Global existence, uniqueness and estimates of the solution to the Navier-Stokes equations

TL;DR

This paper establishes conditions for the global existence and uniqueness of solutions to the Navier-Stokes equations, providing explicit integral representations and energy estimates within a specific function class.

Contribution

It introduces a novel integral equation formulation for Navier-Stokes solutions and proves their uniqueness and existence in a defined function space.

Findings

Uniqueness of solutions in a class with finite norm N_1(v)

Existence of solutions under certain assumptions

Energy estimates for the solutions

Abstract

The Navier-Stokes (NS) problem consists of finding a vector-function $v$ from the Navier-Stokes equations. The solution $v$ to NS problem is defined in this paper as the solution to an integral equation. The kernel $G$ of this equation solves a linear problem which is obtained from the NS problem by dropping the nonlinear term $(v \cdot \nabla)v$. The kernel $G$ is found in closed form. Uniqueness of the solution to the integral equation is proved in a class of solutions $v$ with finite norm $N_1(v)=\sup_{\xi\in \mathbb{R}^3, t\in [0, T]}(1+|\xi|)(|v|+|\nabla v|)\le c (*)$, where $T>0$ and $C>0$ are arbitrary large fixed constants. In the same class of solutions existence of the solution is proved under some assumption. Estimate of the energy of the solution is given.

Equations

$$v^{\prime }+(v,\nabla )v=-\nabla p+\nu \Delta v+f,x\in {\mathbb{R}}^3,t\ge 0,\nabla \cdot v=0,v(x,0)=v_0(x).$$

$$G^{\prime }-\nu \Delta G=\delta (x)\delta (t){\delta }_{jm}-\nabla p_m,\nabla \cdot G=0;G=0,t<0.$$

$$v(x,t)={\int }_0^{t}ds{\int }_{{\mathbb{R}}^3}G(x-y,t-s)[f(y,s)-(v,\nabla )v]dy+{\int }_{{\mathbb{R}}^3}g(x-y,t)v_0(y)dy,$$

$$g(x,t)=\frac{e^{-\frac{|x{|}^2}{4\nu t}}}{(4\nu t\pi {)}^{3/2}},g(x,0)=\delta (x),g^{\prime }-\nu \Delta g=\delta (x)\delta (t);{\int }_{{\mathbb{R}}^3}g(x,t)dx=1.$$

$$H^{\prime }+\nu {\xi }^{2}H=\frac{\delta (t){\delta }_{jm}}{(2\pi {)}^3}-i\xi P_m(\xi ,t),\xi H=0;H(\xi ,t)=0,t<0.$$

$$(2\pi {)}^{-3}\delta (t){\xi }_m=i{\xi }^2{P}_m,P_m:=P_m(\xi ,t)=-i\frac{\delta (t){\xi }_m}{(2\pi {)}^3{\xi }^2},{\xi }^2:={\xi }_j{\xi }_j.$$

$$H_{jm}^{\prime}+\nu\xi^{2}H_{jm}=(2\pi)^{-3}\delta(t)\Big{(}\delta_{jm}-\frac{\xi_{j}\xi_{m}}{\xi^{2}}\Big{)}.$$

$$H=H_{jm}(\xi,t)=(2\pi)^{-3}\Big{(}\delta_{jm}-\frac{\xi_{j}\xi_{m}}{\xi^{2}}\Big{)}e^{-\nu\xi^{2}t};\quad H=0,\,\,t<0.$$

$$G(x,t)=(2\pi)^{-3}\int_{{\mathbb{R}}^{3}}e^{i\xi\cdot x}\Big{(}\delta_{jm}-\frac{\xi_{j}\xi_{m}}{\xi^{2}}\Big{)}e^{-\nu\xi^{2}t}d\xi:=I_{1}+I_{2}.$$

$$I_1={\delta }_{jm}g(x,t);g(x,t):=\frac{e^{-\frac{|x{|}^2}{4\nu t}}}{(4\nu t\pi {)}^{3/2}},t>0;g=0,t<0,$$

$$I_{2}=\frac{\partial^{2}}{\partial x_{j}\partial x_{m}}\Big{(}\frac{1}{|x|2\pi^{3/2}}\int_{0}^{|x|/(4\nu t)^{1/2}}e^{-s^{2}}ds\Big{)},\quad t>0;\quad I_{2}=0,\,t<0.$$

$$G(x,t)=\delta_{jm}g(x,t)+\frac{1}{2\pi^{3/2}}\frac{\partial^{2}}{\partial x_{j}\partial x_{m}}\Big{(}\frac{1}{|x|}\int_{0}^{|x|/(4\nu t)^{1/2}}e^{-s^{2}}ds\Big{)},\,\,t>0.$$

$$\begin{split}Lv=\int_{0}^{t}ds\int_{{\mathbb{R}}^{3}}\Big{[}\delta(x-y)\delta(t-s)\delta_{jm}+\partial_{j}\Big{(}\partial_{m}\big{(}\frac{1}{2\pi^{3/2}|x-y|}\int_{0}^{|x-y|/(4\nu t)^{1/2}}e^{-s^{2}}ds\big{)}\Big{)}\Big{]}\times\\ \Big{(}f(y,s)-(v(y,s),\nabla)v(y,s)\Big{)}dy=f(x,t)-(v(x,t),\nabla)v(x,t)-\nabla p_{m}(x,t),\end{split}$$

$$p_{m}(x,t)=-\int_{0}^{t}ds\int_{{\mathbb{R}}^{3}}\partial_{m}\Big{(}\frac{1}{2\pi^{3/2}|x-y|}\int_{0}^{|x-y|/(4\nu t)^{1/2}}e^{-s^{2}}ds\Big{)}[f(y,s)-(v(y,s),\nabla)v(y,s)]dy,$$

$$N_1(v):=\underset{x\in {\mathbb{R}}^3,t\in [0,T]}{\sup }[(|v(x,t)|+|\nabla v(x,t)|)(1+|x|)],$$

$$\begin{array}{rl}w(x,t)={\int }_0^{t}ds{\int }_{{\mathbb{R}}^3}G(x-y,t-s)[(v_2,\nabla )v_2-(v_1,\nabla )v_1]dy& =\\ {\int }_0^{t}ds{\int }_{{\mathbb{R}}^3}G(x-y,t-s)[(w,\nabla )v_2+(v_1,\nabla )w]dy.& \end{array}$$

$$N_{0}(w)\leq\int_{0}^{t}dsN_{0}\Big{(}\int_{{\mathbb{R}}^{3}}|G(x-y,t-s)|(1+|y|)^{-1}dy\Big{)}N_{0}(w)(N_{1}(v_{2})+N_{1}(v_{1}))\leq c\int_{0}^{t}\frac{N_{0}(w)}{(t-s)^{1/2}}ds.$$

$$N_{0}\Big{(}\int_{{\mathbb{R}}^{3}}|G(x-y,t)|(1+|y|)^{-1}dy\Big{)}\leq ct^{-\frac{1}{2}},$$

$$v(x,t)=-{\int }_0^{t}ds{\int }_{{\mathbb{R}}^3}G(x-y,t-s)(v,\nabla )vdy+F(x,t),$$

$$F(x,t):={\int }_0^{t}ds{\int }_{{\mathbb{R}}^3}G(x-y,t-s)f(y,s)dy+{\int }_{{\mathbb{R}}^3}g(x-y,t)v_0(y)dy.$$

$$v_{n+1}(x,t)=-{\int }_0^{t}B(v_n,\nabla )v_{n}ds+F,v_1=F;Bv:={\int }_{{\mathbb{R}}^3}G(x-y,t-s)v(y,s)dy.$$

$$\underset{n\ge 1}{\sup }{N}_1(v_n)\le c,$$

$$\begin{split}N_{0}(v_{n+1}-v_{n})\leq N_{0}\Big{(}\int_{0}^{t}ds\int_{{\mathbb{R}}^{3}}\frac{|G(x-y,t-s)|}{1+|y|}dy\Big{)}[N_{0}(v_{n}-v_{n-1})(N_{1}(v_{n})+N_{1}(v_{n-1})]&\leq\\ ct^{1/2}N_{0}(v_{n}-v_{n-1}).\end{split}$$

$$N_0(v_{n+1}-v_n)\le c{t}^{1/2}{N}_0(v_n-v_{n-1}).$$

$$v=F-{\int }_0^{\tau }ds{\int }_{{\mathbb{R}}^3}G(x-y,t-s)(v,\nabla )vdy-{\int }_{\tau }^{t}ds{\int }_{{\mathbb{R}}^3}G(x-y,t-s)(v,\nabla )vdy$$

$$F_1-{\int }_{\tau }^{t}ds{\int }_{{\mathbb{R}}^3}G(x-y,t-s)(v,\nabla )vdy,$$

$$\frac{1}{2}\frac{\partial }{\partial t}{\int }_{{\mathbb{R}}^3}|v(x,t){|}^{2}dx+\nu {\int }_{{\mathbb{R}}^3}\nabla v_j(x,t)\cdot \nabla v_j(x,t)dx={\int }_{{\mathbb{R}}^3}f\cdot vdx.$$

$$2\nu {\int }_0^t\Vert \nabla v{\Vert }^{2}dt+E(t)\le E(0)+2{\int }_0^{t}ds|{\int }_{{\mathbb{R}}^3}f\cdot vdx|\le E(0)+2{\int }_0^T\mathcal{N}(f)\mathcal{N}(v)ds.$$

$$(1-ϵ)E_T\le E(0)+\frac{1}{ϵ}({\int }_0^T\mathcal{N}(f)dt{)}^2.$$

$$\frac{1}{\pi }{\int }_0^{\infty }\cos (x_p{\xi }_p)e^{-\nu {\xi }^{2}t}d{\xi }_p=\frac{1}{\pi }\frac{{\pi }^{1/2}}{2(\nu t{)}^{1/2}}{e}^{-\frac{x_p^2}{4\nu t}}=\frac{e^{-\frac{x_p^2}{4\nu t}}}{(4\nu t\pi {)}^{1/2}}.$$

$$I_{2}=\frac{\partial^{2}}{\partial x_{j}\partial x_{m}}\frac{1}{(2\pi)^{3}}\int_{{\mathbb{R}}^{3}}e^{i\xi\cdot x}\frac{e^{-\nu t\xi^{2}}}{\xi^{2}}d\xi=\frac{\partial^{2}}{\partial x_{j}\partial x_{m}}\Big{(}\frac{1}{(2\pi)^{2}}\int_{0}^{\infty}dre^{-\nu tr^{2}}\int_{-1}^{1}e^{ir|x|u}du\Big{)},$$

Full text

The Navier-Stokes (NS) problem consists of finding a vector-function $v$ from the Navier-Stokes equations. The solution $v$ to NS problem is defined in this paper as the solution to an integral equation. The kernel $G$ of this equation solves a linear problem which is obtained from the NS problem by dropping the nonlinear term $(v\cdot\nabla)v$. The kernel $G$ is found in closed form. Uniqueness of the solution to the integral equation is proved in a class of solutions $v$ with finite norm $N_{1}(v)=\sup_{\xi\in{\mathbb{R}}^{3},t\in[0,T]}(1+|\xi|)(|v|+|\nabla v|)\leq c(*)$, where $T>0$ and $C>0$ are arbitrary large fixed constants. In the same class of solutions existence of the solution is proved under some assumption. Estimate of the energy of the solution is given.

1 Introduction

There is a large literature on the Navier-Stokes (NS) problem, ( see [2], [8] and references therein, including the papers by Leray, Hopf, Lions, Prodi, Kato and others). The global existence and uniqueness of a solution was not proved. Various notions of the solution were used, see [2], [3], [6]. In this paper a new notion of the solution is given: the solution to NS problem is defined as a solution to some integral equation. It is proved that this solution is unique, it exists in a certain class of functions globally, i.e., for all $t\geq 0$, it has finite energy, and an estimate of this solution is given.

The NS problem consists of solving the equations

[TABLE]

Vector-functions $v=v(x,t)$, $f=f(x,t)$ and the scalar function $p=p(x,t)$ decay as $|x|\to\infty$ uniformly with respect to $t\in[0,T]$, where $T>0$ is an arbitrary large fixed number, $v^{\prime}=v_{t}$, $\nu=const>0$, $v_{0}$ is given, $\nabla\cdot v_{0}=0$, the velocity $v$ and the pressure $p$ are unknowns, $v_{0}$ and $f$ are known. Equations (1) describe viscous incompressible fluid with density $\rho=1$. We assume that $|f|+|\nabla f|$ and $|v_{0}|+|\nabla v_{0}|$ decay as $O(\frac{1}{(1+|x|)^{a}})$ as $|x|\to\infty$, where $a>3$ and for $f$ the decay with respect to $x$ holds for any $t\geq 0$, uniformly with respect to $t\in[0,T]$, where $T\geq 0$ is an arbitrary large fixed number.

Our approach consists of three steps. First, we construct tensor $G_{jm}(x,t)$, equivalent to Oseen’s thensor in [3], p.62, solving the problem:

[TABLE]

Here $\delta(x)$ is the delta-function, $\delta_{jm}$ is the Kronecker delta, $p_{m}=p(x,t)e_{m}$, $p$ is the scalar, $p(x,t)=0,t<0$; $\nabla p_{m}=\frac{\partial p}{\partial x_{j}}e_{j}e_{m}$, $\{e_{j}\}|_{j=1}^{3}$ is the orthonormal basis of ${\mathbb{R}}^{3}$, $e_{j}e_{m}$ is tensor. Our construction method differs from the one in [3].

Secondly, we prove that solving (1) is equivalent to solving the following integral equation, cf [3], p.62:

[TABLE]

where

[TABLE]

Thirdly, we prove by a new method, using the new assumption (21), see below, that equation (3) has a solution in the space $X$ of functions with finite norm $N_{1}(v):=\sup_{x\in{\mathbb{R}}^{3},t\in[0,T]}\{(|v(x,t)|+|\nabla v(x,t)|)(1+|x|)\}$ and this solution is unique in $X$. In Lemma 1 a new a priori estimate is obtained.

Theorem 1. Problem (1) has a unique solution in $X$. The solution to (1) exists in $X$ provided that $\sup_{n>1}N_{1}(v_{n})<\infty$, where $v_{n}$ is defined in (20). This solution has finite energy $E(t)=\int_{{\mathbb{R}}^{3}}|v(x,t)|^{2}dy$ for every $t\geq 0$.

2 Construction of $G$

Let $G=\int_{{\mathbb{R}}^{3}}H(\xi,t)e^{i\xi\cdot x}d\xi$. Taking Fourier transform of (2) yields

[TABLE]

The $H=H_{jm}$ is a tensor, $1\leq j,m\leq 3$, $P_{m}(\xi,t):=(2\pi)^{-3}\int_{{\mathbb{R}}^{3}}p_{m}(x,t)e^{-i\xi\cdot x}dx,$ $\xi=\xi_{j}e_{j}$, summation is understood here and below over the repeated indices, $1\leq j\leq 3$, $\xi H:=\xi_{j}H_{jm}$. Multiply (5) by $\xi$ from the left, use the equation $\nabla\cdot G=0$ which implies $\xi H=0$, $\xi H^{\prime}=0$, and get

[TABLE]

From (5)-(6) one obtains:

[TABLE]

Thus,

[TABLE]

[TABLE]

The integrals $I_{1},I_{2}$ are calculated in the Appendix:

[TABLE]

[TABLE]

Therefore,

[TABLE]

3 Solution to integral equation for $v$ satisfies NS equations

Apply the operator $L:=\frac{\partial}{\partial t}-\nu\Delta$ to the left side of (3) and use (2) to get

[TABLE]

where

[TABLE]

and $v(x,0)=v_{0}(x)$ because $g(x-y,0)=\delta(x-y)$. Using the formula $\nabla\cdot G=0$, the relation $\int_{{\mathbb{R}}^{3}}g(x-y,t)v_{0}(y)dy=\int_{{\mathbb{R}}^{3}}g(z,t)v_{0}(z+x)dz$ and the formula $\nabla\cdot v_{0}=0$, one checks that $\nabla\cdot v=0$. Thus, a solution to (3) solves (1).

4 Proof of the uniqueness of the solution to $\eqref{e3}$ in the space of functions

with finite norm $N_{1}(v)$

Let $X$ be the space $C^{1}({\mathbb{R}}^{3};1+|x|)$ of vector-functions with the norm

[TABLE]

where $T>0$ is an arbitrary large fixed number, $\nabla$ stands for any first order derivative, and let $N_{0}(v)=\sup_{x\in{\mathbb{R}}^{3}}(|v(x,t)|+|\nabla v(x,t)|)$, $N_{0}(v)\leq N_{1}(v)$, $|v|:=(\sum_{j=1}^{3}|v_{j}|^{2})^{1/2}$. Assume that there are two solutions $v_{1},v_{2}$ to equation (3) with finite norms $N_{1}(v)$ and let $w=v_{1}-v_{2}$. One has

[TABLE]

Therefore

[TABLE]

Since we are proving uniqueness in the set of functions with finite norm $N_{1}(v)$, one has $N_{1}(v_{2})+N_{1}(v_{1})\leq c$, where $c>0$ stands for various estimation constants. It is checked in the Appendix that

[TABLE]

so inequality (16) holds. From (16) by the standard argument one derives that $N_{0}(w)=0$. Thus, $v_{1}=v_{2}$. Uniqueness of the solution to (3) is proved in $X$. $\Box$

5 Proof of the existence of the solution to $\eqref{e3}$ in the set of functions

with finite norm $N_{1}(v)$

Rewrite (3) as

[TABLE]

where $F$ is known:

[TABLE]

Equation (18) is of Volterra type, nonlinear, solvable, as we wish to prove, by iterations:

[TABLE]

Using (17), the formula $\int_{0}^{t}\frac{ds}{s^{1/2}}=2t^{1/2}$, and assuming that

[TABLE]

one gets:

[TABLE]

Therefore,

[TABLE]

If $\tau$ is chosen so that $c\tau^{1/2}<1$, then $B$ is a contraction map on the bounded set $N_{0}(v)\leq R$ for sufficiently large $R$ and $t\in[0,\tau]$. Thus, $v$ is uniquely determined by iterations for $x\in{\mathbb{R}}^{3}$ and $t\in[0,\tau]$. If $t>\tau$ rewrite (18) as

[TABLE]

where $F_{1}$ is a known function since $v$ is known for $t\in[0,\tau]$. To this equation one applies the contraction mapping principle and get $v$ on the interval $[0,2\tau]$. Continue in this fashion and construct $v$ for any $t\geq 0$. Process (20) converges to a solution to equation (3). $\Box$

Let us make a remark concerning the mapping done by the operator in (18). In [4], p. 234, sufficient conditions are given for a singular integral operator to map a class of functions with a known power rate of decay at infinity into a class of functions with a suitable rate of decay at infinity. It follows from [4], Theorem 5.1 on p.234, that if $|f(x)|+|\nabla f|\leq c(1+|x|)^{-a}$, $a>3$, then the part of the operator $G$ in (18), responsible for the lesser decay of the iterations $v_{n}$ at infinity, acts as a weakly singular operator similar to the operator $Q$, $Qf:=\int_{{\mathbb{R}}^{3}}\frac{b(|x-y|)|f(y)|}{|x-y|^{3}}dy\leq c\frac{1}{(1+|x|)^{3}}$ for large $|x|$. This part of $G$ yields the decay $O(\frac{1}{(1+|x|)^{3}})$ at infinity. This decay, in general, cannot be improved. The first iteration $v_{1}$ yields a function decaying with its first derivatives as $O(\frac{1}{(1+|x|)^{3}})$. The second iteration contains a function whose behavior for large $|x|$ is determined by the decay of the functions $F$. Since $(v_{1}\cdot\nabla)v_{1}=O((1+|x|)^{-7})$, the decay of $v_{2}$ is again $O((1+|x|)^{-3})$ because $v_{1}=O((1+|x|)^{-3})$ and $|\nabla v_{1}|=O((1+|x|)^{-4})$. Thus, for $|v_{n}|+|\nabla v_{n}|$ one gets the decay of the order $O((1+|x|)^{-3})$ as $|x|\to\infty$ for every fixed $t\geq 0$, provided that $f$ and $v_{0}$ together with their first derivatives decay not slower than $O((1+|x|)^{-a})$, $a>3$.

6 Energy of the solution

In this Section we prove that the solution has finite energy in a suitable sense and give an estimate of the solution as $t\to\infty$. Let us define the energy by the integral $E(t)=\int_{{\mathbb{R}}^{3}}|v(x,t)|^{2}dy$. If one multiplies (1) by $v$ and integrate over ${\mathbb{R}}^{3}$, then one gets a known conservation law (see [2]):

[TABLE]

Integrating (24) with respect to $t$ over any finite interval $[0,t]$, $0\leq t\leq T$, and denoting $\mathcal{N}(v):=(\int_{{\mathbb{R}}^{3}}|v(x,t)|^{2}dy)^{1/2}=\sqrt{E(t)}$, one gets

[TABLE]

Denote $E_{T}:=\sup_{t\in[0,T]}E(t)$. Maximizing inequality (25) with respect to $t\in[0,T]$, and using the elementary inequality $2ab\leq\epsilon a^{2}+\frac{1}{\epsilon}b^{2}$, $a,b,\epsilon>0$, one derives from (25) the following inequality

[TABLE]

Let $\epsilon=\frac{1}{2}$. Then (25)-(26) allow one to estimate $E_{T}$ and $\nu\int_{0}^{t}\|\nabla v\|^{2}dt$ through $f$ and $v_{0}$. In particular, we have proved the following theorem

Theorem 2. Assume that $\int_{0}^{T}\mathcal{N}(f)dt<\infty$ for all $T>0$ and $E(0)<\infty$. Then $\sup_{t\in[0,T]}E(t)<\infty$ and $\int_{0}^{T}\|\nabla v(x,t)\|^{2}dt<\infty$ for all $T>0$.

Remark 1. It is proved in [5] that in a bounded domain $D\subset{\mathbb{R}}^{3}$ the solution to a boundary problem for equations (1) in a bounded domain with the Dirichlet boundary condition for large $t$ decays exponentially provided that $\int_{0}^{\infty}e^{bt}\mathcal{N}(f)dt<\infty$ for some $b=const>0$.

7 Appendix

1. Integral $I_{1}$. One has $I_{1}=I_{11}I_{12}I_{13}$, where $I_{1p}:=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ix_{p}\xi_{p}-\nu\xi^{2}t}d\xi_{p}.$ No summation in $x_{p}\xi_{p}$ is understood here. One has

[TABLE]

After a multiplication (with $p=1,2,3$) this yields formula (10).

Integral $I_{2}$. One has

[TABLE]

so

[TABLE]

Here we have used formula (2.4.21) from [1]:

[TABLE]

2. One has $\int_{{\mathbb{R}}^{3}}|\nabla g(x,t)|dx\leq(4\nu t\pi)^{-\frac{3}{2}}\int_{{\mathbb{R}}^{3}}e^{-\frac{|x|^{2}}{4\nu t}}\frac{2|x|}{4\nu t}dx=\frac{2}{(\nu t\pi)^{\frac{1}{2}}}.$ This proves the estimate

$\int_{{\mathbb{R}}^{3}}|G(x-y,t)(1+|y|)^{-1}dy\leq ct^{-\frac{1}{2}}$ for the first term of $G$ in (12), namely for $g$. The second term of $G$ is J:=\frac{\partial^{2}}{\partial x_{j}\partial x_{m}}\Big{(}\frac{1}{|x|}\int_{0}^{|x|/(4\nu t)^{1/2}}e^{-s^{2}}ds\Big{)} up to a factor $\frac{1}{2\pi^{3/2}}$.

One checks by direct differentiation that

[TABLE]

Let $b(x):=b(\frac{|x|}{(4\nu t)^{1/2}}):=\int_{0}^{|x|/(4\nu t)^{1/2}}e^{-s^{2}}ds$. Note that $b(s)=O(s)$ as $s\to 0$ and $b(s)\to\frac{\pi^{1/2}}{2}$ as $s\to\infty$. It is sufficient to estimate the term with the strongest singularity, namely,

$J_{1}:=\int_{{\mathbb{R}}^{3}}\frac{b(|x-y|)}{|x-y|^{3}(1+|y|)}dy$. We prove that $\max_{x\in{\mathbb{R}}^{3}}J_{1}\leq c$. The other two terms in the second term of $J$ in (30) are estimated similarly. One has $J_{1}=\int_{{\mathbb{R}}^{3}}\frac{b(|z|)}{|z|^{3}(1+|x+z|)}dy=\int_{|z|\leq 1}+\int_{|z|\geq 1}:=J_{11}+J_{12}$, where $J_{11}\leq c(1+|x|)^{-1},\,\,x\in{\mathbb{R}}^{3}$, because $b(|z|)|z|^{-3}\leq c|z|^{-2}$ for $|z|\leq 1$. Let us estimate $J_{12}$. Note that $1+|y|^{a}\leq(1+|y|)^{a}\leq c(1+|y|^{a})$ for $a\geq 1$. We prove that $J_{12}\leq c\frac{\ln(3+|x|)}{2+|x|}$ for all $x\in{\mathbb{R}}^{3}$. Use the spherical coordinates $|z|=r$, $|x|:=q$, the angle between $x$ and $z$ denote by $\theta$, $\cos\theta=u$, and get for $a=1$:

[TABLE]

Remark 2. In [2] it is shown that the smoothness properties of the solution $v$ are improved when the smoothness properties of $f$ and $v_{0}$ are improved. This also follows from equation (3) by the properties of weakly singular integrals and of the kernel $g(x,t)$.

Let us prove a new a priori estimate.

Lemma 1. If the assumptions of Theorem 2 hold, then $(1+\xi^{2})|\tilde{v}(\xi,t)|^{2}\leq c+ct$ for all $t\geq 0$, where $c>0$ does not depend on $\xi,t$.

Proof. The Fourier transform of (18) yields

[TABLE]

where $H$ is defined in (8),

[TABLE]

over the repeated indices $j$ one sums up, $\mathcal{N}(\tilde{v}):=\|\tilde{v}\|_{L^{2}({\mathbb{R}}^{3})}$, and $\tilde{v}\ast\psi$ denotes the convolution of two functions, so

[TABLE]

where the known estimate $\mathcal{N}(\tilde{v})\leq c$ was used. One has

[TABLE]

It follows from (24) and (25) that

[TABLE]

provided that assumptions of Theorem 2 hold. It follows from (32) that

[TABLE]

Thus,

[TABLE]

provided that $|\xi^{2}||\tilde{F}|^{2}\leq c$. This inequality holds if $f$ and $v_{0}$ are smooth and rapidly decaying. Lemma 1 is proved. $\Box$