deduced in [1] for the terms $I\,,II\,,$ and III , then for them we only carry the bounds calculated in [1] (p. 792–794, step 3) and we limit ourselves to find the bounds for the residual. Hence we set

$$\begin{aligned} I_1+I_2+I_3\le c\varepsilon ^\frac{2}{3}\,. \end{aligned}$$

(93)

For the terms $I_4$ and $I_5$ we employ (66):

$$\begin{aligned} I_4+I_5\le c\varepsilon _1^\frac{4}{9}\left[ \left( {\underset{Q_1}{\int \int }} |v|^6\right) ^\frac{1}{6}+\left( {\underset{Q_1}{\int \int }}\nabla v|^3\right) ^\frac{1}{3}\right] \,. \end{aligned}$$

(94)

Hence, choosing $\varepsilon _2$ in such a way that $c\varepsilon _2<\varepsilon _1^\frac{2}{9}$, via (93) and (94), we complete the proof of step 3.

The lemma is completely proved.

Lemma 11

Let w be a suitable weak solution to problem (55) with assumptions (56) for coefficient v. Let $S>0$ and $x\in \mathbb {R}^3$. Assume the existence of a $t(x)>0$ such that

$$\begin{aligned} {\mathscr {E}}(w,t,x)+\frac{1}{2}\int \limits _{0}^{t}{\mathscr {D}}(w,\tau ,x)d\tau \le S\,, \text{ for } \text{ all } t\in [0,t(x)). \end{aligned}$$

(95)

Then there exists a $\delta (x)\in [0,1)$ such that for all $s\in (0,t(x))$, $w(t,y)\in L^\infty (Q_{\big (\frac{(1-\delta )s}{4}\big )^\frac{1}{2}}(s,x))\,.$ In particular, we get

$$\begin{aligned} |w(t,y)|\le c[(1-\delta )s]^{-\frac{1}{2}} \,, \end{aligned}$$

(96)

for all Lebesgue’s points $(t,y)\in Q_{\big (\frac{(1-\delta )s}{4}\big )^\frac{1}{2}}(s,x)$.

Proof

The function

$$\begin{aligned} f(t,x):=\int \limits _{t}^{t(x)}{\mathscr {D}}(w,\tau ,x)d\tau \end{aligned}$$

(97)

is uniformly continuous on [0, t(x)]. Hence for $\varepsilon >0$ there exists a ${\widetilde{\delta }}(x)\in (0,1]$ such that

$$\begin{aligned} \int \limits _{t(1-{\widetilde{\delta }})}^{t}{\mathscr {D}}(w,\tau ,x)d\tau <\varepsilon \,, \text{ for } \text{ all } t\in [0,t(x)]\,. \end{aligned}$$

(98)

Now for a suitable $\varepsilon$, we can satisfy the assumptions of Lemma 6. Actually, recalling the definition of M(w, t, x, r) given in (58) and applying the Hölder’s inequality, for the first term of (58) we get

$$\begin{aligned} r^{-2}\int \int \limits _{Q_r}(|w|^3+|w||\pi |)\le & {} {\underset{Q_r}{\int \int }}\frac{|w(\tau ,y)|^3 }{|x-y|^2 }\,\,+\frac{|w(\tau ,y)||\pi (\tau ,y)|}{|x-y|^2}\\\le & {} {\underset{Q_r}{\int \int }}\frac{|w(\tau ,y)|^3 }{|x-y|^2 }\,\,+ \Big [{\underset{Q_r}{\int \int }}\frac{|w(\tau ,y)|^3 }{|x-y|^2 } \,\,\Big ]^\frac{1}{3}\Big [{\underset{Q_r}{\int \int }}\frac{|\pi (\tau ,y)|^\frac{3}{2} }{|x-y|^2 } \,\, \Big ]^\frac{2}{3}. \end{aligned}$$

Hence, by virtue of inequalities (25) for the pressure term and via the weighted interpolating inequality (33), we arrive at

$$\begin{aligned} r^{-2}\int \int \limits _{Q_r}(|w|^3+|w||\pi |) \le c\int \limits _{t-r^2}^{t}{\mathscr {E}}^\frac{1}{2}(w,\tau ,x){\mathscr {D}}(w,\tau ,x)d\tau \,. \end{aligned}$$

(99)

Analogously, applying Hölder’s inequality

$$\begin{aligned} r^{-\frac{13}{4}}\int \limits _{t-r^2}^{t} \left( \int \limits _{\,|x-y|<r} |\pi |dy\right) ^\frac{5}{4}d\tau\le & {} r^{-2}\int \limits _{t-r^2}^{t} \left( \int \limits _{\,|x-y|<r} |\pi |^\frac{3}{2}dy\right) ^\frac{5}{6}d\tau \\\le & {} \Big [\int \limits _{t-r^2}^{t}\int \limits _{|x-y|<r} \frac{|\pi |^\frac{3}{2} }{|x-y|^2 }\,\,dy d\tau \Big ]^\frac{5}{6}\,. \end{aligned}$$

Hence, by virtue of inequality (25) and weighted interpolating inequality (33), we arrive at

$$\begin{aligned} r^{-\frac{13}{4}}\int \limits _{t-r^2}^{t}\big (\int \limits _{|x-y|<r} |\pi |dy\big )^\frac{5}{4}d\tau \le c\int \limits _{t-r^2}^{t} {\mathscr {E}}^\frac{1}{2}(w,\tau ,x) {\mathscr {D}}(w,\tau ,x)d\tau \,. \end{aligned}$$

(100)

By virtue of hypothesis (95), via estimates (98)–(100), setting $r^2={\widetilde{\delta }} t$, we arrive at

$$\begin{aligned} M(w,t,x,r)\le cS^\frac{1}{2}\int \limits _{(1-{\widetilde{\delta }})t}^{t}{\mathscr {D}}(w,\tau ,x)d\tau < cS^\frac{1}{2}\varepsilon \,, \text{ for } \text{ all } t\in (0,t(x))\,. \end{aligned}$$

(101)

Hence, choosing $\varepsilon := \varepsilon _1/{cS^\frac{1}{2}}$, we realize condition (61) for w with $r^2={\widetilde{\delta }} t$. Condition (62) is immediate from the assumptions. Hence, setting $\delta :=1-{\widetilde{\delta }}$, the claim of the lemma is a consequence of Lemma 6.

Corollary 2

Let w be a suitable weak solution to problem (55) with assumptions (56) for v. Let $S>0$ and ${\mathbb {E}}\subseteq \mathbb {R}^3$. Assume the existence of a ${\widetilde{T}}>0$ such that

$$\begin{aligned} {\mathscr {E}}(w,t,x)+\frac{1}{2}\int \limits _{0}^{t}{\mathscr {D}}(w,t,x)d\tau \le S\,, \text{ for } \text{ all } (t,x)\in (0,{\widetilde{T}})\times {\mathbb {E}}\,. \end{aligned}$$

(102)

Then for all $x\in {\mathbb {E}}$ there exists a $\delta (x)\in [0,1)$ such that, for all $s\in (0,{\widetilde{T}})$, $w(t,y)\in L^\infty (Q_{\big (\frac{(1-\delta )s}{4}\big )^\frac{1}{2}}(s,x))\,$. In particular, for all $(s,x)\in (0,{\widetilde{T}})\times {\mathbb {E}}$, and for all Lebesgue’s points $(t,y)\in Q_{\big (\frac{(1-\delta )s}{4}\big )^\frac{1}{2}}(s,x)$, we get

$$\begin{aligned} |w(t,y)|\le c[(1-\delta )s]^{-\frac{1}{2}}\,. \end{aligned}$$

(103)

Finally, if $S\le \frac{\varepsilon _1}{2c_1 }$ , with $\varepsilon _1$ as in Lemma 6, then (103) holds with $\delta =0$ too.

Proof

The proof is an immediate consequence of the previous lemma. Actually, the assumption ensures that function (97) can be defined in the following way:

$$\begin{aligned} f(t,x):=\int \limits _{t}^{{\widetilde{T}}}{\mathscr {D}}(\tau ,x,w)d\tau \,. \end{aligned}$$

Hence, for all x, the time interval of uniform continuity of this f(t, x) is $[0,{\widetilde{T}}]$, hence independent of x. This, via estimate (101) and Lemma 6, ensures that (103) holds on $(0,{\widetilde{T}})$. If $S^\frac{3}{2}\le \frac{\varepsilon _1}{2c }$ , then, further the above considerations, by summing estimates (99) and (100), we arrive at

$$\begin{aligned} M(w,t,x,r)\le 2cS^\frac{1}{2}\int \limits _{(1-{\widetilde{\delta }})t}^{t}{\mathscr {D}}(w,\tau ,x)d\tau< 2cS^\frac{3}{2}<\varepsilon _1\,, \text{ for } \text{ all } t\in (0,{\widetilde{T}})\,, \end{aligned}$$

which is verified independently of the uniform continuity, that is in the end independently of $\delta$ too.

Remark 6

We stress that the above lemma and corollary are criteria for the regularity which furnish estimates (96) and (103) respectively. In the sequel we understand that in order to satisfy the assumption of the lemma and of the corollary some suitable conditions of smallness are required for $w_0$. However we stress that $w_0$ is the initial data of a “perturbation” of a suitable weak solution u, thus for u a different result of regularity holds.

5 Proof of Theorems 3–5

Proof of Theorem 3

Setting $w:=u-v$, where v is the $J^{1,2}$-regular solution corresponding to $v_0$, we are in the hypotheses of Lemma 5, hence the weighted energy inequality (41) holds. As a consequence we can apply to w Corollary 2 in order to deduce $w(t,y)\in L^\infty (Q_{\big (\frac{(1-\delta )s}{4}\big )}(s,x))$ as well estimate (103). Hence, by means of the triangular inequality, we easily arrive at estimate (11). The case of $v_0=0$ should be discussed in the same way. For the sake of the brevity we omit the details.

Proof of Theorem 4

Our hypothesis (12) allows us to deduce (41) with $w=u$ , $t^*=\infty$, $\varepsilon _0\le \frac{\varepsilon _1}{2c_1}$ and ${\mathbb {E}}=\{x:|x|>R_0\}$. Hence, by virtue of the last claim of Corollary 2, taking into account that ${\widetilde{T}}=\infty$, via a suitable $c_1$, we obtain (103) with $w=u$, $\delta =0$ and for all $(s,x)\in (0,\infty )\times {\mathbb {E}}$. Considering the latter assumption of (14), $v_0^*$ belongs to $J^{1,2}(\mathbb {R}^3)$, then on some interval $(0,T^*_0)$ there exists the $J^{1,2}$-regular solution $(v^*,\pi _{v^*})$ to problem (1). On cylinder $(0,T^*_0)\times \mathbb {R}^3$ we can consider the difference $(w^*,\pi ^*)$ with $w^*:=u-v^*$ and $\pi ^*:=\pi _u-\pi _{v^*}$ . Set $\varepsilon _0:=\frac{\varepsilon _1}{2c_1 }$  in Lemma 5, then the following weighted energy inequality holds $(t^*\equiv T^*$ below):

$$\begin{aligned} {\mathscr {E}}(w^*,t,x)+\frac{1}{2}\int \limits _{0}^{t}{\mathscr {D}}(w^*,\tau ,x)d\tau \le \frac{\varepsilon _1}{2c_1}, \text{ for } \text{ all } (t,x)\in [0,T^*)\times {\mathbb {E}} \,. \end{aligned}$$

(104)

We set

$$\begin{aligned} v(t,x):=\left\{ \begin{array}{ll}v^*(t,x)\,,&{}\text{ if } (t,x)\in (0,T_0^*)\times \mathbb {R}^3\,, \\ 0&{}\text{ if } t\in [T^*_0,\infty )\times \mathbb {R}^3\,.\end{array}\right. \end{aligned}$$

The above position allow us to consider $(w^*,\pi ^*)$ as a suitable weak solution to problem (55) on the cylinder $(0,T^*_0)\times \mathbb {R}^3$. By virtue of (104) we satisfy the assumptions of Corollary 2 for all $(t,x)\in (0,T^*)\times \{|x|\le R_0\}$. The last claim of Corollary 2 ensures (103) with $\delta =0$. This last, via the triangular inequality, allows us to arrive at (15). In order to complete the proof of the theorem we have to prove estimate (16). So that we have to argue for $t\ge T^*$. As already claimed, estimate (103) holds for u, thus estimate (16) holds for all $t>T^*$ and $|x|> R_0$. In particular they hold for $t\in [T^*,\max \{T_0,T^*\})$. Hence (16) is true for $t\in [T^*,\max \{T_0,T^*\})$ and $|x|>R_0$. Taking into account the former assumption of (14), by virtue of the first claim of Corollary 2, we get (103) with $\delta (x)$. This last allows us to arrive at (11). Hence we can consider a covering of $[T^*,\max \{T_0,T^*\}]\times \{|x|\le R_0\}$ by means of a family of open parabolic neighborhood $Q_{\big (\frac{(1-\delta (x))s}{4} \big )^\frac{1}{2}}(s,x)$, $(s,x)\in [T^*,\max \{T_0,T^*\}]\times \{|x|\le R_0\}$. Since the set $[T^*,\max \{T_0,T^*\}]\times \{|x|\le R_0\}$ is a compact, then from the covering we realize a finite family of parabolic cylinder ${\widetilde{Q}}_h$, $h=1,\dots ,M,$ which covers the compact set. Hence any $(t,x)\in [T^*,\max \{T_0,T^*\}]\times \{|x|\le R_0\}$ belongs to some ${\widetilde{Q}}_h$, thus, with a constant $\overline{c}$ independent of (t, x), we get

$$\begin{aligned} |u(t,x)|\le \overline{c}r_h^{-\frac{1}{2}}+|v(t,x)|\le \overline{c}{T^*}^{-\frac{1}{2}}+|v(t,x)|\,, \end{aligned}$$

which proves (16) .

The difference between Theorems 4 and 5 is in the fact that in the latter the hypotheses do not provide for the existence of $v_0$ and $v_0^*$ satisfying estimates (14). In the following lemma we show that, under the only assumption $u_0\in J^2(\mathbb {R}^3)$, there exist suitable $v_0$ and $v_0^*$ such that (14) holds on a subset ${\mathbb {F}}_\sigma \subseteq B_{R'_0}$ whose complement set in $B_{R'_0}$ has Lebesgue measure less than $\sigma$, where $\sigma >0$ is arbitrarily chosen.

Lemma 12

Let $u_0$ be in $J^2(\mathbb {R}^3)$ and let $\varepsilon >0$. For all $R>0$ and $\sigma >0$ there exist a closed set ${\mathbb {F}}\subseteq B_{R}$ and $\mathfrak {g}\in J^{1,2}(\mathbb {R}^3)$ such that

$$\begin{aligned} |B_R-{\mathbb {F}}|<\sigma , \text{ and } \sup _{{\mathbb {F}}}\int \limits _{\mathbb {R}^3}\frac{|u_0-\mathfrak {g}|^2}{|x-y| }\,dy<\varepsilon \,. \end{aligned}$$

(105)

Proof

Let us consider the mollified sequence $\{J_m[u_0]\}$ converging to $u_0$ in $J^2(\mathbb {R}^3)$. As consequence, by virtue of the Hardy–Littlewood–Sobolev theorem, the sequence

$$\begin{aligned} \{\psi _m(x)\}\,, \text{ with } \psi _m(x):=\int \limits _{\mathbb {R}^3}\frac{|u_0-J_m[u_0]|^2}{|x-y|}\,dy\,, \text{ converges } \text{ to } 0 \text{ a.e. } \text{ in } B_R. \end{aligned}$$

By virtue of the Severini–Egorov theorem, for all $\sigma >0$ there exists a closed set ${\mathbb {F}}\subseteq B_R$ such that

$$\begin{aligned} |B_R-{\mathbb {F}}|<\sigma \text{ and } \psi _m\rightarrow 0 \text{ uniformly } \text{ on } {\mathbb {F}}\,. \end{aligned}$$

The lemma is proved.

Now, we are in a position to prove Theorem 5.

Proof of Theorem 5

The first claim, that is (13) on $(0,\infty )\times \{x:|x|>R'_0\}$, holds as proved in Theorem 4. By virtue of Lemma 12 we realize estimate (105) on some closed set ${\mathbb {F}}_\sigma$, that we twice consider. A first time we choose $\varepsilon \le \frac{\varepsilon _1}{2c_1 }$ . In such case we denote by $v_0^*$ the function $\mathfrak {g}$ in (105). By means of $v_0^*$ we deduce (41) with $\varepsilon \le \varepsilon _1/2c_1$. Thus, via Corollary 2, we obtain (103) with $\delta =0$ on some interval $(0,T^*)\times {\mathbb {F}}_\sigma$. Since $w=u-v^*$, where $v^*$ is the $J^{1,2}$-regular solution corresponding to $v_0^*$, we deduce the validity of (18). A second time we choose $\varepsilon \le 1/(4c)^2$. In such case we denote by $v_0$ the function $\mathfrak {g}$ in (105). By means of $v_0$ we deduce (41) with $\varepsilon _0=1/(4c)^2$. Thus, via Corollary 2, we obtain (103) for all $(s,x)\in (0,T_0)\times {\mathbb {F}}_\sigma$. A priori we have to consider $T_0\ne T^*$. If $T^*<T_0$, since $w=u-v$, where v is the $J^{1,2}$-regular solution corresponding to $v_0$, we deduce the validity of (19). Actually, we have a family of open set of the kind $Q_{[(1-\delta )t/4]^{-\frac{1}{2}}}(t,x)$ covering the compact set $[T^*,T_0]\times {\mathbb {F}}_\sigma$. Hence, we can extract a finite family ${\widetilde{Q}}_1,\ldots ,{\widetilde{Q}}_m$ in order to deduce (19) with $T^*\le \min _{h\in \{1,\ldots ,m\}}r_h^2$.

6 Proof of Theorem 6

The proof is based on a duality argument classical in PDEs that goes back to the Holmgren uniqueness theorem for the system of the Cauchy-Kowalewskaya type and considered by Foias in [6] for the Navier–Stokes equations using as adjoint the linear Stokes Cauchy problem. We are able to repeat the Foias arguments, but without extra assumptions on the solutions.

We denote by $\psi$ the solution to the Cauchy problem:

$$\begin{aligned} \psi _t-\varDelta \psi= & {} -\nabla \pi _\psi \,,\quad \nabla \cdot \psi =0\, \text{ in } (0,T)\times \mathbb {R}^3\,, \nonumber \\ \psi= & {} \psi _0\in {\mathscr {C}}_0(\mathbb {R}^3) \text{ on } \{0\}\times \mathbb {R}^3\,. \end{aligned}$$

(106)

It is well known that $\psi$ is a smooth solution with $\psi \in C([0,T);J^p(\mathbb {R}^3))$, for all $p\in (1,\infty )$. Moreover, the following estimates hold:

$$\begin{aligned} \,\,q\ge p\,,\,||\psi (t)||_q\le & {} ct^{-\frac{3}{2}\left( \frac{1}{p}-\frac{1}{q}\right) }||\psi _0||_p\,,\; \text{ for } \text{ all } t>0\,, \nonumber \\ ||\nabla \psi (t)||_q\le & {} c_1t^{-\frac{1}{2}-\frac{3}{2}\left( \frac{1}{p}-\frac{1}{q}\right) }||\psi _0||_p\,, \text{ for } \text{ all } t>0\,. \end{aligned}$$

(107)

In the case of $p=q=2$, in (107) we have $c=1$ and $c_1=1/\sqrt{2}$.

For $t>0$, we set ${\widehat{\psi }}(\tau ,x):=\psi (t-\tau ,x)$ provided that $(\tau ,x)\in (0,t)\times \mathbb {R}^3$. It is well known that ${\widehat{\psi }}$ is a solution backward in time with ${\widehat{\psi }}(t,x)=\psi _0(x)$.

Lemma 13

Let u be a sws. Then relation iii. of Definition 2, firstly written for $\varphi \in C^{1}_0([0,T)\times \mathbb {R}^3)$, can be extended to all $\varphi \in C([0,T);J^{1,2}(\varOmega ))$ with $\varphi _t\in L^2(0,T;L^2(\varOmega ))$.

Proof

This is a classical result essentially due to Prodi [15]. It is based on the integrability properties of a sws and a classical density argument.

Proof of Theorem 6

We denote by $(w,\pi _w)$ the difference of two solutions $(u,\pi _u)$ and $(\overline{u},\pi _{\overline{u}})$ furnished by Theorem 4. Hence, for all $t>s\ge 0$, $(w,\pi _w)$ solves the integral equation:

$$\begin{aligned}&\int \limits _{s}^{t}\Big [(w,\varphi _\tau )-(\nabla w,\nabla \varphi )+(u\cdot \nabla \varphi ,w)+( w\cdot \nabla \varphi ,\overline{u})\Big ]d\tau \nonumber \\&\quad +(w(s),\varphi (s))=(w(t),\varphi (t))\,, \end{aligned}$$

(108)

for all $\varphi \in C([0,T);J^{1,2}(\varOmega ))$ with $\varphi _t\in L^2(0,T;L^2(\varOmega ))$ . For given $t>0$ and $s>0$, we set ${\widehat{\psi }}$ in place of $\varphi$. Integrating by parts, we get

$$\begin{aligned} \int \limits _{s}^{t}\Big [ (u\cdot \nabla {\widehat{\psi }},w)+( w\cdot \nabla {\widehat{\psi }},\overline{u})\Big ]d\tau +(w(s),{\widehat{\psi }} (s))=(w(t),\psi _0)\,. \end{aligned}$$

(109)

Taking estimate (15) into account for both the solutions, applying Hölder’s inequality and estimates (107) for $p=q=2$, for all $t\in (0,T^*)$, we deduce the estimate

$$\begin{aligned} |(w(t),\psi _0)|\le & {} ||w(s)||_2||\psi (t-s)||_2 + \int \limits _{s}^{t} \Big [{\frac{\sqrt{2}c_0\varepsilon ^\frac{2}{3}_1 }{\sqrt{\tau }}}+ ||v^{*}(\tau )||_\infty \Big ] ||w(\tau )||_2||\nabla \psi (t-\tau )||_2d\tau \\\le & {} ||w(s)||_2||\psi _0||_2 + ||\psi _0||_2\int \limits _{s}^{t} \Big [{\frac{\sqrt{2}c_0\varepsilon ^\frac{2}{3}_1 }{\sqrt{\tau }}}+ ||v^{*}(\tau )||_\infty \Big ] ||w(\tau )||_2(t-\tau )^{-\frac{1}{2}}d\tau \,, \end{aligned}$$

for all $\psi _0\in {\mathscr {C}}_0(\mathbb {R}^3)$. Letting $s\rightarrow 0$, since $||w(0)||_2=0$, we get

$$\begin{aligned} ||w(t)||_2\le \int \limits _{s}^{t} \Big [\frac{\sqrt{2}c_0\varepsilon ^\frac{2}{3}_1}{\sqrt{\tau }}+||v^{*}(\tau )||_\infty \Big ] ||w(\tau )||_2(t-\tau )^{-\frac{1}{2}}d\tau \,. \end{aligned}$$

Recalling that $\int \limits _{0}^{t} (t-\tau )^{-\frac{1}{2}}\tau ^{-\frac{1}{2}}d\tau =\pi$, and observing that v, via the embedding theorem, belongs to $\in L^4(0,T^*;L^\infty (\mathbb {R}^3))$, we can deduce the uniqueness on $(0,t^*)\subseteq (0,T^*)$ provided that $\sqrt{2}c_0\varepsilon _1^\frac{2}{3}\pi <1$ and $t^*$ is sufficiently small. Now, we have to discuss the uniqueness for $t\in [t^*,T_0)$. On the other hand by virtue of (16) and Theorem 8, both the solutions are local regular on $(0,T_0)$, therefore from some $t_0\in (0,t^*)$ we have to discuss the uniqueness of two regular solutions. Thus the claim of the uniqueness trivially follows.

7 Proof of Theorem 7

In the hypotheses of Theorem 7 and by virtue of estimates (18) and (19), we state that, for all $\eta >0$, the sws $u\in L^r(\eta ,T_0;L^s(\mathbb {R}^3))$. Hence, by virtue of Theorem 8, u is a local regular solution for all $t\in (0,T_0)$. This proves the first part of the theorem.

Let $\overline{u}$ be another sws ensured by Theorem 5 and corresponding to $u_0$. We set $w=\overline{u}-u$. Let h be the smooth function defined in (20). We write $w=hw+(1-h)w$. Hence we get $||w||_2\le ||hw||_2+||(1-h)w||_2$. We separately estimate hw and $(1-h)w$. We set $\overline{h}:=1-h$. The term $\overline{h}w$ is estimated by duality starting from the integral equation

$$\begin{aligned} (w(t),\varphi (t))= & {} (w(s),\varphi (s))\nonumber \\&+\int \limits _{s}^{t}\Big [(w,\varphi _\tau )-(\nabla w,\nabla \varphi )+(\overline{u}\cdot \nabla \varphi , w)+(w\cdot \nabla \varphi ,u)+(\pi _w,\nabla \cdot \varphi )\Big ]d\tau ,\nonumber \\ \end{aligned}$$

(110)

that we consider for all $\varphi \in C([0,T);W^{1,2}(\varOmega ))$ with $\varphi _t\in L^2(0,T;L^2(\varOmega ))$ and $\varphi =0$ in a neighborhood of T. Let $\psi$ be the solution to the heat equation with initial data $\psi _0\in C_0^\infty (\mathbb {R}^3)$, and ${\widehat{\psi }}$ the solution $\psi$ written backward in time on $(0,t)\times \mathbb {R}^3$. We substitute $\varphi$ with $\overline{h}{\widehat{\psi }}$. Integrating by parts, formula (110) becomes

$$\begin{aligned} (w(t)\overline{h},\psi _0)= & {} \int \limits _{0}^{t}\Big [(\overline{u}\cdot \nabla \psi ,\overline{h} w)+(w\cdot \nabla \psi ,\overline{h}u)-(\nabla \pi ^1,\overline{h}\psi )+(\overline{h}\pi ^2,\nabla \cdot \psi )\nonumber \\&+(\overline{u}\cdot \nabla \overline{h},\psi \cdot w)+(w\cdot \nabla \overline{h},\psi \cdot u)+(\pi ^2\nabla \overline{h},\psi )\Big ]d\tau =:{{\underset{i=1}{\overset{7}{\sum }}}}I_i,\nonumber \\ \end{aligned}$$

(111)

where, via the representation (31), we have mean $\pi _w$ decomposed in $\pi ^1+\pi ^2$. Recalling that Theorem 5 ensures estimates (13) and (18) and (19) for u and $\overline{u}$ corresponding to the same data $u_0$, united with the properties related to function h given in (20), for all $\tau \in (0,T_0)$, we have

$$\begin{aligned} ||\overline{h}u(\tau )||_\infty +||\overline{h} \overline{u}(\tau )||_\infty \le \frac{2c_0\varepsilon _1^\frac{2}{3}}{\sqrt{\tau }}+ A(\tau ) \end{aligned}$$

(112)

where we set

$$\begin{aligned} A(\tau ):=\Bigg [ \chi _{[0,T^*]}(\tau )||v^*(\tau )||_\infty +\chi _{[T^*,t]}(\tau )\Bigg (\frac{c}{(T^*)^{\frac{1}{2}} }\,\,\,+||v(\tau )||_\infty \Bigg )\Bigg ]\,. \end{aligned}$$

Hence, applying Hölder’s inequality, we get

$$\begin{aligned} |I_1+I_2|\le & {} \int \limits _{0}^{t}(||\overline{h} u ||_\infty + ||\overline{h}\overline{u}||_\infty )||\nabla \psi (t-\tau )||_2||w||_2d\tau \\\le & {} ||\psi _0||_2\int \limits _{0}^{t}\Big [\frac{\sqrt{2}c_0\varepsilon _1'}{\sqrt{\tau }}+A(\tau )\Big ] (t-\tau )^{-\frac{1}{2}}||w(\tau )||_2d\tau \,. \end{aligned}$$

Via estimates (32), we obtain the following estimates:

$$\begin{aligned} |I_3(t)|= & {} \left| \int \limits _{0}^{t} (\nabla \pi _1,\overline{h}{\widehat{\psi }})d\tau \right| \le ||\psi _0||_2\int \limits _{0}^{t}||\nabla \pi _1||_2d\tau \nonumber \\\le & {} c||\psi _0||_2\int \limits _{0}^{t}||w||_2\big [||u||_3+||\overline{u}||_3\big ]B(t,\tau )d\tau \\ |I_4(t)|= & {} \left| \int \limits _{0}^{t} (\pi _2,\overline{h}\nabla \cdot \psi )d\tau \right| \le c||\psi _0||_2 \int \limits _{0}^{t}||\pi _2||_2(t-\tau )^{-\frac{1}{2}}d\tau \\\le & {} c||\psi _0||_2\int \limits _{0}^{t}\Big [\frac{\sqrt{2}c_0\varepsilon '_1}{\sqrt{\tau }}+A(\tau )\Big ] (t - \tau )^{-\frac{1}{2}}||w||_2 + ||w||_2\big [||u||_3+ ||\overline{u}||_3\big ]B(t,\tau )d\tau \end{aligned}$$

where we set $B(t,\tau ):=1+(t-\tau )^{-\frac{1}{2}}$ and we took estimate (112) into account. Finally, recalling that $\nabla \overline{h}$ has compact support, applying Hölder’s inequality, and employing (29) for the pressure field, we arrive at

$$\begin{aligned} |I_5+I_6+I_7|\le & {} c\int \limits _{0}^{t}||\psi ||_\infty \big [(||u||_2+||\overline{u}||_2) ||w||_2+||\pi _w||_{{\mathcal {H}}^1}\big ]d\tau \\\le & {} c||\psi _0||_2||u_0||_2\int \limits _{0}^{t}(t-\tau )^{-\frac{3}{4}} ||w(\tau )||_2d\tau . \end{aligned}$$

The above estimates and formula (111), uniformly in $t\in (0,T_0)$, ensure

$$\begin{aligned} ||\overline{h} w(t)||_2\le \sqrt{2}\pi c_0\varepsilon _1^\frac{2}{3} \sup _{(0,t)}||w(t)||_2+ \int \limits _{0}^{t}\Big [\frac{A(\tau )}{(t-\tau )^{\frac{1}{2}}}+\frac{||u_0||_2}{(t-\tau )^{\frac{3}{4}}}\,\Big ]||w(\tau )||_2d\tau \,. \end{aligned}$$

(113)

Now we estimate $||hw(t)||_2$. For this task we employ Leray’s technique adapted to sws. We consider $\phi =h^2$ in the energy relation of each weighted energy relations related to the sws u and $\overline{u}$. So that from (2) we get

$$\begin{aligned}&\int \limits _{\mathbb {R}^3}|hu(t)|^2dx+2\int \limits _{\sigma }^{t}\int \limits _{\mathbb {R}^3}|h\nabla u|^2dxd\tau \le \int \limits _{\mathbb {R}^3}|u(\sigma )h|^2dx \nonumber \\&\quad +\int \limits _{\sigma }^{t}\int \limits _{\mathbb {R}^3}|u|^2\varDelta h^2dxd\tau +\int \limits _{\sigma }^{t}\int \limits _{\mathbb {R}^3}(|u|^2+2\pi _u) u\cdot \nabla h^2 dxd\tau , \end{aligned}$$

(114)

for all $t\ge \sigma$, for $\sigma =0$ and a.e. in $\sigma \ge 0$, and analogously for $\overline{u}$

$$\begin{aligned}&\int \limits _{\mathbb {R}^3}|h\overline{u}(t)|^2dx+2\int \limits _{\sigma }^{t}\int \limits _{\mathbb {R}^3}|h\nabla \overline{u}|^2 dxd\tau \le \int \limits _{\mathbb {R}^3}|h\overline{u}(\sigma )|^2dx \nonumber \\&\quad +\int \limits _{\sigma }^{t}\int \limits _{\mathbb {R}^3}|\overline{u}|^2\varDelta h^2dxd\tau +\int \limits _{\sigma }^{t}\int \limits _{\mathbb {R}^3}(|\overline{u}|^2+2\pi _{\overline{u}}) \overline{u}\cdot \nabla h^2 dxd\tau , \end{aligned}$$

(115)

for all $t\ge \sigma$, for $\sigma =0$ and a.e. in $\sigma \ge 0$. Since u is $J^{1,2}$-regular, we substitute the test function $\varphi$ with $h^2u$ in the weak formulation of $\overline{u}$

$$\begin{aligned}&\int \limits _{s}^{t}\Big [(\overline{u},u_\tau h^2)-(\nabla \overline{u},h^2\nabla u)-(\overline{u}\cdot \nabla \overline{u},h^2 u)\Big ]d\tau \\&\quad +(\overline{u}(s),h^2 u(s) )=(\overline{u}(t),h^2u(t))+G(s,t)\,, \end{aligned}$$

where we set

$$\begin{aligned} G(s,t):=\int \limits _{s}^{t}\Big [(\nabla \overline{u},\nabla h^2\otimes u)-(\pi _{\overline{u}}\nabla h^2 \cdot u)\Big ]d\tau \,. \end{aligned}$$

Analogously, employing the regularity of u, we set $\overline{u}h^2\in L^2(0,T;J^{1,2}(\mathbb {R}^3))$ as test function for the equation of $(u,\pi _u)$. Recalling that, for all $\psi \in L^2(\mathbb {R}^3)$, $(u(t),\psi )$ is continuous function of t, we get

$$\begin{aligned}&\int \limits _{s}^{t}\Big [(u_\tau ,\overline{u} h^2)+(h^2\nabla \overline{u},\nabla u)+(u\cdot \nabla u,h^2\overline{u})\Big ]d\tau =G_1(s,t)\,, \end{aligned}$$

where we set

$$\begin{aligned} G_1(s,t):=\int \limits _{s}^{t}\Big [-(\nabla u,\nabla h^2\otimes \overline{u})+(\pi _{u}\nabla h^2 \cdot \overline{u})\Big ]d\tau \,. \end{aligned}$$

Making the difference between the two weak formulations we get

$$\begin{aligned} -2\int \limits _{s}^{t}(\nabla \overline{u},h^2\nabla u)= & {} \int \limits _{s}^{t}\Big [(\overline{u}\cdot \nabla \overline{u},h^2u)+(u\cdot \nabla u,h^2\overline{u}) +G(s,t)-G_1(s,t)\Big ]d\tau \\&-(\overline{u}(s),h^2 u(s) )+(\overline{u}(t),h^2u(t))\,. \end{aligned}$$

Now, multiplying for 2 the last relation and summing it with the weighted energy relations (114) and (115), we get

$$\begin{aligned}&||hw(t)||_2^2+2\int \limits _{s}^{t}||h\nabla w(\tau )||_2^2d\tau \le ||hw(s)||_2^2 \\&\quad +\int \limits _{s}^{t}\Big [(|\overline{u}|^2\overline{u}+|u|^2u,\nabla h^2)+ 2(\overline{u}\cdot \nabla \overline{u},h^2u)+2(u\cdot \nabla u,h^2\overline{u})\Big ]d\tau \\&\quad +\int \limits _{s}^{t}(|w(\tau )|^2,\varDelta h^2)d\tau -4\int \limits _{s}^{t}(\pi _{\overline{u}}-\pi _u,\nabla h \cdot wh)d\tau =:||w(s)h||_2^2+{{\underset{i=1}{\overset{3}{\sum }}}}I_i\,. \end{aligned}$$

We estimate separately the integrals $I_i$, $i=1,2,3$ . Considering the mollified $\overline{u}_\varepsilon$, set $w_\varepsilon :=\overline{u}_\varepsilon -u$, via an integrating by parts, we get

$$\begin{aligned} 2(\overline{u}\cdot \nabla \overline{u},uh^2)+(|\overline{u}|^2,\overline{u}\cdot \nabla h^2)= & {} 2\lim _\varepsilon \Big [(\overline{u}_\varepsilon \cdot \nabla \overline{u},(u-\overline{u})h^2)\Big ]\nonumber \\= & {} -2\lim _\varepsilon (\overline{u}_\varepsilon \cdot \nabla \overline{u},wh^2)\,, \end{aligned}$$

(116)

$$\begin{aligned} 2( u\cdot \nabla u,\overline{u}h^2)-2(u\cdot \nabla u,uh^2)= & {} 2( u\cdot \nabla u,wh^2)\,. \end{aligned}$$

(117)

Summing the right hand sides of (116) and (117), we arrive at

$$\begin{aligned} -2\lim _\varepsilon (\overline{u}_\varepsilon \cdot \nabla \overline{u}-u\cdot \nabla u,wh^2)= & {} -2\lim _\varepsilon ((\overline{u}_\varepsilon -u)\cdot \nabla \overline{u}+u\cdot \nabla (\overline{u}-u),wh^2)\nonumber \\= & {} -2\lim _\varepsilon (w_\varepsilon \cdot \nabla \overline{u}+u\cdot \nabla w,wh^2)\,. \end{aligned}$$

(118)

Hence, via (116)–(118), for $I_1$ we obtain

$$\begin{aligned} I_1= & {} 2\int \limits _{s}^{t}\Big [-\lim _\varepsilon \big [(w_\varepsilon \cdot \nabla w,wh^2)+(w_\varepsilon \cdot \nabla u,wh^2)\big ]-(u\cdot \nabla w,wh^2) \Big ]d\tau \\= & {} \int \limits _{s}^{t}\Big [\lim _\varepsilon \big [(w_\varepsilon ,\nabla h^2 |w|^2)+2(w_\varepsilon \cdot \nabla h^2, u\cdot w)\big ]\\&+(u\cdot \nabla h^2,|w|^2)+2(w_\varepsilon \cdot \nabla w,uh^2) \Big ]d\tau \\= & {} \int \limits _{s}^{t} \Big [(w\cdot \nabla h^2,|w|^2)+ 2(w\cdot \nabla h^2, u\cdot w)\\&+(u\cdot \nabla h^2,|w|^2)+2(w\cdot \nabla w,uh^2)\Big ]d\tau \,, \end{aligned}$$

where we taken into account that u enjoys the $J^{1,2}$-regularity. Recalling the assumption on u related to the support ${\mathbb {H}}$ of h, by applying Holder’s inequality, we get

$$\begin{aligned} |I_1|\le & {} c\int \limits _{s}^{t} ||w||_2||w||_3||h w||_6d\tau +c\int \limits _{s}^{t}||u||_3||w||_2||h w||_6d\tau \\&+c \int \limits _{s}^{t} ||u||_{L^s({\mathbb {H}})}||h w||_{\frac{2s}{s-2}}||h \nabla w||_2d\tau . \end{aligned}$$

Thus via the Gagliardo–Nirenberg inequality, we get

$$\begin{aligned} |I_1|\le & {} c\int \limits _{s}^{t}(||w||_3+||u||_3)||w||_2\big [||h \nabla w||_2+||\nabla hw||_2\big ] d\tau \\&+c\int \limits _{s}^{t} ||u||_{L^s({\mathbb {H}})}||wh||_2^\frac{s-3}{s}\big [||h \nabla w||_2^\frac{3}{s}+||\nabla hw||_2^\frac{3}{s}\big ]||h\nabla w||_2d\tau \end{aligned}$$

Since $\varDelta h^2\in L^\infty (\mathbb {R}^3)$, we get

$$\begin{aligned} |I_2|\le c\int \limits _{s}^{t}||w(\tau )||_2^2d\tau \,. \end{aligned}$$

Finally, recalling (27), we write

$$\begin{aligned} (\pi _{\overline{u}}-\pi _u,\varDelta g)=-(w\otimes w+u\otimes w+w\otimes u,\nabla \nabla g)\,, \text{ for } \text{ all } g\in C_0^\infty (\mathbb {R}^3)\,, \end{aligned}$$

via (28), by applying Hölder’s inequality, we easily arrive at

$$\begin{aligned} |I_3|\le & {} c\int \limits _{s}^{t} ||\pi _{\overline{u}}-\pi _u||_\frac{6}{5}||hw||_6d\tau \le c\int \limits _{s}^{t} \big [2||u||_3 ||w||_2+||w||_3||w||_2\big ]||hw||_6d\tau \\\le & {} c\int \limits _{s}^{t} \big [||u||_3+||w||_3\big ]||w||_2\big [||h\nabla w||_2+2||w\nabla h||_2\big ] d\tau \,. \end{aligned}$$

Collecting the estimates related to $I_i$, $i=1,2,3,$ and substituting on the right-hand side of the energy relation related to wh, applying Young’s inequality, we get

$$\begin{aligned} ||h w(t) ||_2^2&+\int \limits _{s}^{t}||h\nabla w(\tau )||_2^2d\tau \le ||hw(s)||_2^2 \\&+\,c\int \limits _{s}^{t}\Big [||u||_{L^s({\mathbb {H}})}^r||w||_2^2+||u||_{L^s({\mathbb {H}})}^2||w||_2^2+||w||_2^2 \\&+\,\big [||u||_3 +||w||_3\big ]^2||w||_2^2+\big [||w||_3+||u||_3\big ]||w||_2^2 \Big ]d\tau \,. \end{aligned}$$

Letting $s\rightarrow 0$ in the above inequality, and recalling the assumption on u and employing the energy inequality, we get the estimate

$$\begin{aligned} ||h w(t)||_2\le k(t)\sup _{(0,t)}||w(\tau )||_2\,, \end{aligned}$$

(119)

where k(t) is a continuous function with the limit property $\lim _{t\rightarrow 0}k(t)=0$ . Estimate (113) and estimate (119) furnish

$$\begin{aligned} ||w(t)||_2\le \big [\sqrt{2}\pi c_0\varepsilon _1^\frac{2}{3}+k(t)\big ] \sup _{(0,t)}||w(t)||_2+ \int \limits _{0}^{t}\Big [\frac{A(\tau )}{(t-\tau )^{\frac{1}{2}}}+\frac{||u_0||_2}{(t-\tau )^{\frac{3}{4}}}\,\Big ]||w(\tau )||_2d\tau . \end{aligned}$$

Since $\sqrt{2}\pi c_0\varepsilon _1^\frac{2}{3}<1$, for a suitable $t^*\in (0,T_0)$ we get $w(t)=0$ for all $t\in [0,t^*]$. Then, recalling that in the beginning of the section we claimed that u is regular, for $t>0$, the Leray arguments furnish the coincidence of u and $\overline{u}$ on $[t^*,T_0)$ .