About Bellman Principle and Solution Properties for Navier–Stokes Equations in the 3D Cauchy Problem

Abstract

Without belittling the achievements of many mathematicians in the studying of the Navier-Stokes equations, the real ways opened J. Leray and O.A. Ladyzhenskaya. The main goal of this work is to compare the smoothness property of a weak solution in the Cauchy problem after some moment if it is known solution regularity until this moment with the optimality property in the Bellman principle. Naturally, all these are connected with the existence problem of blow up solution in the Cauchy problem for Navier-Stokes equations in space attracting a lot of attention up to now. The smoothness control and controlling parameters can be varied. It is important to control the dissipation of kinetic energy to the fix moment or rate of change of kinetic energy square or the summability of velocity gradient to the fixed point in time and so on. There are possible other control parameters due to a weak solution.

1. Introduction

The well-known principle of dynamic programming (Bellman principle) postulates that whatever the state of the system at any step and the control at this step selected, subsequent controls should be selected optimal relative to the state to which the system will arrive at the end of this step. We would like to get something similar for the smoothness properties of a weak solution in the Cauchy problem for the Navier-Stokes equations in space, i.e., we would like to find out the conditions under which the existing regularity of the solution in a short time interval (see Refs. [1, 2]), and some important parameters (control parameters) of this solution at a certain point in time, while this smoothness is still preserved, it will be able to ensure the existence of a global smooth extension of this solution, or at least of a local smooth extension for some guaranteed longer time interval. Let us explain this. The most important characteristic of a fluid flow is kinetic energy at every point in time. If kinetic energy changing at the fixed point in time is not large, then we have no phenomenon collapse. Hence, we obtain the one from the controlling parameters. We can also control a rate of change of kinetic energy square. Here, the situation is the same. If at the initial time or the later point in time this rate of change is not large, then we have no phenomenon blow up again. Really, instead of the point time controlling, we can apply some new characteristics which are described as the mean value of some quantity over the fixed time interval. The result will be the same. This generality shows that comparison with Bellman principle is relevant.

In this way, the first steps were undertaken in Ref. [3] where numerical parameters were introduced as control parameters. Indirectly, they control the solution smoothness over a longer time interval. This point of view differs from the classical methods in studying the smoothness properties of weak solutions (see Refs. [1, 2, 4, 5, 6]) where the main tools are connected with embedding lemmas and multiplicative inequalities. In fact, embedding lemmas and multiplicative inequalities are the main tools up to now.

Here, another interesting aspect should be noted. It is related to the asymptotics of smooth solutions at infinity and integral identities for solenoid fields (see, for example, [7]). It may be a new tool for new apriori estimates.

2. Notations

We consider a motion of an ideal incompressible fluid and the simplest problem that is Cauchy problem in space $\left(n=3\right)$ which is described by equations:

$$\frac{\partial u}{\partial t}+\sum _{i=1}^3{u}_i\frac{\partial u}{\partial x_i}=\nu \mathrm{\Delta }u-\nabla P,\mathit{div}u=0,u\left(0,x\right)=\phi \left(x\right),$$E1

where $u=u\left(t,x\right)=\left(u_1\left(t,x\right),u_2\left(t,x\right),u_3\left(t,x\right)\right)$ is velocity vector. Symbols $t$ and $x=\left(x_1,x_2,x_3\right)$ are time and spatial variables, respectively. A function $P=P\left(t,x\right)$ is pressure function and a vector field $\phi \left(x\right)=\left({\phi }_1\left(x\right),{\phi }_2\left(x\right),{\phi }_3\left(x\right)\right)$ is initial data; a constant $\nu$ is viscosity coefficient.

By symbols $\Delta$ and $\nabla$, we denote Laplace operator and gradient operator on spatial variables, respectively. In particular, $\nabla u$ is Jacobi matrix on spatial variables.

Next, we apply the following notation:

$$\begin{array}{c}\mid \phi \mid =\sqrt{\sum _{i=1}^3{\phi }_i^2},\mid \nabla \phi \mid =\sqrt{\sum _{i=1}^3{\left|\nabla {\phi }_i\right|}^2},\\ \parallel u\left(t,\cdot \right){\parallel }_p^p={\int }_{R^3}{\left|u(t,x)\right|}^p\mathit{dx},\parallel \nabla u\left(t,\cdot \right){\parallel }_p^p={\int }_{R^3}{\left|\nabla u(t,x)\right|}^p\mathit{dx},p>1.\end{array}$$E2

Lebesgue class $L_p\left(R^3\right)$ is defined as a set of vector fields in $R^3$ with finite norm $\parallel v{\parallel }_2$. In particular, and this is very important, for solutions $u$ of problem (1), the kinetic energy at moment $t$ is expressed by mean (energy integral) $\parallel u\left(t,\cdot \right){\parallel }_2^2$ (see Refs. [1, 6]).

In addition, we suppose that initial data $\phi$, as vector field $\phi \in C_{6/5,3/2}^{\infty }\left(R^3\right)$, that is, $\phi$ is infinitely differentiable mapping. It belongs to Lebesgue class $L_{6/5}\left(R^3\right)$, and the first partial derivatives $\nabla \phi \in L_{3/2}\left(R^3\right)$ and the rest derivatives belong to class $L_r\left(R^3\right)$ for any $r>1$ (class $C_{6/5,3/2}^{\infty }\left(R^3\right)$, its properties, and usefulness are described in Ref. [3]). In particular, it implies the following inclusions $\phi \in L_2\left(R^3\right),\nabla \phi \in L_2\left(R^3\right)$ (see Ref. [3], Lemma 32). For us, this class is interesting only because for any fixed $t$ solution $u\left(t,\cdot \right)$ of Cauchy problem (1) belongs to class $C_{6/5,3/2}^{\infty }\left(R^3\right)$ (see Ref. [3], Theorems 2, 6).

The well-known classical results belonging to O. A. Ladyzhenskaya and J. Serrin (see Refs. [1, 2]) show an existence of time interval $\left[0,T\right)$ where the solution of problem (1) is regular in zone $\left[0,T\right)\times R^3$. Denote by $T_{\star }$ the least upper bound of these $T$. If $T_{\star }<\infty$, then the solution of problem (1) is called a blow up solution, and the fluid flow describing of this solution is called a turbulent flow. In this case, we have the collapse phenomenon.

Now, describe the control parameters knowing the values of which we can be sure that there will be no collapse or it will not be in a guaranteed time interval. Following [3] (see formulae (68), (69), (87), and (5) from [3] respectively), we define these parameters $\lambda ,\mu ,\epsilon$ as parameters, which control solution smoothness and a number $T_0$ by equalities:

$$l\left(\phi \right)=\parallel \phi {\parallel }_2\cdot \parallel \nabla \phi {\parallel }_2,\lambda ={\left(\frac{4\sqrt[4]{3}}{3a_1}\right)}^2\frac{{\nu }^2}{l\left(\phi \right)}=\frac{81{\nu }^2}{8l\left(\phi \right)},\lambda \left(t\right)=\frac{81{\nu }^2}{8l\left(u\left(t,\cdot \right)\right)},$$E3

where $u\left(t,x\right)$ is the solution of Cauchy problem (1),

$$\mu =\frac{T_{\ast }}{T_0},$$E4

$$\parallel u\left(T_0,\cdot \right){\parallel }_2^2=\parallel \phi {\parallel }_2^2\left(1-{\mathit{\epsilon \lambda }}^2\right),$$E5

$$T_0={\left(\frac{9}{4}\right)}^4\frac{{\nu }^3}{\parallel \nabla \phi {\parallel }_2^4}.$$E6

Here, $\left[0,T_0\right)$ is that time interval (it is not necessarily optimal) where every weak solution (see definition in Refs. [1, 4, 6]) of problem (1) is regular (i.e., smooth) and it satisfies condition:

$$\parallel \nabla u\left(t,\cdot \right){\parallel }_2^2\le \frac{\parallel \nabla \phi {\parallel }_2^2}{\sqrt{1-\frac{t}{T_0}}}$$E7

(see Lemma 39 from Ref. [3]).

If $T_{\star }$ is finite (see Ref. [3], Lemma 50, Theorems 6–7), then for these parameters, there are fulfilled inequalities: $\lambda <1,0<\epsilon <1$ and

$${\tau }^2\left(\epsilon \right)=\frac{1}{4}{\left(\epsilon +\frac{1}{\epsilon }\right)}^2<\mu <{\lambda }^{-4}.$$E8

From Leray’s estimates (see Ref. [8]), it follows that every blow up solution of problem (1) satisfies the condition:

$${\int }_0^{T_{\ast }}\parallel \nabla u\left(t,\cdot \right){\parallel }_2^4\mathit{dt}=+\infty$$E9

for finite $T_{\ast }$. Nevertheless, this weak solution for every $T<T_{\ast },T>0,$ satisfies the inequality:

$${\int }_0^T\parallel \nabla u\left(t,\cdot \right){\parallel }_2^4\mathit{dt}<+\infty ,$$E10

that implies solution smoothness on set $\left[0,T\right]\times R^3$ (see Ref. [2].)

3. Main results

If initial data $\phi \in C_{6/5,3/2}^{\infty }\left(R^3\right)$ and parameter $\lambda \ge 1$, then Cauchy problem (1) has a global regular solution (see Theorem 7 from Ref. [3]). In other words, we have no any collapse in this case. Note, from the definition of parameter $\lambda$, condition $\lambda \ge 1$ means that the rate of change of kinetic energy square at initial point $t=0$ is negligible. It implies the determinism from the begining. If it is not there, we introduce new parameters and tools. Therefore, the following two results are very useful to compare with formula (9). The first of them it gives the sufficient condition of determinism by another tool. Moreover, it is important to note here that we do not require finiteness of mixed norm

$$\parallel u{\parallel }_{p,q}={\left({\int }_0^T\parallel u\left(t,\cdot \right){\parallel }_p^q\mathit{dt}\right)}^{1/q},p\ge 3,q\ge 2,3/p+2/q\le 1.$$E11

over time interval $\left[0,T\right]$, where Lebesgue norm is defined by formula (2). Every weak solution of problem (1) with finite mixed norm is smooth and unique (see Refs. [1, 2, 4, 6]).

Theorem 1. Let $\phi \in C_{6/5,3/2}^{\infty }\left(R^3\right)$ be initial data of problem (1). Parameter $\lambda <1$ and mean $T_0$ are defined by formulae (3) and (6), respectively. Suppose that

$$\frac{1}{T_0}{\int }_0^{T_0}\parallel \nabla u\left(t,\cdot \right){\parallel }_2^4\mathit{dt}\le \parallel \nabla \phi {\parallel }_2^{4}ln\frac{1}{1-{\lambda }^4},$$E12

where $u$ is a smooth solution of problem (1) on the time interval $\left[0,T_0\right)$. Then, this solution has a global regular extension on the set $\left[0,\infty \right)\times R^3.$ Moreover, the following estimates are fulfilled:

$$\begin{array}{l}\parallel \nabla u\left(T_0,\cdot \right){\parallel }_2^2\le \frac{{\lambda }^2\parallel \nabla \phi {\parallel }_2^2}{\sqrt{1-{\lambda }^4}},\lambda \left(T_0\right)>1,\\ \parallel \nabla u\left(t,\cdot \right){\parallel }_2^2\le \frac{{\lambda }^2\left(T_0\right)}{{\lambda }^2\left(T_0\right)-1}\parallel \nabla u\left(T_0,\cdot \right){\parallel }_2^2\end{array}$$E13

for all $t>T_0$, where $\lambda \left(T_0\right)$ is defined in formula (3).

Proof. For the first time, we note that there exists a number $\xi \in \left(0,T_0\right]$ satisfying inequality

$$\parallel \nabla u\left(\xi ,\cdot \right){\parallel }_2^2\le \frac{{\lambda }^2\parallel \nabla \phi {\parallel }_2^2}{\sqrt{1-{\lambda }^4\frac{\xi }{T_0}}}.$$E14

Let us suppose the opposite. Then, on interval $\left[0,T_0\right]$, we have the following inequality:

$$\parallel \nabla u\left(t,\cdot \right){\parallel }_2^4>\frac{{\lambda }^4\parallel \nabla \phi {\parallel }_2^4}{1-{\lambda }^4\frac{t}{T_0}}.$$E15

Integrating it over this interval, we obtain the estimate contradicting theorem condition for the mean value.

Therefore, Eq. (14) is true. Rewriting it by the following way:

$$\frac{T_0}{{\lambda }^4}\le \xi +\frac{c{\nu }^3}{\parallel \nabla u\left(\xi ,\cdot \right){\parallel }_2^4}={\tau }_1\left(\xi \right),c={\left(\frac{9}{4}\right)}^4,$$E16

we note that, for all $t\ge \xi ,t<T_{\ast }$, ${\tau }_1\left(\xi \right)\le {\tau }_1\left(t\right)$ because function ${\tau }_1$ is not decreasing (see Lemma 45, formula (85) from Ref. [3]).

Therefore, if $T_{\ast }$ is finite, then we obtain inequality $\frac{1}{{\lambda }^4}\le \mu$. This contradicts to inequality (8). Hence, $\mu =\infty$. Then, solution $u$ of problem (1) is global and regular.

From formula (16) and monotonicity of function ${\tau }_1$, it follows immediately that, for all $t,\xi \le t<T_{\ast }$, the next inequality is fulfilled:

$$\parallel \nabla u\left(t,\cdot \right){\parallel }_2^2\le \frac{{\lambda }^2\parallel \nabla \phi {\parallel }_2^2}{\sqrt{1-{\lambda }^4\frac{t}{T_0}}}.$$E17

Hence, we have the first inequality of Theorem 1.

Let us prove the second estimate. Suppose the opposite. Then, $\lambda \left(T_0\right)\le 1$. In this case, for solution $u\left(t,x\right)$_, function ${\tau }_2\left(t\right)=\parallel u\left(t,\cdot \right){\parallel }_2^2\left({\lambda }^2\left(t\right)-1\right)$ is not decreasing function (see Ref. [3], inequality (77)). Hence, for $0\le t\le T_0$_, we obtain estimates:

$$\parallel u\left(t,\cdot \right){\parallel }_2^2\left({\lambda }^2\left(t\right)-1\right)\le \parallel u\left(T_0,\cdot \right){\parallel }_2^2\left({\lambda }^2\left(T_0\right)-1\right)\le 0.$$E18

Then, $\lambda \left(t\right)\le 1.$

Hence, for all $t$, $0\le t\le T_0$, we have the estimate:

$$4c{\mathrm{\nu }}^4\le \parallel u\left(t,\cdot \right){\parallel }_2^2\parallel \nabla u\left(t,\cdot \right){\parallel }_2^2$$E19

where constant $c$ from Eq. (16). It is the strong inequality in some neighborhood of point $t=0$ because $\lambda \left(0\right)=\lambda <1$ and functions ${\eta }_1\left(t\right)=\parallel \nabla u\left(t,\cdot \right){\parallel }_2,{\eta }_4\left(t\right)=\parallel u\left(t,\cdot \right){\parallel }_2$ are continuous (see Lemma 36 in Ref. [3]).

Therefore, integrating Eq. (19) over interval $\left[0,T_0\right]$, we extract a strong estimate:

$$4c{\mathrm{\nu }}^4{T}_0<{\int }_0^{T_0}\parallel u\left(t,\cdot \right){\parallel }_2^2\parallel \nabla u\left(t,\cdot \right){\parallel }_2^2\mathit{dt}=\frac{1}{4\nu }\left(\parallel \phi {\parallel }_2^4\parallel -\parallel u\left(T_0,\cdot \right){\parallel }_2^4\right).$$E20

Hence,

$$\parallel u\left(T_0,\cdot \right){\parallel }_2^4<\parallel \phi {\parallel }_2^4\left(1-{\lambda }^4\right).$$E21

Apply this inequality and inequality (17) for $t=T_0$, then from (19), we obtain the strong estimate

$$4c{\nu }^4<\parallel \phi {\parallel }_2^2\parallel \nabla \phi {\parallel }_2^2{\lambda }^2=4c{\nu }^4.$$E22

Contradiction. The second inequality is proved.

The third estimate follows from Theorem 10 (see Ref. [3]) if we consider $u\left(t+T_0,x\right)$ as the Cauchy problem solution with initial data $u\left(T_0,x\right)$. Theorem 1 is proved.

The following statement is connected with a local extension. The model is the same as above (see Theorem 1).

Theorem 2. Let $\phi \in C_{6/5,3/2}^{\infty }\left(R^3\right)$ be initial data in problem (1). Parameter $\lambda <1$ and mean $T_0$ are defined by formulae (3) and (6), respectively. Suppose that

$$\frac{1}{T_0}{\int }_0^{T_0}\parallel \nabla u\left(t,\cdot \right){\parallel }_2^4\mathit{dt}\le \parallel \nabla \phi {\parallel }_2^{4}ln\frac{1}{1-{\lambda }^2},$$E23

where $u$ is a smooth solution of problem (1) on time interval $\left[0,T_0\right)$. Then, this solution has a local smooth extension on set $\left[0,\frac{T_0}{{\lambda }^2}\right)\times R^3.$ In addition, it is true the following estimate:

$$\parallel \nabla u\left(t,\cdot \right){\parallel }_2^2\le \frac{\lambda \parallel \nabla \phi {\parallel }_2^2}{\sqrt{1-{\lambda }^2\frac{t}{T_0}}}$$E24

for every moment $t,T_0\le t<\frac{T_0}{{\lambda }^2}$.

Proof. This theorem is proved by the same way as Theorem 1. Here, there exists a number $\xi \in \left(0,T_0\right]$ satisfying condition

$$\parallel \nabla u\left(\xi ,\cdot \right){\parallel }_2^2\le \frac{\lambda \parallel \nabla \phi {\parallel }_2^2}{\sqrt{1-{\lambda }^2\frac{\xi }{T_0}}}.$$E25

As in Theorem 1 above, we prove it from the opposite. The monotonicity of function ${\tau }_1$ from proof of this theorem implies inequality:

$$\frac{T_0}{{\lambda }^2}\le \xi +\frac{c{\nu }^3}{\parallel \nabla u\left(\xi ,\cdot \right){\parallel }_2^4}={\tau }_1\left(\xi \right)\le {\tau }_1\left(t\right),\xi <t.$$E26

Hence, we have estimate (24). Theorem 2 is proved.

Remark. The number $\frac{T_0}{{\lambda }^2}=\frac{\parallel \phi {\parallel }_2^2}{4\nu \parallel \nabla \phi {\parallel }_2^2}$ is interesting because it does not depend on from constants in apriori estimates for solutions. This is the first. Second, it influences on estimates for kinetic energy of turbulence flows at moment close to initial (see Ref. [9], Theorem 1).

Theorem 3. Let $\phi \in C_{6/5,3/2}^{\infty }\left(R^3\right)$ be initial data in problem (1). Parameter $\lambda <1$ and $\sqrt{2}{\lambda }^2\ge 1$. In addition, number $T_0$ and parameter $\epsilon$ are defined by formulae (5) and (6), respectively. Suppose $\epsilon \le \sqrt{2}-1$. Then, a weak solution $u$ of problem (1) is global and regular.

Proof. If solution $u$ of problem (1) is blow up solution, then from Eq. (8) and theorem conditions, we obtain for parameter $\mu$ following inequalities: $\mu <{\lambda }^{-4}\le 2$. Hence, and still one estimate from Eq. (8), we have inequality: $\frac{1}{2}\left(\epsilon +\frac{1}{\epsilon }\right)<\sqrt{2}$.

Therefore, $\epsilon >\sqrt{2}-1$. Contradiction. Theorem 3 is proved.

Theorem 4. Let $\phi \in C_{6/5,3/2}^{\infty }\left(R^3\right)$ be initial data in problem (1). Parameters $\lambda <1$ and $\epsilon$ are defined by formulae (3) and (6), respectively. Suppose that number $T_0$ from (5) and a weak solution $u$ of problem (1) satisfies inequality

$$\parallel u\left(\frac{T_0}{{\lambda }^4},\cdot \right){\parallel }_2^2\ge \parallel \phi {\parallel }_2^2\left(1-\tau \left(\epsilon \right){\lambda }^2\right),$$E27

where function $\tau \left(\epsilon \right))$ from Eq. (8). Then, solution $u$ is global and regular.

Proof. Suppose the opposite. Then,

$$T_{\ast }<\frac{T_0}{{\lambda }^4}$$

and

$$\parallel u\left(T_{\ast },\cdot \right){\parallel }_2^2\ge \parallel \phi {\parallel }_2^2\left(1-\tau \left(\epsilon \right){\lambda }^2\right).$$E28

On the other hand, we have

$$\parallel u\left(T_{\ast },\cdot \right){\parallel }_2^2\le \parallel \phi {\parallel }_2^2\left(1-\sqrt{\mu }{\lambda }^2\right).$$E29

(see Ref. [3], Lemma 49). Comparing them, we obtain the contradiction with (8). Theorem 4 is proved.

Theorems 3 and 4 describe the influence on the appearance of collapse by kinetic energy estimates. For application of them, the other facts are considered lower.

4. Kinetic energy estimates and control of solution smoothness

Now, we consider the effect of kinetic energy estimates on the lifetime of smooth solution $u$ in problem (1). Choose $T_0$ from Eq. (6), $t\in \left(0,T_0\right)$ and integrate (7) over $\left[0,t\right]$. Then, we obtain

$$\parallel u\left(t,\cdot \right){\parallel }_2^2\ge \parallel \phi {\parallel }_2^2\left(1-{\lambda }^2+{\lambda }^2\sqrt{1-\frac{t}{T_0}}\right)\ge \parallel \phi {\parallel }_2^2\left(1-{\lambda }^2\sqrt{\frac{t}{T_0}}\right)$$E30

(see Lemma 41 in Ref. [3]). If mean $\parallel u\left(T_0,\cdot \right){\parallel }_2^2$ is not close to minimum, then we have no blow up solution. This is described by the following theorem.

Theorem 5. Let $\phi \in C_{6/5,3/2}^{\infty }\left(R^3\right)$ be initial data in problem (1). Parameter $\lambda <1$ and mean $T_0$ are defined by formulae (3) and (6), respectively. Suppose that

$$\parallel u\left(T_0,\cdot \right){\parallel }_2^2\ge \parallel \phi {\parallel }_2^2\sqrt{1-{\lambda }^4},$$E31

where $u$ is a smooth solution of problem (1) on the time interval $\left[0,T_0\right)$. Then, solution $u$ can be extended as the global and regular.

Proof. Suppose the opposite. Then, for every $t\in \left[0,T_0\right)$_, it is fullfilled inequality $\lambda \left(t\right)<1$ (see (3)) for finitite mean $T_{\ast }$ (see Ref. [3], Lemma 50, Theorems 6–7). Then, we have (19)) where the inequality is strong for all $t$. Hence, integrating it over $\left[0,T_0\right)$_, we obtain (21). This contradicts the condition from Theorem 5. It is proved.

For the first time, this statement is given in Ref. [3].

The following theorem shows lifetime of smooth solution in problem (1) if the kinetic energy dissipation is increasing.

Theorem 6. Let $\phi \in C_{6/5,3/2}^{\infty }\left(R^3\right)$ be initial data in problem (1). Parameter $\lambda <1$ and mean $T_0$ are defined by formulae (3) and (6), respectively. Suppose that

$$\parallel u\left(T_0,\cdot \right){\parallel }_2^2\ge \parallel \phi {\parallel }_2^2\left(1-\lambda +\lambda \sqrt{1-{\lambda }^2}\right),$$E32

where $u$ is a smooth solution of problem (1) on the time interval $\left[0,T_0\right)$. Then, solution $u$ can be extended a local and regular on set $[0,\frac{T_0)}{{\lambda }^2}\times R^3$.

Proof. We introduce a function

$$\omega \left(t\right)=\parallel u\left(t,\cdot \right){\parallel }_2^2-\parallel \phi {\parallel }_2^2\left(1-\lambda +\lambda \sqrt{1-{\lambda }^2\frac{t}{T_0}}\right).$$E33

Since function $\omega$ is smooth and $\omega \left(0\right)=0,\omega \left(T_0\right)\ge 0$_, then there exists a number $\xi \in \left(0,T_0\right)$ satisfying condition ${\omega }^{\text{'}}\left(\xi \right)\ge 0$. Hence, it follows inequality (25), which implies (26). From monotonicity of function ${\tau }_1$ in formula (26), we get the main statement. Theorem 6 is proved.

Finally, for illustration, we show some facts due to blow up solutions and kinetic energy (other aspects connecting with estimates of norm $\parallel u\left(t,\cdot \right){\parallel }_p,p\ge 3,$ are covered in Refs. [4, 5, 8]).

Theorem 7. Let $\phi \in C_{6/5,3/2}^{\infty }\left(R^3\right)$ be initial data. Suppose parameter $\lambda <1$ and $u$ is blow up solution of problem (1). If there exists a number $t_0$ from interval $\left(0,{\tau }^2\left(\epsilon \right)T_0\right)$ with condition

$$\parallel u\left(t_0,\cdot \right){\parallel }_2^2=\parallel \phi {\parallel }_2^2\sqrt{1-\frac{t_0}{\mu T_0}},$$E34

where function $\tau =\tau \left(\epsilon \right)$ from Eq. (8). Then, parameter $\mu$ from formula (4) satisfies the estimate:

$$\mu \ge \frac{1}{{\lambda }^2\left(2-{\lambda }^2\right)},$$E35

that is, solution $u$ is regular on strip domain $\left[0,\frac{T_0}{{\lambda }^2\left(2-{\lambda }^2\right)}\right)\times R^3$.

The proof is given in Ref. [9], Theorem 1.3.

Here, we must note the critical level of kinetic energy. Naturally, it is described by equality

$$\parallel u\left(t_0,\cdot \right){\parallel }_2^2=\parallel \phi {\parallel }_2^2\left(1-{\lambda }^2\sqrt{\frac{t_0}{T_0}}\right)$$E36

because the last inequality in Eq. (30) and the following theorem are true (see Ref. [3], Theorem 8).

Theorem 8. Let $\phi \in C_{6/5,3/2}^{\infty }\left(R^3\right)$ be initial data. Suppose parameter $\lambda <1$ and a vector field $u$ is a solution of problem (1), on an interval $\left[0,T\right)$, inequality

$$\parallel u\left(t,\cdot \right){\parallel }_2^2\ge \parallel \phi {\parallel }_2^2\left(1-{\lambda }^2\sqrt{\frac{t}{T_0}}\right)$$E37

is fullfilled. Then, the weak solutions $u\left(t,x\right)=\left(u_1\left(t,x\right),u_2\left(t,x\right),u_3\left(t,x\right)\right)$ and $P=P\left(t,x\right)$ are regular on strip domain $\left[0,T\right)\times R^3$.

Proof. Suppose $T_{\ast }<T$. Then, from condition, it follows

$$\parallel u\left(T_{\ast },\cdot \right){\parallel }_2^2\ge \parallel \phi {\parallel }_2^2\left(1-\sqrt{\mu }{\lambda }^2\right).$$E38

Blow up solution of problem (1) satisfies inequalities:

$$\parallel \phi {\parallel }_2^2{\lambda }^2\sqrt{\mu -\frac{t}{T_0}}+\parallel u\left(T_{\ast },\cdot \right)\parallel \le \parallel u\left(t,\cdot \right){\parallel }_2^2\le \parallel \phi {\parallel }_2^2\left(1-\sqrt{\mu }{\lambda }^2+{\lambda }^2\sqrt{\mu -\frac{t}{T_0}}\right)$$E39

(see Ref. [3], Lemma 49). Comparing these estimates, we get identity:

$$\parallel u\left(t,\cdot \right){\parallel }_2^2=\parallel \phi {\parallel }_2^2\left(1-\sqrt{\mu }{\lambda }^2+{\lambda }^2\sqrt{\mu -\frac{t}{T_0}}\right).$$E40

Differentiate this identity at point $t=0$. After simple calculations, we obtain equality $\mu =1$. This contradicts to the left-hand inequality in Eq. (8) because dissipation parameter $\epsilon <1$ (see Ref. [3], Lemma 44). The smoothness pressure function $P$ follows from inclusion $u\left(t,\cdot \right)\in C_{6/5,3/2}^{\infty }$ (see Ref. [3], Theorem 6). The theorem is proved.

5. Conclusions

Summarizing above, we underline the first controlling parameter $\lambda$ defined at the initial time by formula (3). If this parameter $\lambda \ge 1$, then we have no blow up solution.

Next, if $\lambda <1$, then we introduce another controlling parameter $\epsilon$ which is defined at point time $T_0$ by formula (5). If at this point time, the kinetic energy satisfies the inequality

$$\parallel u\left(T_0,\cdot \right){\parallel }_2^2\ge \parallel \phi {\parallel }_2^2\sqrt{1-{\lambda }^4}$$E41

then we do not have phenomenon collapse again, as in the previous case. But if the kinetic energy changes in the boundaries

$$\parallel \phi {\parallel }_2^2\left(1-\lambda +\lambda \sqrt{1-{\lambda }^2}\right)\le \parallel u\left(T_0,\cdot \right){\parallel }_2^2<\parallel \phi {\parallel }_2^2\sqrt{1-{\lambda }^4}$$E42

then the guaranteed time interval without collapse is $\left[0,\frac{T_0}{{\lambda }^2}\right)$. In general, it is given by formula (8).

A more refined result is connected with the following lower estimate. If kinetic energy satisfies inequality

$$\parallel u\left(t,\cdot \right){\parallel }_2^2\ge \parallel \phi {\parallel }_2^2\left(1-{\lambda }^2\sqrt{\frac{t}{T_0}}\right)$$E43

for every $t,0\le t\le T,$ then the weak solution $u$ of problem (1) is regular on time interval $\left[0,T\right)$. Now, no examples where kinetic energy could satisfy the strong opposite inequality:

$$\parallel u\left(t,\cdot \right){\parallel }_2^2<\parallel \phi {\parallel }_2^2\left(1-{\lambda }^2\sqrt{\frac{t}{T_0}}\right)$$E44

yet. Therefore, it may be perspective for the further researches. This is the first.

Second, the control may not be point time-based, but on average. For example, if parameter $\lambda <1$ and for velocities gradient, the mean value satisfies the inequality

$$\frac{1}{T_0}{\int }_0^{T_0}\parallel \nabla u\left(t,\cdot \right){\parallel }_2^4\mathit{dt}\le \parallel \nabla \phi {\parallel }_2^{4}ln\frac{1}{1-{\lambda }^4},$$E45

then we have no collapse. The existence of smooth solution on time interval $\left[0,\frac{T_0}{{\lambda }^2}\right)$ is described by the inequality of Theorem 2.

In other words, we are doing a subsequent smmothness control in the same way as we are finding optimality control in Bellman principle. Here, the controlling parameters may be various, and, therefore, the researches will be interesting also in this way.

The most important results are described by Theorems 1–2, 5–6, and 8.

Acknowledgments

No external funding was received for this study.

Conflict of interest

The author declares no conflict of interest.

Notes/thanks/other declarations

Dedicated to the bright memory of Academician Ju. G. Reshetnyak.

The author is grateful to all reviewers for their critical comments. Many thanks.