On the Stability of Steady-State Solutions to the Navier–Stokes Equations in the Whole Space

Abstract

We prove asymptotic stability of steady-state solutions to the Navier–Stokes equations in the whole space. One of the novelties of this work consists in considering perturbations that show a distinct pointwise behavior (in space and time) and correspond to initial data only belonging to suitable weighted Lebesgue spaces.

1 Introduction

Stability of steady-state solutions to the Navier–Stokes equations in the exterior, $\Omega$, of a three-dimensional bounded region is among the classical and most studied problems in mathematical fluid mechanics. Actually, the first, rigorous study goes back to more than half century ago, owing to the work of Heywood [10]. The main finding -as expected on physical ground- is that steady-state is asymptotically stable, provided its “magnitude” is sufficiently “small.” Since then, many significant papers have been published, where the problem is addressed under different assumptions either on the steady-state or else the functional properties of the dynamical perturbations, or both. The list of contributors is too long to include here and we refer the reader to the review article [8], especially Section 10.4, and the bibliography therein.

In spite of the abundant literature, it seems that there is room left for further and interesting investigation. In fact, the class of perturbations to the basic state considered in the papers referred to as above, is contained in Lebesgue (or even “larger”) spaces. Now, if, on the one hand, this allows for a rather general type of perturbations, on the other hand it may have less impact in the application and/or numerical computation, where the knowledge of a specific and controlled pointwise behavior would be more advantageous.

Objective of this paper is to perform a preliminary study in the latter direction. Actually, we will analyze the case when $\Omega$ is the whole three-dimensional space and prove an asymptotic stability theorem in a class of perturbations where, in particular, their pointwise behavior in space and time is entirely controlled by the initial data; see Theorem 2 and Corollary 1. In order to achieve this goal, we begin to define the class of steady-state solutions, whose asymptotic stability we intend to investigate. Such solutions are characterized by velocity fields decaying like $|x|^{-2}$ at large distances, and their existence and uniqueness is proved for “small” and suitable corresponding body force; see Theorem 1. We next write the evolution equation for the perturbation field as an integral equation (see (5.4)) and study existence and uniqueness of corresponding solutions in appropriate weighted Lebesgue spaces, $L_\alpha ^p$, with a (spatial) weight that grows algebraically at large distances from the origin; see (2.1). We thus prove existence by standard successive approximations procedure, that we show to converge provided the magnitudes of the initial data and of the steady-state are restricted in suitable norms. A crucial tool for the above procedure to work is a result on convolutions in weighted spaces due to Kerman [11], and refined in [15]; see Lemma 2. Furthermore, we prove uniqueness by a duality argument. Our method is close to that used in the well-known Holmgren theorem, as proposed by Foias [6] in the Navier–Stokes setting, but suitably modified to fit in our context.

As mentioned earlier on, an important consequence of our main result -stated in Theorem 2- is that we are able to control the pointwise space-time behavior of perturbations. More precisely, we show that they decay in space as $|x|^{-1+\frac{3}{p}}$ and time as $t^{-\frac{3}{2p}}$, $p>3$; see Corollary 1. It is worth remarking that these results are obtained without assuming any pointwise behavior on the initial data, but only that the latter belongs to a suitable weighted Lebesgue space (see (2.5)). Our finding should be contrasted with classical results (see [2, 12]) and those ones obtained in the case of weak solutions [3, 5], where an asymptotic pointwise behavior is imposed at the outset on the initial data. This behavior is preserved for $t>0$ and it can also be converted into a time asymptotic behavior by means of parabolic scaling $|x|\sim t^\frac{1}{2}$. In fact, our results are more in the spirit of those obtained in [9], where the initial datum is given in a weighted $L^2$ space. The fundamental difference is that, unlike [9], a priori our solutions, at all $t\ge 0$, are not requested to have finite energy and are perturbations not to the rest but to a non-trivial steady-state.

For the sake of the brevity, we omit the details concerning regularity properties of our solution. In fact, once we have established a (local) $L^\infty$-bound, they follow via standard arguments developed for the integral equation (5.3).

The plan of the paper is the following. In Sect. 2 we formulate the problem, and state and comment our main results. In Sect. 3 we prove existence and uniqueness of steady-state solutions in a suitable function class where the velocity field decays like $|x|^{-2}$ for large |x|. In the following Sect. 4 we begin the study of the perturbation problem by presenting a number of preliminary lemmas. The main objective of Sect. 5 is to show several basic estimates in weighted spaces for convolutions of the Oseen fundamental solution with functions belonging to weighted spaces as well. In the last Sect. 6, we give a proof of our main result, Theorem 2, to get existence of solutions by applying successive approximation procedure to the integral equation (5.3) and, successively, discussing their uniqueness. In the same section, as a corollary to this finding, we furnish the desired pointwise space-time behavior.

2 Formulation of the Problem and Statement of the Main Results

We introduce some notation. For given $p>3$ and $\alpha \in {\mathbb {R}}$, by $L^p_\alpha =L^p_\alpha ({\mathbb {R}}^3)$ we denote the following weighted Lebesgue space

$$\begin{aligned} \{u: \,u|x|^{\alpha }\in L^p(\mathbb {R}^3)\}\,, \end{aligned}$$

(2.1)

endowed with the natural norm

$$\begin{aligned} \Vert u\Vert _{L^p_\alpha }:=\Vert u|x|^\alpha \Vert _p\,. \end{aligned}$$

Moreover, by $D^{1,q}({\mathbb {R}}^3)$, $1<q<\infty$, we denote the completion of $C_0^\infty ({\mathbb {R}}^3)$ in the Lebesgue gradient norm $\Vert \nabla u\Vert _q$. We also set

$$\begin{aligned} {[\!|{u}|\!]}_\beta :=\sup _{x\in {\mathbb {R}}^3}[(1+|x|^\beta )|u(x)|]\,,\ \ \beta >0\,. \end{aligned}$$

Finally, we indicate by ${\mathscr {C}}_0$ the class of vector functions, $\varphi \in C_0^\infty ({\mathbb {R}}^3)$ with $\textrm{div}\,\varphi =0$. All other notation is rather standard.

Consider the steady-state Navier–Stokes equations in the whole space corresponding to a given body force f:

$$\begin{aligned} w\cdot \nabla w+\nabla \pi _w=\Delta w +f\,,\quad \nabla \cdot w=0\,,\hbox { in }\mathbb {R}^3\,. \end{aligned}$$

(2.2)

Our first objective is to determine the class of solutions to (2.2) where our stability results apply. Precisely, we show the following theorem.

Theorem 1

Let $f=\textrm{div}\,F$, where F is a second-order tensor field such that

$$\begin{aligned} {\mathcal {D}}:={[\!|{F}|\!]}_2+{[\!|{f}|\!]}_3<\infty \,. \end{aligned}$$

Then, there is an absolute constant $C_0>0$ such that, if

$$\begin{aligned} {\mathcal {D}}<C_0\,, \end{aligned}$$

(2.3)

problem (2.2) has one and only one solution $(w,\pi _w)$ with w satisfying

$$\begin{aligned} {[\!|{w}|\!]}_2\le C_1{\mathcal {D}}\,, \end{aligned}$$

where $C_1>0$ is another absolute constant.

The main goal of this paper is to investigate the asymptotic stability of the steady-state solutions of the theorem above, in the class of perturbations $(v,\pi )$ with $v\in L^\infty (\mathbb {R}^3)\cap L^p_\alpha (\mathbb {R}^3)$.

We consider special perturbations $u_0$ of the kinetic field w, that is, such that $u_0-w:=v_0\in L^p_\alpha$, and look for sufficient conditions for the global existence of corresponding perturbation solutions $(v,\pi )$, ensuring the stability in the $L^p_\alpha$-norm and attractiveness in the $L^\infty$-norm of the given steady motion.

To this end, we commence by observing that the perturbation $(v,\pi )$ to the velocity w satisfies the following Cauchy problem

$$\begin{aligned} \begin{array}{l} v_t+v\cdot \nabla (v+w)+w\cdot \nabla v+\nabla \pi =\Delta v\,,\quad \nabla \cdot v=0\,,\hbox { in }(0,T)\times \mathbb {R}^3\,,\quad \\ \displaystyle v=v_0\hbox { on }\{0\}\times \mathbb {R}^3\,. \end{array} \end{aligned}$$

(2.4)

Set ${w^{(0)}:={[\!|{w}|\!]}_1}$, $w^{(1)}:=||w||_3$ and $w^{(2)}:={[\!|{w}|\!]}_2$. The main result of our paper can be stated as follows.

Theorem 2

Let

$$\begin{aligned} v_0\in L^p_\alpha (\mathbb {R}^3)\,, \ p>3\,, \ \alpha :=1-\frac{3}{p}. \end{aligned}$$

(2.5)

There is an absolute constant $c>0$ such that if

$$\begin{aligned} \begin{array}{c} c\big [(w^{(0)})^{1-\frac{3}{p}}(w^{(1)})^\frac{3}{p}+w^{(2)}\big ]+2\big ({(1+c)||v_0||_{L^p_\alpha }}\big )^\frac{1}{2}<1\,,\quad \\ \displaystyle 4c\big [(w^{(0)})^{1-\frac{3}{p}}(w^{(1)})^\frac{3}{p}+w^{(2)}\big ]<1\,, \end{array} \end{aligned}$$

(2.6)

then there exits a unique solution $(v,\pi )$ to problem (2.4) defined and smooth for all $t>0$ such that

$$\begin{aligned} \lim _{t\rightarrow 0}(v(t),\varphi )=(v_0,\varphi )\,,\hbox { for all }\varphi \in {\mathscr {C}}_0(\mathbb {R}^3)\,, \end{aligned}$$

(2.7)

$$\begin{aligned} t^\frac{1}{2}||v(t)||_\infty + ||v(t)||_{L^p_\alpha }\le c||v_0||_{L^p_\alpha }\,,\hbox { for all }t>0 \end{aligned}$$

(2.8)

and

$$\begin{aligned} t^\frac{1}{2}||\pi (t)||_{L^p_\alpha }\le c||v_0||_{L^p_\alpha }\big [||v_0||_{L^p_\alpha }+2w^{(2)}\big ]\,,\hbox { for all } t>0\,, \end{aligned}$$

(2.9)

where c is a constant independent of v.

Corollary 1

Any solution of Theorem 2 has the following pointwise behavior:

$$\begin{aligned} |v(t,x)|\le c \big [||v_0||_{L^p_\alpha }+||v_0||_{L^p_\alpha }^2\big ]t^{-\frac{3}{2p}} |x|^{-\alpha }\,,\hbox { for all }(t,x)\,. \end{aligned}$$

(2.10)

In particular, we achieve that for all $R>0$

$$\begin{aligned} |v(t,x)|\le c||v_0||_{L^p_\alpha }t^{-\frac{1}{2}}\hbox { if }|x|\le R\,\hbox { and }|v(t,x)|\le c \big [||v_0||_{L^p_\alpha }+||v_0||_{L^p_\alpha }^2\big ]t^{-\frac{3}{2p}}|x|^{-\alpha }\,,\hbox { if }|x|>R\,,\end{aligned}$$

(2.11)

where c is a constant independent of $v_0$ and (t, x).

We would like to make a number of remarks concerning the above results.

Remark 1

Since, as is easily shown, all norms $w^{(i)}$, $i=0,1,2$, are bounded above by ${[\!|{w}|\!]}_2$, in view of Theorem 2.2 the assumptions (2.6) can be restated by requiring that ${\mathcal {D}}$ and $\Vert v_0\Vert _{L^p_\alpha }$ are below a given absolute constant.

Remark 2

The $L^p$-weighted approach used in Theorem 2 has two noteworthy aspects. On the one hand, the perturbation is allowed to be singular at a point $x_0$ (in our assumption the origin O). On the other hand, it provides a pointwise and uniform spatial behavior of the solution, for $t>0$, without making a similar request on the initial data, but only requiring a weighted summability property.

Remark 3

The metric of $L^p_\alpha$ is not comparable with that of $L^3$ for $p>3$. Instead, with respect the Besov’s spaces $B^{-\frac{2}{s}}_{r,\infty }$ with $\frac{2}{s}+\frac{3}{r}=1$, we have $L^p_\alpha \subset B^{-\frac{2}{s}}_{r,\infty }$. Actually, $L^p_\alpha$, with $\alpha =1-\frac{3}{p}$, is another example of scaling invariant norm (in the sense proposed in [1]) for the global existence with small data. In this regards we notice that for $p=2$ the above space was considered in [4] in the different context of sufficient conditions for the existence of regular solutions. For $p=3$ , $L^3_\alpha \equiv L^3$ (that is $\alpha =0$) and the result goes back to the well known theory.

Remark 4

In [4], the weighted $L^2_{\frac{1}{2}}$-space is restricted to the so called Kato class. In this way the authors are able to give an estimate of the interval of existence (a priori finite) without employing auxiliary functions (density property). The interest of the Kato class is connected with the regularity criterium proposed in [1].

Remark 5

We think that an analogous approach can be used to investigate the case $p\in (2,3)$ and furnish a new regularity criterium. This study will be object of future work.

3 Proof of Theorem 1

In this section we will provide a proof of Theorem 1. To this end, we need some preparatory results contained in the next lemma.

Lemma 1

Let the assumption of Theorem 1 be satisfied. Then, given $\beta \in (0,1)$, there exists $C_0=C_0(\beta )>0$ such that if (2.3) holds, problem (2.2) has one and only one solution $(w,\pi _w)$ satisfying

$$\begin{aligned} {[\!|{w}|\!]}_{1+\beta }\le C_1\,{\mathcal {D}}\,. \end{aligned}$$

Proof

From [7, Theorem X.9.1] we know that, under the stated assumption on f there exists one and only one solution $(w,\pi _w)\in D^{1,q}({\mathbb {R}}^3)\times L^q({\mathbb {R}}^3)$, for all $q>3/2$, such that

$$\begin{aligned} {[\!|{w}|\!]}_1\le c\,{[\!|{F}|\!]}_2\,, \end{aligned}$$

(3.12)

where, here and hereafter, c denotes a positive absolute constant. We now recall the following classical representation for the velocity field, valid for all $x\in {\mathbb {R}}^3$ [7, Theorem X.5.2]

$$\begin{aligned} w_j(x)=\int _{{\mathbb {R}}^3} U_{ij}(x-y)f(y)dy- \int _{{\mathbb {R}}^3} w_l(y)w_i(y)D_lU_{ij}(x-y)dy \end{aligned}$$

(3.13)

where $\left( U,q\right)$ is the Stokes fundamental solution given by (in 3D)

$$\begin{aligned} U_{ij}(x)=\frac{-1}{8\pi |x|}\left[ \delta _{ij}+\frac{x_ix_j}{|x|^2}\right] \,,\ \ q_j(x)=\frac{1}{4\pi }\frac{x_j}{|x|^3}\,,\ \ i,j=1,2,3\,. \end{aligned}$$

(3.14)

By [14, Lemma 2.5], we get

$$\begin{aligned} \left| \int _{{\mathbb {R}}^3} U_{ij}(x-y)f(y)dy\right| \le c\,{\mathcal {D}}\,(1+|x|)^{-2}\,. \end{aligned}$$

(3.15)

Moreover, setting for $\rho >0$

$$\begin{aligned} \Omega _{\rho }:=\{|x|\le \rho \}\,,\ \ \Omega ^{\rho }:={\mathbb {R}}^3-{\Omega _{\rho }} \,, \end{aligned}$$

by (3.12) and (3.14), we have, on the one hand, by the inequality $|x-y|\ge {\frac{1}{2}}|x|$, $|x|\ge 2R_0$, $y\in \Omega _{R_0}$

$$\begin{aligned} |s_j(x)|:=\left| \int _{\Omega _{R_0}}w_l(y)w_i(y)D_lU_{ij}(x-y)dy\right| \le c\,{\mathcal {D}}^2\,|x|^{-2}\,,\ \ |x|\ge 2R_0\,, \end{aligned}$$

and, on the other hand, by the Hölder inequality,

$$\begin{aligned} |s_j(x)|\le \left( \int _{\Omega _{R_0}}|x-y|^{-8/3}dy\right) ^{3/4}\Vert w\Vert _{8,\Omega _{R_0}}^2\le c\,{\mathcal {D}}\,,\ \ |x|\le 2R_0 \end{aligned}$$

We thus conclude

$$\begin{aligned} |s(x)|\le c\,{\mathcal {D}}\,|x|^{-2}\,, \ \ x\in \Omega ^{R_0}\,. \end{aligned}$$

(3.16)

Therefore, (3.13) implies

$$\begin{aligned} w_j(x)=- \int _{\Omega ^{R_0}} w_l(y)w_i(y)D_lU_{ij}(x-y)dy +{{\mathfrak {S}}}_j(x)\,,\ \ x\in \Omega ^{R_0}\,, \end{aligned}$$

(3.17)

where

$$\begin{aligned} {\mathfrak {S}}_j:=\int _{{\mathbb {R}}^3} U_{ij}(x-y)f(y)dy+s_j(x) \end{aligned}$$

where, by (3.15) and (3.16)

$$\begin{aligned} |\mathbf{{\mathfrak {S}}}(x)|\le c\,{\mathcal {D}}\, |x|^{-2}\,,\ \ x\in \Omega ^{R_0}\,. \end{aligned}$$

(3.18)

Consider next the equation

$$\begin{aligned} \textsf{w}_j(x)=-{\mathcal {r}} \int _{\Omega ^{R_0}} \textsf{w}_l(y)w_i(y)D_lU_{ij}(x-y)dy +{{\mathfrak {S}}}_j(x)\,,\ \ x\in \Omega ^{R_0}\,, \end{aligned}$$

(3.19)

in the unknown field $\textsf{w}:\Omega ^{R_0}\mapsto {\mathbb {R}}^3$, and set

$$\begin{aligned} n_\alpha (w):=\sup _{x\in \Omega ^{R_0}}\left( |x|^{1+\alpha }|w(x)|\right) \,. \end{aligned}$$

By (3.17), w is a solution to (3.19), and, in fact, it is the only solution in the class of those $\textsf{w}$ with $n_0(\textsf{w})<\infty$, provided ${\mathcal {D}}$ is sufficiently small. To show this, from (3.19) we see that the difference $u:=w-\textsf{w}$ satisfies

$$\begin{aligned} u_j(x)=- \int _{\Omega ^{R_0}} u_l(y)w_i(y)D_lU_{ij}(x-y)dy \,,\ \ x\in \Omega ^{R_0}\,, \end{aligned}$$

which, by the properties of the tensor U, implies

$$\begin{aligned} |u(x)|\le {[\!|{w}|\!]}_1\,n_0(u)\int _{{\mathbb {R}}^3}\frac{dy}{|x-y|^2|y|^2}\,. \end{aligned}$$

(3.20)

Since (see e.g., [7, Lemma II.9.2])

$$\begin{aligned} \int _{{\mathbb {R}}^3}\frac{dy}{|x-y|^2|y|^{\gamma }}\le \frac{c}{|x|^{\gamma -1}}\,\ \ \gamma \in (1,3)\,,\ \ x\in {\mathbb {R}}^3-\{0\}\,, \end{aligned}$$

(3.21)

with $c=c(\gamma )$, by (3.20) and (3.12) we deduce

$$\begin{aligned} n_0(u)\le c\,{[\!|{w}|\!]}_1n_0(u)\le c\,{\mathcal {D}}\, n_0(u) \,, \end{aligned}$$

from which the stated uniqueness property follows for ${\mathcal {D}}$ less than an absolute constant. Thus, in order to show the lemma it is enough to prove that, for any $\alpha \in (0,1)$, under the stated assumption there exists a solution w to (3.19) with $n_\alpha (w)<\infty$. This property is, in turn, obtained by an elementary fixed point argument based on the contraction-mapping theorem. Actually, for $\delta >0$, let

$$\begin{aligned} {\mathcal {E}}_{\alpha ,\delta }(\Omega ^{R_0})=\{z:\ n_\alpha (z)\le \delta \}\,, \end{aligned}$$

and consider the map $M:z\in \mathcal E_{\alpha ,\delta }(\Omega ^{R_0})\mapsto \textsf{w}$, where $\textsf{w}$ satisfies

$$\begin{aligned} \textsf{w}_j(x)=- \int _{\Omega ^{R_0}} z_l(y)w_i(y)D_lU_{ij}(x-y)dy +{{\mathfrak {S}}}_j(x)\,,\ \ x\in \Omega ^{R_0}\,. \end{aligned}$$

(3.22)

Combining (3.12), (3.18), (3.21), (3.22) and the assumption on z, we find

$$\begin{aligned} n_\alpha (\textsf{w})\le c\,{[\!|{w}|\!]}_0\,n_\alpha (z)+n_\alpha (\mathbf{{\mathfrak S}})\le c\,{\mathcal {D}}\,\delta +n_\alpha (\mathbf {{\mathfrak {S}}})\,. \end{aligned}$$

Thus, if we take $\delta :=2n_\alpha ({{{\mathfrak {S}}}})$, from this inequality we get $n_\alpha (\textsf{w})\le \delta$, provided $c\,\mathcal D\le 1/2$. Moreover, by taking ${\mathfrak {S}}=0$ in (3.22), and proceeding in analogous fashion, we find

$$\begin{aligned} n_\alpha (\textsf{w})\le c\,{\mathcal {D}}\,n_\alpha (z)\le n_\alpha (z)/2\,, \end{aligned}$$

which proves that M is a contraction in $\mathcal E_{\alpha ,\delta }(\Omega ^{R_0})$. Consequently, (3.19) has a solution $\textsf{w}\in {\mathcal {E}}_{\alpha ,\delta }(\Omega ^{R_0})$, with $n_\alpha (\textsf{w})\le c\,{\mathcal {D}}$, and the proof of the lemma is completed. $\square$

Proof of Theorem 1

Fix $\beta \in (1/2,1)$. From the previous lemma, we know that

$$\begin{aligned} |w(x)|\le c\,{\mathcal {D}}\,{(1+|x|^{-1-\beta }})\,,\ \ \beta \in (1/2,1)\,. \end{aligned}$$

(3.23)

In view of (3.15), in order to show the result, we only have to prove that the last integral on the right-hand side of (3.13) decays like $1/|x|^2$, as $|x|\rightarrow \infty$. To this end, set $|x|=2R$ and split the above integral as follows

$$\begin{aligned} \int _{\Omega _R}w_l(y)w_i(y)D_lU_{ij}(x-y)dy+\int _{\Omega ^R}w_l(y)w_i(y)D_lU_{ij}(x-y)dy:=I_R(x)+I^R(x)\,. \end{aligned}$$

(3.24)

Since $|x-y|\ge |x|/2$ for $y\in \Omega _R$, by the properties of U and by (3.23) we immediately find

$$\begin{aligned} |I_R(x)|\le c\int _{\Omega _R}\frac{|w|^2}{|x-y|^2}\le {c\,\mathcal D}^2\,{|x|^{-2}}\,. \end{aligned}$$

(3.25)

Moreover, again from (3.23), we deduce

$$\begin{aligned} |I^R(x)|\le \frac{\mathcal D^2}{|x|^2}\int _{{\mathbb {R}}^3}\frac{dy}{|x-y|^2|y|^{2\beta }}\,, \end{aligned}$$

which, by (3.21) implies that

$$\begin{aligned} |I^R(x)|\le {c{\mathcal {D}}^2}\,{|x|^{-2}}\,. \end{aligned}$$

The theorem is then a consequence of this latter inequality and of (3.24), (3.25). $\square$

4 On the Resolution of the Perturbation Problem

In the present and next section, we shall collect a number of preparatory lemmas that we will eventually employ, in Sect. 6, to furnish a proof of Theorem 2 and its corollary. We begin with some results concerning convolution integrals. To this end, by K[g](x) we mean the convolution product

$$\begin{aligned} K[g](x):=\int \limits _{\mathbb {R}^3}K(x-y)g(y)dy\,, \end{aligned}$$

with a similar notation for functions depending on (t, x). With the letter c we shall denote a constant whose value is independent of (t, x) and v. Only a finite number of constants will enter our computations, the greatest of which is the one involved in the statement of Theorem 2.

The following result is crucial to our aims.

Lemma 2

Suppose $r,q\in (1,\infty )$ and $q_\circ \in (1,\infty ]$, $\frac{1}{r}\le \frac{1}{q_\circ }+\frac{1}{q}$. Then we get

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

(4.1)

if, and only if,

$$\begin{aligned} \begin{array}{c}\frac{1}{r}=\frac{1}{q_\circ }+\frac{1}{q}+\frac{\alpha ' +\beta +\gamma }{n}-1\,\\ \alpha '<\frac{n }{q_0'},\,\;\beta<\frac{n}{q'},\,\;\gamma<\frac{n}{r}\,\; \text{ if } q_\circ<\infty \,,\\ \alpha ' \in (0,n),\,\;\beta<\frac{n}{q'},\,\;\gamma <\frac{n}{r}\,\; \text{ if } q_\circ =\infty \,,\end{array} \end{aligned}$$

(4.2)

and

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

(4.3)

Proof

See Theorem A$^\prime$ in [15]. $\square$

Let

$$\begin{aligned} H(t,z):=(4\pi t)^{-\frac{3}{2}}\exp [-|z|^2\big /(4t)]\,,\ \ t>0\,, \end{aligned}$$

(4.4)

be the fundamental solution to the heat equation, and let $\varphi _{0}=\varphi _{0}(x)$. Then, $H[\varphi _{0}]$ is the solution to the heat equation. The following result holds.

Lemma 3

Let $\varphi _{0}\in C_0(\mathbb {R}^3)$, $\alpha :=\frac{3}{p}-1$, $p>3$. Then,

$$\begin{aligned} \begin{array}{l}||H[\varphi _{0}](t)||_{L^{p'}_\alpha }\le ||\varphi _{0}||_{L^{p'}_\alpha }\,\hbox {for all }t>0\,,\quad \\ \displaystyle ||\nabla H[\varphi _{0}](t)||_{L^r_{-\gamma }}\le ct^{-\frac{1}{2}-\frac{\varepsilon }{2}}||\varphi _{0}||_{L^{p'}_\alpha }\,,\hbox { for all }t>0\,, \end{array} \end{aligned}$$

(4.5)

where $\frac{1}{r}:=\frac{p-1-\varepsilon }{p}$, $\gamma =-\alpha (1+\varepsilon )$, $\varepsilon >0$, and c is independent of $\varphi _{0}$.

Proof

Clearly $H[\varphi ](t,x)$ is a smooth solution to the heat equation, so that estimate (4.5)$_1$ easily follows from the equation after a straightforward integration by parts. Also, estimate (4.5)$_2$ follows at once from (4.1) in Lemma 2 with $\beta :=3(1-\frac{1}{q})-\varepsilon \,,$ once we observe that $||\nabla H(t)|z|^\beta ||_q\le c\,t^{-\frac{1}{2}-\frac{\varepsilon }{2}}\,.$ $\square$

We next consider Poisson equation

$$\begin{aligned} \Delta \pi =-\nabla a\cdot (\nabla b)^\top \,,\hbox { in }\mathbb {R}^3\,, \end{aligned}$$

(4.6)

where a, b are given fields with $\nabla \cdot a=\nabla \cdot b=0$, and $``\top ''$ denotes transpose.

Lemma 4

If $a\in L^\infty \cap L^p_{\alpha }$ and $b\in L^p_\alpha$, $p>3$, $\alpha :=1-\frac{3}{p}$, then there exists a unique solution to (4.6) such that

$$\begin{aligned} ||\pi ||_{L^\frac{p}{1+\varepsilon }_{\alpha (1+\varepsilon )}}\le c||a||_\infty ^{1-\varepsilon }||a||_{L^p_\alpha }^\varepsilon ||b||_{L^p_\alpha }\,,\end{aligned}$$

(4.7)

with a constant c independent of a and of b.

Proof

We consider the solution given by $\pi (x):=-D_{x_ix_j}{\mathscr {E}}[a_ib_j](x)$, where $\mathscr {E}(z)=(4\pi )^{-1}|z|^{-1}$ is the fundamental solution to Laplace equation. Applying the classical Calderon–Zygmund theorem on singular integrals with the weight $|x|^{\alpha }$ [18], we show

$$\begin{aligned} ||\pi ||_{L^\frac{p}{1+\varepsilon }_{\alpha (1+\varepsilon )}}\le c\Big [\int \limits _{\mathbb {R}^3} |a|^{\frac{p}{1+\varepsilon }}|b|^{\frac{p}{1+\varepsilon }}|x|^{\alpha p}dx\Big ]^\frac{1+\varepsilon }{p}\,. \end{aligned}$$

Applying Hölder inequality, we get

$$\begin{aligned} ||\big [|a||b|\big ]^\frac{1}{1+\varepsilon }||_{L^ p_\alpha }\le ||b||_{L^p_\alpha }\Big [\int \limits _{\mathbb {R}^3}|a|^\frac{p}{\varepsilon }|x|^{\alpha p}dx\Big ]^\frac{\varepsilon }{p}\,, \end{aligned}$$

which leads to (4.7) provided that we pull out of the integral $L^\infty$-norm of $|a(x)|^{p\frac{1-\varepsilon }{\varepsilon }}\,.$

$\square$

Following [17] we show the following.

Lemma 5

Let $\xi _0>0$ and $c>0$. Let $\gamma \in (0,1)$ and assume $(1-\gamma )^2-4c\xi _0>0$. Let $\{\xi _m\}$ be a non-negative sequence of real numbers such that

$$\begin{aligned} \xi _m\le \xi _0+c\xi _{m-1}^2+\gamma \xi _{m-1}\,. \end{aligned}$$

(4.8)

Indicated by $\xi$ the minimum solution of the algebraic equation

$$\begin{aligned} c\xi ^2-(1-\gamma )\xi +\xi _0=0\,, \end{aligned}$$

then $\xi _{m}\le \xi$, for all $m\in \mathbb {N}$.

Proof

Since $\xi :=\frac{2\xi _0}{(1-\gamma )+\big [(1-\gamma )^2-4c\xi _0\big ]^\frac{1}{2}}>\xi _0$ , via (4.8) for $m=1$, we get

$$\begin{aligned} \xi _1< \xi _0+c\xi ^2+\gamma \xi =\xi _0+c\xi ^2-(1-\gamma )\xi +\xi =\xi \,. \end{aligned}$$

By the induction procedure we arrive at the thesis.$\square$

5 Integral Equation Formulation and Preliminary Results

The first objective of this section is to rewrite the perturbation problem (2.4) as an integral equation in the space $L^p_\alpha$. Successively, we will prove some preparatory results on the involved convolution integrals.

We begin to recall that the Oseen tensor, E(s, z), fundamental solution to the Stokes system, is defined by

$$\begin{aligned} \begin{array}{l}E_{ij}(s,z):=-H(s,z)\delta _{ij}+D_{z_iz_j}\phi (s,z)\,,\quad \\ \displaystyle \phi (s,z):={\mathscr {E}}(z)s^{-\frac{3}{2}}\int \limits _{0}^{|z|} \exp [-{a^2}/{4s^\frac{3}{2}}]da\,, \end{array} \end{aligned}$$

where $H(t,z):=(4\pi t)^{-\frac{3}{2}}\exp [-{|z|^2}/{4t}]$ is the fundamental solution to the heat equation, and ${\mathscr {E}}(z)$ fundamental solution to Laplace equation. The following estimates are well known (cf. [12], or [16]):

$$\begin{aligned} |D_s^k D^h_z E(s,z)|\le c(|z|+s^\frac{1}{2})^{-3-h-k},\hbox { for all }s>0 \hbox { and }z\in \mathbb {R}^3\,, \end{aligned}$$

(5.1)

$$\begin{aligned} \begin{array}{l}\hbox {for all }\theta \in (0,1),\hbox { uniformly in }(s,z)\quad \\ \displaystyle \big |D^h_zE(s,z)-D^h_z E(s,\overline{z})|\le c{|z-\overline{z}|^\theta }\big [(|z|+s^\frac{1}{2})^{-(3+h+1)\theta }+(|\overline{z}|+\!s^\frac{1}{2})^{-(3+h+1)\theta }\big ]\quad \\ \displaystyle \times \big [(|z|+s^\frac{1}{2})^{-(3+h)(1-\theta )}+(|\overline{z}|+\!s^\frac{1}{2})^{-(3+h)(1-\theta )}\big ]\,,\quad \\ \displaystyle \big |D^k_sE(s,z)-D^k_sE(\overline{s},z)\big |\le c{|s-\overline{s} |^\frac{\theta }{2}}\big [(|z|+s^\frac{1}{2})^{-(3+h+1)\theta }+(|z|+\overline{s}^\frac{1}{2})^{-(3+{k}+1)\theta }\big ]\quad \\ \displaystyle \times \big [(|z|+s^\frac{1}{2})^{-(3+k)(1-\theta )}+(|z|+\!\overline{s}^\frac{1}{2})^{-(3+h)(1-\theta )}\big ]\,, \end{array} \end{aligned}$$

(5.2)

where $D^h_z$ is the symbol of the partial derivatives with respect to $z_i$-variable $\alpha _i$ times, $i=1,2,3$, and $h=\alpha _1+\alpha _2+\alpha _3$.

A solution to problem (2.4) can be searched as a solution to the following nonlinear integral equation

$$\begin{aligned} v(t,x)=H[v_0](t,x)+E[v\cdot \nabla v+w\cdot \nabla v+v\cdot \nabla w](t,x)\,, \end{aligned}$$

(5.3)

where $E[v\cdot \nabla v]$ denotes the convolution of the fundamental Oseen tensor with the convective term on the cylinder $(0,T)\times \mathbb {R}^3$ and, we recall, H(t, z) is given in (4.4). Setting, for $i=1,2,3$,

$$\begin{aligned} \nabla _xE_i[a\otimes b](t,x):=\int \limits _{0}^{t}\int \limits _{\mathbb {R}^3} D_{x_j}E_{ih}(t-\tau ,x-y) (a_hb_j)(y,\tau ) dyd\tau \,, \end{aligned}$$

we deduce, after an integration by parts over ${\mathbb {R}}^3$ of the nonlinear term, that (5.3) can be rewritten as follows

$$\begin{aligned} v(t,x)=H[v_0](t,x)-\nabla _xE[v\otimes v+v\otimes w+w\otimes v](t,x)\,,\,\hbox { for all }(t,x)\in (0,T)\times \mathbb {R}^3\,. \end{aligned}$$

(5.4)

Lemma 6

Let $a\in L_{\alpha }^p$. Then for the convolution product H[a] the following estimate holds

$$\begin{aligned} t^\frac{1}{2}||H[a](t)||_\infty \le c||a||_{L^p_\alpha }\,,\hbox { for all }t>0\,,\end{aligned}$$

(5.5)

Proof

Applying Hölder inequality, we get

$$\begin{aligned} |H[a](t,x)|\le ct^{-\frac{3}{2p}}\Big [\int \limits _{\mathbb {R}^3}H(t,x-y)|y|^{-\frac{p-3}{p-1}}dy\Big ]^\frac{1}{p'}||a||_{L^p_\alpha }=:ct^{-\frac{3}{2p}}\mathcal G^\frac{1}{p'}(t,x)||a||_{L^p_\alpha }\,, \end{aligned}$$

(5.6)

for all $(t,x)\!\in \! (0,T)\times \mathbb {R}^3.$ The function ${\mathcal {G}}$ is the solution to the heat equation corresponding to the initial datum $|y|^{-\frac{p-3}{p-1}}\in L^{3\frac{p-1}{p-3}}_w(\mathbb {R}^3)$, where $L^s_w$ denotes the weak $L^s$ space. By the properties of the solution to the heat equation in Lorentz spaces (e.g., [13, 19]) we thus deduce that

$$\begin{aligned} ||{\mathcal {G}}(t)||_\infty \le ct^{-\frac{p-3}{2(p-1)}},\hbox { for all }t>0, \end{aligned}$$

which, combined with (5.6), leads to (5.5). $\square$

Lemma 7

Let $a\in L^p_\alpha (\mathbb {R}^3)$. Then,

$$\begin{aligned} ||H[a](t)||_{L^p_\alpha }\le ||a||_{L^p_\alpha }\,,\hbox { for all }t>0\,, \end{aligned}$$

(5.7)

with

$$\begin{aligned} \lim _{t\rightarrow 0}||H[a](t)-a||_{L^p_\alpha }=0\,. \end{aligned}$$

(5.8)

Proof

For $0\le \alpha p' <1$, $p'\le 2$ conjugate exponent of p, estimate (5.7) is an immediate consequence of Lemma 3. By duality one deduce the same for $p\ge 2$. The property (5.8) follows from the weak convergence in $t=0$ of H[a](t) to a, and the estimate (5.7). $\square$

Lemma 8

Let $\sup _{(0,T)}\!\!\big [t^\frac{1}{2}||a(t)||_\infty +t^\frac{1}{2}||b(t)||_\infty \big ]\!<\infty$ and $\sup _{(0,T)}\!\!\big [||a(t)||_{L^p_\alpha }+||b(t)||_{L^p_\alpha }\big ]\!<\infty$. Then there exists a constant c independent of a and of b such that

$$\begin{aligned} t^\frac{1}{2}||\nabla E[a\otimes b](t)||_\infty \!\le \displaystyle c\Big [\sup _{(0,t)}\tau |||a(\tau )|| b(\tau )|||_\infty +\sup _{(0,t)}\tau ^\frac{1}{2}||a(\tau )||_\infty ||b(\tau )||_{L^p_\alpha }\Big ]\,,\hbox { for all }t>0\,. \end{aligned}$$

(5.9)

Proof

From (5.1) we get

$$\begin{aligned} \begin{array}{ll} \displaystyle |\nabla E[a\otimes b](t,x)|&{}\displaystyle \le \int \limits _{\frac{t}{2}}^{t}\int \limits _{\mathbb {R}^3}\frac{ |a(\tau ,y)||b(\tau ,y)|}{(|x-y|^2+t-\tau )^{2}}dyd\tau \displaystyle +\int \limits _{0}^{\frac{t}{2}}\int \limits _{\mathbb {R}^3}\frac{|a(\tau ,y)||b(\tau ,y)|}{(|x-y|^2+t-\tau )^2}dyd\tau \quad \\ {} &{}\displaystyle =:I_1(t)+I_2(t)\,.\end{array} \end{aligned}$$

By our hypotheses we get

$$\begin{aligned} I_1(t)\le c\int \limits _{\frac{t}{2}}^{t}\frac{1}{\tau }\sup \tau |||a(\tau )||b(\tau )|||_\infty \int \limits _{\mathbb {R}^3}(|z|^2+t-\tau )^{-2}dzd\tau \le ct^{-\frac{1}{2}}\sup _{(\frac{t}{2},t)}\tau |||a(\tau )||b(\tau )|||_\infty \,. \end{aligned}$$

Applying Hölder inequality, we get

$$\begin{aligned} I_2(t)\displaystyle \le c\int \limits _{0}^{\frac{t}{2}}||a(\tau )||_\infty \Big [\int \limits _{\mathbb {R}^3}(|x-y|^2+(t-\tau ))^{-2p'}|y|^{-\frac{p-3}{p-1}}dy\Big ]^\frac{1}{p'}||b(\tau )||_{L^p_\alpha }d\tau =:I_{21}+I_{22}. \end{aligned}$$

The following hold:

$$\begin{aligned} \begin{array}{ll}I_{21}&{}\displaystyle \le c\sup _{(0,t)}\tau ^\frac{1}{2}||a(\tau )||_\infty ||b(\tau )||_{L^p_\alpha }\int \limits _{0}^{\frac{t}{2}}\tau ^{-\frac{1}{2}}(t-\tau )^{-2}\Big [\int \limits _{|y|<(t-\tau )^\frac{1}{2}}|y|^{-\frac{p-3}{p-1}}dy\Big ]^\frac{1}{p'}\\ {} &{}\displaystyle \le ct^{-\frac{1}{2}}\sup _{(0,t)}\tau ^\frac{1}{2}||a(\tau )||_\infty ||b(\tau )||_{L^p_\alpha }\,;\end{array} \\ \begin{array}{ll}I_{22}&{}\displaystyle \le c\sup _{(0,t)}\tau ^\frac{1}{2}||a(\tau )||_\infty ||b(\tau )||_{L^p_\alpha }\int \limits _{0}^{\frac{t}{2}}\tau ^{-\frac{1}{2}}(t-\tau )^{-\frac{p-3}{2p}}\Big [\int \limits _{|y|>(t-\tau )^\frac{1}{2}}(|x-y|^2+(t-\tau ))^{-2p'}dy\Big ]^\frac{1}{p'}\\ {} &{}\displaystyle \le ct^{-\frac{1}{2}}\sup _{(0,t)}\tau ^\frac{1}{2}||a(\tau )||_\infty ||b(\tau )||_{L^p_\alpha }\,.\end{array} \end{aligned}$$

These jointly with the one for $I_1$ complete the estimate for the right-hand side of (5.9). $\square$

Lemma 9

Let $\sup _{(0,T)}t^\frac{1}{2}||b(t)||_\infty \!<\infty$ and $\sup _{(0,T)}||b(t)||_{L^p_\alpha }<\infty$. Assume also $a=a(x)$ with ${[\!|{a}|\!]}_2<\infty$. Then, there exists a constant c independent of a and of b such that

$$\begin{aligned} t^\frac{1}{2}||\nabla E[a\otimes b](t)||_\infty \!\le \displaystyle c{[\!|{a}|\!]}_2\sup _{(0,t)}\tau ^\frac{1}{2}||b(\tau )||_\infty \text{ for } \text{ all } t>0\,. \end{aligned}$$

(5.10)

Proof

With the help of (5.1) we get

$$\begin{aligned} \begin{array}{ll} \displaystyle |\nabla E[a\otimes b](t,x)|&{}\displaystyle \le \int \limits _{\frac{t}{2}}^{t}\int \limits _{\mathbb {R}^3}\frac{ |a(y)||b(\tau ,y)|}{(|x-y|^2+t-\tau )^{2}}dyd\tau \displaystyle +\int \limits _{0}^{\frac{t}{2}}\int \limits _{\mathbb {R}^3}\frac{|a(y)||b(\tau ,y)|}{(|x-y|^2+t-\tau )^2}dyd\tau \quad \\ {} &{}\displaystyle =:I_1(t)+I_2(t)\end{array}\,, \end{aligned}$$

which, in turn, by Hölder inequality and (3.21), furnishes

$$\begin{aligned} \begin{array}{ll}I_1(\tau )&{}\displaystyle \le c\sup _{(0,t)}\tau ^\frac{1}{2}||b(\tau )||_\infty \!\int \limits _{\frac{t}{2}}^{t}\!\tau ^{-\frac{1}{2}}\!\int \limits _{\mathbb {R}^3} \frac{|a(y)|}{(|x-y|^2+t-\tau )^2}dyd\tau \le ct^{-\frac{1}{2}} \sup _{(0,t)}\tau ^\frac{1}{2}||b(\tau )||_\infty \int \limits _{\mathbb {R}^3}\frac{|a(y)|}{|x-y|^2}dy\quad \\ {} &{}\displaystyle \le ct^{-\frac{1}{2}} \sup _{(0,t)}\tau ^\frac{1}{2}||b(\tau )||_\infty {[\!|{a}|\!]}_2\,;\end{array} \\ \begin{array}{ll}I_2(\tau )&{}\displaystyle \le c\sup _{(0,t)} \tau ^\frac{1}{2}||b(\tau )||_\infty \!\int \limits _{0}^{\frac{t}{2}}\!\tau ^{-\frac{1}{2}}\!\!\int \limits _{\mathbb {R}^3}\! \frac{|a(y)|}{(|x-y|^2\!+t-\tau )^2}dyd\tau \le ct^{-1}\sup _{(0,t)} \tau ^\frac{1}{2}||b(\tau )||_\infty \!\int \limits _{0}^{\frac{t}{2}}\!\!\tau ^{-\frac{1}{2}}\!\!\int \limits _{\mathbb {R}^3}\!\!\frac{|a(y)|}{|x-y|^2}dy \quad \\ {} &{}\displaystyle \le ct^{-\frac{1}{2}}\sup _{(0,t)}\tau ^\frac{1}{2}||b(\tau )||_\infty {[\!|{a}|\!]}_2\,.\end{array} \end{aligned}$$

Summing the estimates of $I_1$ and $I_2$, we arrive at (5.10). $\square$

Lemma 10

Let $\sup _{(0,T)}\big [t^\frac{1}{2}||a(t)||_\infty +||a(t)||_{L^p_\alpha }+||b(t)||_{L^p_\alpha }\big ]<\infty$. Then there exists a constant c independent of a(t, x) and b(t, x) such that

$$\begin{aligned} ||\nabla E[a\otimes b](t)||_{L^p_\alpha }\le c\big [\sup _{(0,t)}\tau ^\frac{1}{2}||a(\tau )||_\infty \big ] ^{1-\varepsilon }\sup _{(0,t)}||a(\tau ) ||_{L^p_\alpha }^\varepsilon ||b(\tau )||_{L^p_\alpha }, \hbox { for all }t\in (0,T)\,. \end{aligned}$$

(5.11)

Proof

By virtue of estimates (5.1), for arbitrary $\varepsilon >0$, we get

$$\begin{aligned} \begin{array}{l} \displaystyle |\nabla E[a\otimes b](t,x)|\le c \Big [\sup _{(0,t)}\tau ^\frac{1}{2}||a(\tau )||_\infty \Big ]^{1-\varepsilon }\int \limits _{0}^{t}\tau ^{-\frac{1-\varepsilon }{2}}\int \limits _{\mathbb {R}^3}\frac{|a(\tau ,y)|^\varepsilon |b(\tau ,y)|}{(|x-y|^2+t-\tau )^2}dyd\tau \\ \displaystyle =c \Big [\sup _{(0,t)}\tau ^\frac{1}{2}||a(\tau )||_\infty \Big ]^{1-\varepsilon }\int \limits _{0}^{t}\tau ^{-\frac{1-\varepsilon }{2}}I(\tau ,x)d\tau \,, \end{array} \end{aligned}$$

(5.12)

where we set

$$\begin{aligned} I(t,\tau ,x):= \int \limits _{\mathbb {R}^3}\frac{|a(\tau ,y)|^\varepsilon |b(\tau ,y)|}{(|x-y|^2+t-\tau )^2}dy\,. \end{aligned}$$

Applying to this last term Lemma (2) with

$$\begin{aligned} r:=p,\, \gamma :=-(1-\frac{3}{p}),\;\;q_0:=\frac{p}{{1+\varepsilon }},\,\alpha :=(1-\frac{3}{p})(1+\varepsilon )\,,\;\; q\ge \frac{3p}{3+2p-\varepsilon p}, \,\beta :=3(1-\frac{1}{q})-\varepsilon \,, \end{aligned}$$

we get

$$\begin{aligned} ||I(t,\tau )||_{L^p_\alpha }\le c||(|z|+t-\tau )^{-4}|z|^\beta ||_q|||a(\tau )|^\varepsilon |b(\tau )||y|^{(1-\frac{3}{p})(1+\varepsilon )}||_{\frac{p}{1+\varepsilon }}=:J_1(t-\tau )J_2(\tau )\,. \end{aligned}$$

(5.13)

We readily show

$$\begin{aligned} \displaystyle J_1^q(t-\tau )\displaystyle \le c\int \limits _{\mathbb {R}^3}\frac{ |z|^{\beta q}}{(|z|+\sqrt{t-\tau })^{4q}} dz \le c(t-\tau )^{-\frac{q}{2}(1+\varepsilon )}\,, \end{aligned}$$

whereas, by Hölder inequality,

$$\begin{aligned} J_2(\tau )\le ||a(\tau )||_{L^p_\alpha }^\varepsilon ||b(\tau )||_{L^p_\alpha }\,. \end{aligned}$$

Collecting the last two estimates, we arrive at

$$\begin{aligned} ||I(\tau )||_{L^p_\alpha }\le c(t-\tau )^{-\frac{1}{2}(1+\varepsilon )}||a(\tau )||_{L^p_\alpha }^\varepsilon ||b(\tau )||_{L^p_\alpha }\,. \end{aligned}$$

Therefore from (5.12) and the generalized Minkowski inequality, we infer

$$\begin{aligned} \begin{array}{ll}||\nabla E[a\otimes b](t)||_{L^p_\alpha }&{}{}\displaystyle \le c\sup _{(0,t)}\big [\tau ^\frac{1}{2}||a(\tau )||_\infty \big ] ^{1-\varepsilon }\int \limits _{0}^{t}\tau ^{-\frac{1-\varepsilon }{2}}||I(t,\tau )||_{L^p_\alpha }d\tau \\ {} &{}{}\displaystyle \le c\sup _{(0,t)}\big [\tau ^\frac{1}{2}||a(\tau )||_\infty \big ] ^{1-\varepsilon } \sup _{(0,t)}||a(\tau ) ||_{L^p_\alpha }^\varepsilon ||b(\tau )||_{L^p_\alpha }\,,\end{array} \end{aligned}$$

which furnishes (5.11). $\square$

In the proof of Theorem 2, we will employ the following lemma with $a\equiv w(x)$. However, for the sake of generality, we will show it under the more general assumption $a=a(t,x)$.

Lemma 11

Let $\sup _{(0,T)}\big [t^\frac{1}{2}||b(t)||_\infty +||b(t)||_{L^p_\alpha }\big ]<\infty$. Assume also $\sup _{(0,T)}{[\!|{a(t)}|\!]}_1<\infty$, and $a\in L^\infty (0,T;L^3))$. Then there exists a constant c independent of a(t, x) and b(t, x) such that

$$\begin{aligned} \begin{array}{l}\displaystyle ||\nabla E[a\otimes b](t)||_{L^p_\alpha }\le c\sup _{(0,T)}{[\!|{a(t)}|\!]}_1^{1-\frac{3}{p}}\big [\sup _{(0,t)}\tau ^\frac{1}{2}||b(\tau )||_\infty \big ]^{1-\varepsilon }\sup _{(0,t)}||b(\tau ) ||_{L^p_\alpha }^\varepsilon ||a(\tau )||_{3}^\frac{3}{p},\quad \\ \displaystyle \hbox { for all }t\in (0,T)\,.\end{array} \end{aligned}$$

(5.14)

Proof

We shall use the arguments employed in the previous lemma. In fact, we may show (5.12) with $I(t,\tau ,x)$ substituted by

$$\begin{aligned} {\widetilde{I}}(t,\tau ,x):= \int \limits _{\mathbb {R}^3}\frac{|a(\tau ,y)||b(\tau ,y)|^\varepsilon }{(|x-y|^2+t-\tau )^2}dy\,. \end{aligned}$$

Making the same estimate employed for (5.13), we obtain

$$\begin{aligned} ||{\widetilde{I}}(t,\tau )||_{L^p_\alpha }\le c ||(|z|+t-\tau )^{-4}|z|^\beta ||_q |||a(\tau )||b(\tau )|^\varepsilon |y|^{(1-\frac{3}{p})(1+\varepsilon )}||_{\frac{p}{1+\varepsilon }}=:{\widetilde{J}}_1(t-\tau ){\widetilde{J}}_2(\tau )\,. \end{aligned}$$

Being $J_1\equiv {\widetilde{J}}_1$, we give the estimate

$$\begin{aligned} {\widetilde{J}}_1^q(t-\tau )\le c(t-\tau )^{-\frac{q}{2}(1+\varepsilon )}\,. \end{aligned}$$

As for ${\widetilde{J}}_2$, by employing the assumptions on a, we get

$$\begin{aligned} {\widetilde{J}}_2(\tau )\le ||a(\tau )||_{L^p_\alpha }||b(\tau )||^\varepsilon _{L^p_\alpha }\le \sup _{(0,T)}{[\!|{a(t)}|\!]}_1^{1-\frac{3}{p}}||a(\tau )||_3^\frac{3}{p}||b(\tau )||^\varepsilon _{L^p_\alpha }\,, \end{aligned}$$

so that, we deduce

$$\begin{aligned} \begin{array}{ll}||\nabla E[a\otimes b](t)||_{L^p_\alpha }&{}\displaystyle \le c\sup _{(0,t)}\tau ^\frac{1}{2}||b(\tau )||_\infty ^{1-\varepsilon }\int \limits _{0}^{t}\tau ^{-\frac{1-\varepsilon }{2}}||{\widetilde{I}}(t,\tau )||_{L^p_\alpha }d\tau \\ {} &{}\displaystyle \le c\sup _{(0,T)}{[\!|{a(t)}|\!]}_1^{1-\frac{3}{p}} \sup _{(0,t)}\tau ^\frac{1}{2}||b(\tau )||_\infty ^{1-\varepsilon }\sup _{(0,t)}||b(\tau ) ||_{L^p_\alpha }^\varepsilon ||a(\tau )||_{3}^\frac{3}{p}\,, \end{array} \end{aligned}$$

which furnishes (5.14).$\square$

6 Proof of Theorem 2 and Corollary 1

The proof of the theorem will be achieved by the following approximation scheme with $m\ge 1$:

$$\begin{aligned} v^m(t,x)=H[v_0](t,x)-\nabla _x E\big [(v^{m-1}\otimes v^{m-1}+w\otimes v^{m-1}+v^{m-1}\otimes w\big ](t,x)\,, \end{aligned}$$

(6.1)

whose investigation requires the proof of a number of preliminary lemmas. Set

$$\begin{aligned} |||u(t)|||:=\sup _{(0,t)}\tau ^\frac{1}{2}||u(\tau )||_\infty +\sup _{(0,t)}||u(\tau )||_{L^p_\alpha }\,. \end{aligned}$$

(6.2)

Lemma 12

Let $v_0\in L^p_\alpha$ with $\textrm{div}\,v_0=0$, and let $v^0(t,x):=H[v_0](t,x)$. Then, there exists a constant c, independent of $v_0$ and $m\in \mathbb {N}$, such that for the sequence defined by (6.1) the following estimate holds

$$\begin{aligned} \begin{array}{l} |||v^m(t)|||\le (1+c)||v_0||_{L^p_\alpha }+c|||v^{m-1}(t)|||^2+c\Big [ \big [w^{(0)}\big ]^{1-\frac{3}{p}}\big [w^{(1)}\big ]^\frac{3}{p}+w^{(2)}\Big ]|||v^{m-1}(t)|||\,, \end{array} \end{aligned}$$

(6.3)

for all $t>0\,.$

Proof

From (6.1) with $m=1$, and Lemmas 8–9, and Lemmas 10–11, for all $t>0$, we deduce

$$\begin{aligned} \begin{array}{lll} s^\frac{1}{2}&{}||v^1(s)||_\infty &{}\displaystyle \le s^\frac{1}{2}||v^0(s)||_\infty +c\Big [\sup _{(0,s)}\tau ^\frac{1}{2}||v^0(\tau )||_\infty \!+\sup _{(0,s)} ||v^0(\tau )||_{L^p_\alpha }\!\Big ]^2\!+w^{(2)}\!\sup _{(0,s)}\tau ^\frac{1}{2}||v^0(\tau )||_\infty \,, \quad \\ \displaystyle &{}||v^1(s)||_{L^p_\alpha }&{}\displaystyle \le ||v^0(s)||_{L^p_\alpha }+c\big [\sup _{(0,s)}\tau ^\frac{1}{2}||v^0(\tau )||_\infty +\sup _{(0,s)}||v^0||_{L^p_\alpha }\big ] ^2\quad \\ \displaystyle &{}&{}+c\big [w^{(0)}\big ]^{1-\frac{3}{p}}\big [w^{(1)}\big ]^\frac{3}{p}\big [\sup _{(0,s)}\tau ^\frac{1}{2}||v^0(\tau )||_\infty +\sup _{(0,s)}||v^0(\tau ) ||_{L^p_\alpha }\big ], \end{array} \end{aligned}$$

(6.4)

where c is a constant independent of t. Owing to Lemmas 6–7, we infer $|||v^0(t)|||\le (1+c)||v_0||_{L^p_\alpha }$, and so, summing side-by-side the two inequalities in (6.4)$_3$ and taking the sup on $s\in (0,t)$, we show

$$\begin{aligned} |||v^1(t)|||\le (1+c)||v_0||_{L^p_\alpha }+ c|||v^0(t)|||^2+c \Big [ \big [w^{(0)}\big ]^{1-\frac{3}{p}}\big [w^{(1)}\big ]^\frac{3}{p}+w^{(2)}\Big ]|||v^0(t)|||\,, \end{aligned}$$

with a constant c independent of $v_0$ and t. As a result, for $m=1$, (6.1) is well defined and estimate (6.3) is true. Then, by induction one proves the desired estimate for all $m\in \mathbb {N}$. $\square$

Lemma 13

Let $\{v^m\}$ be the sequence defined in (6.1) corresponding to $v_0\in L^p_\alpha$. If (2.6) holds with the constant c given in (6.3), then there exists a function v in the class defined by (6.2) such that

$$\begin{aligned} \lim _m|||v^m(t)-v(t)|||=0,\hbox { for all }t\in (0,\infty )\,, \end{aligned}$$

and v is a solution to (5.4) on $(0,\infty )$.

Proof Under the assumption (2.6), by Lemma 5 we get, uniformly in $m\in \mathbb {N}$,

$$\begin{aligned} |||v^m(t)|||\le \!\frac{2(1+c)||v_0||_{L^p_\alpha }}{1+\big ((1-\gamma )^2-4(1+c)||v_0||_{L^p_\alpha }\big ]\big )^\frac{1}{2}}=:A\,,\hbox { for all }t>0\,, \end{aligned}$$

(6.5)

where we set $\gamma :=c\gamma _0$ with $\gamma _0:=[w^{(0)}]^{1-\frac{3}{p}}[w^{(1)}]^\frac{3}{p}+w^{(2)}$. Hence, estimate (6.5) ensures that, for all $t>0$ the sequence $\{|||v^m|||_{(t,\rho )}\}$ is bounded. Set $W^{m}:=v^{m}-v^{m-1}$, $m\ge 0$ and $v^{-1}\equiv 0$. From (6.1) it follows that

$$\begin{aligned} W^{m+1}(t,x)=-\nabla _x E[W^m\otimes v^{m}](t,x)-\nabla _x E[v^{m-1}\otimes W^{m}]+E[w\otimes W^m+W^m\otimes w](t,x)\,. \end{aligned}$$

Also, from the assumption( 2.6)$_1$ and (6.5), we deduce $\gamma <1$ and $A<\frac{1}{2}$. Thus, setting $a:=\max \{\gamma ,\frac{1}{4}\}$, and employing the results of Lemmas 8–9, and Lemmas 10–11, in conjunction with (6.5), we easily prove the sequence of estimates

$$\begin{aligned} |||W^1|||_{(t,\rho )} \le 4ca^2, \dots , |||W^m|||_{(t,\rho )}\le a(4ca)^m,\dots \,. \end{aligned}$$

(6.6)

Consequently, in view of (2.6)$_2$, we get the convergence of $\{v^m\}$ to a function v in the class (6.2), which ensures, in particular, that the limit function v satisfies the integral equation (5.4). $\square$

Proof of Thorem 2

In view of the previous lemma, in order to conclude the proof of the theorem, it remains to show the existence of a pressure field satisfying the stated properties, the way in which v assumes the initial data and, finally, the uniqueness of the obtained solution. We begin to associate to the solution v determined in the previous lemma, a pressure field defined by

$$\begin{aligned} \pi (t,x):=-D_{x_i}\int \limits _{\mathbb {R}^3}D_{x_j}\mathscr {E}(x-y)v_i(t,y)\big [v_j(t,y)+2w_j(x)\big ]dy\,. \end{aligned}$$

(6.7)

By the theory of singular integrals with weight [18], one shows

$$\begin{aligned} ||\pi (t)||_{L^p_\alpha }\le c|||v|^2+2|v||w|||_{L^p_\alpha }\,,\hbox { for all }t>0\,, \end{aligned}$$

which implies

$$\begin{aligned} ||\pi (t)||_{L^p_\alpha }\le c\Vert v(t)\Vert _\infty \left( \Vert v(t)\Vert _{L^p_\alpha }+2\Vert w\Vert _{L^p_\alpha }\right) \,,\hbox { for all }t>0\,. \end{aligned}$$

(6.8)

Since

$$\begin{aligned} ||w||_{L^p_\alpha }\le c\,{[\!|{w}|\!]}_2 \end{aligned}$$

(6.9)

the inequality (2.9) for the pressure follows from (2.8), ((6.8)) and ((6.9)). We also observe that, by virtue of classical arguments, the uniform convergence of the sequence of continuous functions $\{v^m\}$ ensures that the limit function v is continuous in $(t,x)\in (\eta ,T)\times \mathbb {R}^3$, for all $\eta >0$, and also differentiable together with the associated pressure field $\pi$ defined by (6.7). The existence proof of the pressure is thus completed.

We next show the property (2.7). This will be done by using the representation (5.3). In view of (5.8), we only have to show that the integral involving the Oseen term tends weakly to zero in $t=0$. Multiplying by $\varphi \in {\mathscr {C}}_0$ this integral and integrating over ${\mathbb {R}}^3$, we get (with obvious meaning of the symbols)

$$\begin{aligned} \begin{array}{ll}\displaystyle (\varphi , \nabla E[v\otimes (v+w)+w\otimes v])&{}{}=\displaystyle \int \limits _{0}^{t}\big ( v(\tau )\otimes (v(\tau )+w)+w\otimes v(\tau )\big ), \nabla {\widetilde{E}}[ \varphi ](t-\tau ))d\tau \\ {} &{}{}\displaystyle =:J_1(t,v,v)+J_2(t,v,w) \end{array}\,, \end{aligned}$$

where ${\widetilde{E}}[\varphi ]:=\int \limits _{\mathbb {R}^3} E(t-\tau ,x-y)\cdot \varphi (x)dx$. Employing the regularity of $\varphi$ and Lemma 2 with exponents

$$\begin{aligned} \frac{1}{r}=\frac{p-2}{p},\,q\in (1,\frac{3}{2}),\, \frac{1}{q}+\frac{1}{q_\circ }=\frac{4}{3},\, \gamma =2(1-\frac{3}{p}),\,\alpha =\beta =0\,, \end{aligned}$$

we get the following estimate:

$$\begin{aligned} ||\nabla {\widetilde{E}}[\varphi ](t-\tau )||_{L^{\frac{p}{p-2}}_{-2\alpha }}\le c(t-\tau )^{-2+\frac{3}{2q}}||\varphi ||_{q_\circ }\,. \end{aligned}$$

Hence, applying Hölder inequality, we get

$$\begin{aligned} |J_1(t)|\le \int \limits _{0}^{t} |||v(\tau )|^2||_{L^\frac{p}{2}_{2(1-\frac{3}{p})}} ||\nabla {\widetilde{E}}[\varphi ](t-\tau )||_{L^\frac{p}{p-2}_{-2(1-\frac{3}{p})}}\le c\sup _{(0,T)}||v(\tau )||_{L^p_{1-\frac{3}{p}}}^2||\varphi ||_{q_\circ }t^\frac{1}{2}\,. \end{aligned}$$

Analogously, from estimate (6.9) and Hölder inequality, we get

$$\begin{aligned} |J_2(t)|\le \int \limits _{0}^{t} ||v(\tau )||_{L^ p_{1-\frac{3}{p}}}||w||_{L^ p_{1-\frac{3}{p}}}||\nabla {\widetilde{E}}[\varphi ](t-\tau )||_{L^\frac{p}{p-2}_{-2(1-\frac{3}{p})}}\le c\sup _{(0,T)}||v(\tau )||_{L^p_{1-\frac{3}{p}}}{[\!|{w}|\!]}_2||\varphi ||_{q_\circ }t^\frac{1}{2}\,. \end{aligned}$$

Hence, $\lim _{t\rightarrow 0}[J_1(t,v,v)+J_2(t,v,w)]=0$, and the desired limit property follows.

In order to prove uniqueness, we employ a duality argument introduced in [6] for the Navier–Stokes equations. Actually, we are going to prove that the following homogeneous problem:

$$\begin{aligned} \begin{array}{l} \displaystyle (u(t),\varphi (t))=\int \limits _{0}^{t}\big [(\overline{v}\cdot \nabla \varphi ,u)+(u\cdot \nabla \varphi , v)+(u\otimes w,\nabla \varphi +(\nabla \varphi )^T)\\ \displaystyle +(\pi _u,\nabla \cdot \varphi )+(\varphi _{\tau }+\Delta \varphi , u)\big ]d\tau \,, \end{array} \end{aligned}$$

(6.10)

with $t^\frac{1}{2}{\bar{v}},t^\frac{1}{2} v\in L^\infty (0,T;L^\infty (\mathbb R^3 ))$, $\overline{v}, v\in L^\infty (0,T;L^p_{-\alpha })$, $u:=\bar{v}-v$, and $\varphi \in C^2((0,T)\times \mathbb {R}^3)$ admits only the null solution. For $\varphi _{0}\in C_0^2(\mathbb {R}^3)$ we consider the solution $H[\varphi _{0}](t,x)$ to the Cauchy problem of the heat equation. Writing backward in time the heat solution $\varphi :=H[\varphi _{0}](t,x)$, that is ${{\widehat{\varphi }}} (\tau ,x):=\varphi (t-\tau ,x)$ with $\tau \in (0,t)$, and substituting it into (6.10), we get

$$\begin{aligned} (u(t),\varphi _{0})=\int \limits _{0}^{t}\big [(\overline{v}\cdot \nabla {{\widehat{\varphi }}},u)+(u\cdot \nabla {{\widehat{\varphi }}}, v)+(u\otimes w,\nabla {{\widehat{\varphi }}}+(\nabla {{\widehat{\varphi }}})^T)+(\pi _u,\nabla \cdot {{\widehat{\varphi }}})\big ]d\tau =:{\underset{i=1}{\overset{4}{\sum }}}\int \limits _{0}^{t}I_i(t)\,.\qquad \end{aligned}$$

(6.11)

We estimate the quantities on the right-hand side. Applying Hölder inequality and taking into account estimate (4.5)$_2$ for $\nabla {{\widehat{\varphi }}}(t-\tau ,x)$ along with the properties of the solution ${\bar{v}}$, we obtain

$$\begin{aligned} \big |I_1(t)+I_2(t)\big |\le c ||u||_{L^p_\alpha }||\nabla {{\widehat{\varphi }}}||_{L^\frac{p}{p-1-\varepsilon }_{-\alpha (1+\varepsilon )}}\big [||\overline{v}||_{L^{\frac{p}{\varepsilon }}_{\alpha \varepsilon }}+||v||_{L^{\frac{p}{\varepsilon }}_{\alpha \varepsilon }}\big ]\le c(t-\tau )^{-\frac{1}{2}-\frac{\varepsilon }{2}}\tau ^{-\frac{1}{2}+\frac{\varepsilon }{2}}||u||_{L^p_\alpha }||v_0||_{L^p_\alpha }||\varphi _{0}||_{L^{p'}_{-\alpha }}\,. \end{aligned}$$

We next consider $I_3(t)$. Firstly, we note that

$$\begin{aligned} ||w||_{L^{\frac{p}{\varepsilon }}_{\alpha \varepsilon }}\le [w^{(0)}]^{\frac{p-3\varepsilon }{p}}[w^{(1)}]^\frac{3\varepsilon }{p}. \end{aligned}$$

Hence, applying Hölder inequality, we get

$$\begin{aligned} |I_3(t)|\le 2 ||u||_{L^p_\alpha }||\nabla {{\widehat{\varphi }}} ||_{\frac{p}{p-1-\varepsilon }}||w||_{L^\frac{p}{\varepsilon }_{\alpha \varepsilon }}\le c[w^{(0)}]^{\frac{p-3\varepsilon }{p}}[w^{(1)}]^\frac{3\varepsilon }{p} ||u||_{L^p_\alpha }(t-\tau )^{-\frac{1}{2}}||\varphi _{0}||_{L^{p'}_{-\alpha }} \end{aligned}$$

In order to estimate the term $I_4(t)$, we observe that $\pi _u$ is a solution to equation (4.6) with $a\otimes b:=\bar{v}\otimes u+u\otimes v+2u\otimes w$. Hence, employing Hölder inequality and estimate (4.7) for $\pi _u$, and following the previous arguments for the estimates of $I_i,i=1,2,3$, we deduce

$$\begin{aligned} \big |I_4(t)\big |\le c||\pi ||_{L^\frac{p}{1+\varepsilon }_{\alpha (1+\varepsilon )}}||\nabla {{\widehat{\varphi }}}||_{\frac{p}{p-1-\varepsilon }}\le c||u||_{L^p_\alpha }\big [||\overline{v}||_{L^p_\alpha }^{1-\varepsilon }||\overline{v}||_\infty ^\varepsilon +||v||_{L^p_\alpha }^{1-\varepsilon }||v||_\infty ^\varepsilon + [w^{(0)}]^{\frac{p-3\varepsilon }{p}}[w^{(1)}]^\frac{3\varepsilon }{p}\big ]||\nabla {{\widehat{\varphi }}}||_{\frac{p}{p-1-\varepsilon }}\,. \end{aligned}$$

We also get

$$\begin{aligned} |I_4(t)|\le c(t-\tau )^{-\frac{1}{2}-\frac{\varepsilon }{2}}\Big [\tau ^{-\frac{1}{2}+\frac{\varepsilon }{2}} ||u||_{L^p_\alpha }||v_0||_{L^p_\alpha }+[w^{(0)}]^{\frac{p-3\varepsilon }{p}}[w^{(1)}]^\frac{3\varepsilon }{p}\Big ]||\varphi _{0}||_{L^{p'}_{-\alpha }}\,. \end{aligned}$$

Thus, from (6.11), and the above estimates for $I_i(t),i=1,\dots ,4$, we obtain

$$\begin{aligned} \begin{array}{l}\displaystyle |(u(t),\varphi _{0})|\le c||v_0||_{L^p_\alpha }||\varphi _{0}||_{L^{p'}_{-\alpha }} \int \limits _{0}^{t} ||u(\tau )||_{L^p_\alpha }(t-\tau )^{-\frac{1}{2}-\frac{\varepsilon }{2}} \tau ^{\frac{\varepsilon }{2}-\frac{1}{2}}d\tau \\ \displaystyle +c||\varphi _{0}||_{L^{p'}_{-\alpha }}[w^{(0)}]^{\frac{p-3\varepsilon }{p}}[w^{(1)}]^\frac{3\varepsilon }{p} \int \limits _{0}^{t}(t-\tau )^{-\frac{1}{2}}||u(\tau )||_{L^p_\alpha }d\tau \,. \end{array} \end{aligned}$$

Since $\varphi _{0}\in L^{p'}_{-\alpha }$ is arbitrary, from the latter we get

$$\begin{aligned} ||u(t)||_{L^p_\alpha }\le c||v_0||_{L^p_\alpha } \int \limits _{0}^{t} ||u(\tau )||_{L^p_\alpha }(t-\tau )^{-\frac{1}{2}-\frac{\varepsilon }{2}}\tau ^{\frac{\varepsilon }{2}-\frac{1}{2}}d\tau +c[w^{(0)}]^{\frac{p-3\varepsilon }{p}}[w^{(1)}]^\frac{3\varepsilon }{p}\int \limits _{0}^{t}(t-\tau )^{-\frac{1}{2}}||u(\tau )||_{L^p_\alpha }d\tau \,, \end{aligned}$$

which, in turn, implies

$$\begin{aligned} ||u(t)||_{L^p_\alpha }\le A\sup _{(0,T)}||u(\tau )||_{L^p_\alpha },\hbox { for all }t\in (0,T)\,, \end{aligned}$$

with $A=A(t):=c(t ^\frac{1}{2}[w^{(0)}]^{\frac{p-3\varepsilon }{p}}[w^{(1)}]^\frac{3\varepsilon }{p}+||v_0||_{L^p_\alpha })\,.$ Thus, if $c\,||v_0||_{L^p_\alpha }<1$, we can find $\overline{t}\in (0,T)$ such that $A(\overline{t})<1$, and, by the previous inequality, we infer uniqueness on $(0,\overline{t}]$. For $t\ge \overline{t}$, $||v(t)||_\infty$ is not singular, so that one deduces the uniqueness property in the whole interval $\in [\overline{t},T)$, which completes the proof. $\square$

Proof of Corollary 1

We consider the integral equation (5.4) for v. With obvious meaning of the symbols, we write

$$\begin{aligned} v(t,x)={\underset{i=1}{\overset{4}{\sum }}}I_i(t,x)\,. \end{aligned}$$

(6.12)

Our goal is to estimate each term in the sum by the right-hand side of (2.10). We set $A:=||v_0||_{L^p_\alpha }$. The first term of the sum is the solution to the heat equation. Hence, applying Hölder inequality, we get

$$\begin{aligned} |I_1(t,x)|\le ||H(t,x)||_{L^{p'}_{-\alpha }} ||v_0||_{_{{\small L^p_\alpha }}}\,. \end{aligned}$$

Recalling that (as easily proved from (4.4))$|H(t,z)|\le c(|z|+t^\frac{1}{2})^{-3}$, we show the following estimate

$$\begin{aligned} \begin{array}{ll}||H(t,x)||_{L^{p'}_{-\alpha }}^{p'}&{}\le \displaystyle \int \limits _{|y|<\frac{|x|}{2}}H^{p'}(t,x-y)|y|^{-\alpha p'}dy+\int \limits _{|y|>\frac{|x|}{2}}H^{p'}(t,x-y)|y|^{-\alpha p'}dy\quad \\ {} &{}\displaystyle \le c|x|^{-3}t^{-\frac{3}{2}\frac{1}{p-1}}\int \limits _{|y|<\frac{|x|}{2}}|y|^{-\alpha p'}dy +c|x|^{-\alpha p'}\int \limits _{\mathbb {R}^3}H^{p'}(t,z)dz\quad \\ {} &{}\displaystyle \le c |x|^{-\alpha p'}t^{-\frac{3}{2}\frac{1}{p-1}}\,, \end{array} \end{aligned}$$

where c is a constant independent of (t, x). Hence estimate for $I_1$ easily follows. For $\varepsilon \in (0,1)$, we set $p(\varepsilon ):=\frac{p}{1+\varepsilon }$ and denote by $p'(\varepsilon )$ its coniugate exponent. Empolying Hölder inequality, we get

$$\begin{aligned} \begin{array}{ll} |I_2(t,x)|&{}\displaystyle \le \big [\sup _{(0,t)}\tau ^\frac{1}{2}||v(\tau )||_\infty \big ]^{1-\varepsilon }\int \limits _{0}^{t}||\nabla E(t-\tau ,x)||_{L^{p'(\varepsilon )}_{-\alpha (1+\varepsilon ) }}||v(\tau )||_{L^p_{\alpha }}^{1+\varepsilon }\tau ^{-\frac{1}{2}+\frac{\varepsilon }{2}}d\tau \quad \\ {} &{}\displaystyle \le cA^2\int \limits _{0}^{t}||\nabla E(t-\tau ,x)||_{L^{p'(\varepsilon )}_{-\alpha (1+\varepsilon )}}\tau ^{-\frac{1}{2}+\frac{\varepsilon }{2}}d\tau \le cA^2t^{-\frac{3}{2p}}|x|^{-\alpha }\,. \end{array} \end{aligned}$$

(6.13)

To show this, we use the following estimates on $\nabla E$:

$$\begin{aligned} \begin{array}{ll}||\nabla E(t-\tau ,x)||_{L^{p'(\varepsilon )}_{-\alpha }}^{p'(\varepsilon )}&{} \le \displaystyle \int \limits _{\mathbb {R}^3}\big [|x-y|+(t-\tau )^\frac{1}{2}\big ]^{-4p'(\varepsilon )}|y|^{-\alpha (1+\varepsilon ) p'(\varepsilon )}dy\quad \\ {} &{}\displaystyle \le ct^{-\frac{3}{2}\frac{1}{p-1-\varepsilon }-p'(\varepsilon )\frac{1+\varepsilon }{2 }} \int \limits _{\mathbb {R}^3}\big [|x-y|+(t-\tau )^\frac{1}{2}\big ]^{-4p'(\varepsilon )+\frac{3}{p-1-\varepsilon }+{(1+\varepsilon )}{p'(\varepsilon )}}|y|^{-\alpha (1+\varepsilon )p'(\varepsilon )}dy \,, \end{array} \end{aligned}$$

which, in turn, since $4p'(\varepsilon )-\frac{3}{p-1-\varepsilon }-{(1+\varepsilon )}{p'(\varepsilon )}<3$ and $\alpha (1+\varepsilon )p'(\varepsilon )<3$, via [7, Lemma II. 9.2], allows us to prove

$$\begin{aligned} \begin{array}{ll}||\nabla E(t-\tau ,x)||_{L^{p'(\varepsilon )}_{-\alpha }}^{p'(\varepsilon )}&{}\displaystyle \le ct^{-\frac{3}{2}\frac{1}{p-1-\varepsilon }-p'(\varepsilon )\frac{1+\varepsilon }{2}}\int \limits _{\mathbb {R}^3}|x-y|^{-4p'(\varepsilon ) +\frac{3}{p-1-\varepsilon }+{(1+\varepsilon )}{p'(\varepsilon )}}|y|^{-\alpha (1+\varepsilon )p'(\varepsilon )}dy\quad \\ {} &{}\displaystyle = ct^{-\frac{3}{2}\frac{1}{p-1-\varepsilon }-p'(\varepsilon )\frac{1+\varepsilon }{2}}|x|^{-\alpha p'(\varepsilon )}\,.\end{array} \end{aligned}$$

The latter entails the validity of (6.13) with c independent of (t, x). Finally, by a similar fashion, we can derive similar estimate for $I_3$ and $I_4$. Precisely, for $i=3,4$, we get

$$\begin{aligned} |I_i(t,x)|\le \sup _{(0,\infty )}\tau ^\frac{1}{2}||v(\tau )||_\infty \int \limits _{0}^{t}\tau ^{-\frac{1}{2}}||\nabla E(t-\tau ,x)w||_1d\tau \,. \end{aligned}$$

Recalling the assumption on w and (5.1), we prove:

$$\begin{aligned} ||\nabla E(t-\tau ,x)w||_1\le c w^{(0)}(t-\tau )^{-\frac{1}{2}-\frac{3}{2p}} \int \limits _{\mathbb {R}^3}|x-y|^{-3+\frac{3}{p}}|y|^{-1}dy\le cw^{(0)}(t-\tau )^{-\frac{1}{2}-\frac{3}{2p}}|x|^{-\alpha }\,. \end{aligned}$$

Hence, we obtain the desired estimate for both $I_i$, $i=3,4$, with a constant c independent of (t, x). Collecting the estimates for $I_i$ and using them on the right-hand side of (6.12), completes the proof of (2.10). In order to prove (2.11), it is enough to combine the estimate just proved with the one given in (2.9) for $\Vert v(t)\Vert _\infty$. $\square$