1.2 Exact solution of hydrodynamic equations

Fundamental turbulence problem was unsolved during many years by virtue of the absence of analytical, time-dependent, smooth-at-all-time solutions of the nonlinear hydrodynamic equations. A few exact solutions are known in hydrodynamics, but none of these solutions is time-dependent and defined in unbounded space or in space with periodic boundary conditions [38–40].

The importance of this problem is determined by stability and predictability problems in all fields of science where solutions and methods of hydrodynamics are used. In this connection in 2000, the problem of the existence of smooth timedependent hydrodynamic solutions was stated as one of the seven Millennium Prize Problems (MPPs) by the Clay Institute of Mathematics [27]. MPPs relate only to incompressible flows "since it is well known that the behavior of compressible flows is abominable" [41].

Here we show that even for a compressible case, it is possible to obtain exact analytical, time-dependent, smooth-at-all-time solutions of Hopf equation (4) (which gives also new class solution also for vortex typ. 2D and 3D Euler equation) when any viscosity of super threshold friction is taken into account [22–26].

With the aim to introduce effective volume viscosity (in addition to external friction in (4)), let us consider the n-dimensional Hopf equation (4) in the moving with velocity Við Þt coordinate system, where Við Þt is a random Gaussian deltacorrelated in-time velocity field for which the relations hold:

$$\begin{aligned} \left< V\_i(t) V\_j(\tau) \right> &= 2\nu \delta\_{\vec{\eta}} \delta(t - \tau) \\ \left< V\_i(t) \right> &= \mathbf{0} \end{aligned} \tag{9}$$

In (9) δij is the Kronecker delta, δ is Dirac-Heaviside delta function, and the coefficient ν characterizes the action of the viscosity forces. In the general case, the coefficient ν can be a function of time when describing the effective turbulent viscosity, but also it can coincide with the constant kinematic viscosity coefficient when the random velocity field considered corresponds to molecular fluctuations. We will restrict our attention to the consideration of the case of constant coefficient ν in (9).

Thus, the initial equation (4) (for the case μ ¼ 0) takes the form:

$$\frac{\partial \mu\_i}{\partial t} + \left(u\_j + V\_j(t)\right) \frac{\partial u\_i}{\partial \mathbf{x}\_j} = \mathbf{0} \tag{10}$$

As shown in Appendix, in the case of an arbitrary dimensionality of the space (n = 1, 2, 3, etc.), Eq. (10) has the following exact solution (see also [22–26]):

$$u\_i(\overrightarrow{\boldsymbol{\omega}},t) = \int d^n \xi \boldsymbol{u}\_{0i}(\overrightarrow{\boldsymbol{\xi}}) \delta \left( \overrightarrow{\boldsymbol{\xi}} - \overrightarrow{\boldsymbol{\kappa}} + \overrightarrow{B} \ (t) + t \overrightarrow{u\_0} \ (\overrightarrow{\boldsymbol{\xi}}) \right) \det \boldsymbol{\hat{A}} \tag{11}$$

where BiðÞ¼ <sup>t</sup> <sup>Ð</sup><sup>t</sup> 0 dt1Við Þ <sup>t</sup><sup>1</sup> , <sup>A</sup>^ � Anm <sup>¼</sup> <sup>δ</sup>nm <sup>þ</sup> <sup>t</sup> <sup>∂</sup>u0<sup>n</sup> <sup>∂</sup>ξ<sup>m</sup> , det <sup>A</sup>^ is the determinant of the matrix A^ , and u0<sup>i</sup> x !� � is an arbitrary smooth initial velocity field. The solution (11)

Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory… DOI: http://dx.doi.org/10.5772/intechopen.80813

satisfies Eq. (10) only at such times for which the determinant of the matrix A^ is positive for any values of the spatial coordinates det A^ > 0.

In the case of the potential initial velocity field, the solution (11) is potential for all successive instants of time, corresponding to a zero-vortex field. On the contrary, in the case of nonzero initial vortex field, the solution also determines the evolution of velocity with a nonzero vortex field. In [42] the potential solution to

the two-dimensional Hopf equation (4) (or when B ! ¼ 0 in (12)) was obtained only in the Lagrangian representation which also exactly follows from (11) for n = 2. It is important to understand that here in (11) we have a solution in Euler variables, which is firstly obtained in [22] for n = 2 and n = 3. From the solution of (10) or (4) in Lagrangian variables, it is unreal to obtain a solution of (4) or (10) in Euler variables. From the other side, it is easy to obtain a solution in Lagrangian variables if we have a solution in Euler variables as in (11).

For example, in the one-dimensional case (n = 1) in (11), we have det <sup>A</sup>^ <sup>¼</sup> <sup>1</sup> <sup>þ</sup> <sup>t</sup> du<sup>01</sup> dξ<sup>1</sup> , and the solution (11) coincides exactly with the solutions obtained in [43, 44]. The solution (11) can be obtained if we use the integral representation for the implicit solution of Eq. (10) in the form uk x !; t � � <sup>¼</sup> <sup>u</sup>0<sup>k</sup> <sup>x</sup> ! � <sup>B</sup> ! ðÞ�<sup>t</sup> t u! <sup>x</sup> !; t � � � � with the use of the Dirac delta function (see Appendix or [22, 23]).

After averaging over the random field Bið Þt (with the Gaussian probability density), from (11) we can obtain the exact solution in the form:

$$\langle u\_i \rangle = \int d^n \xi \mu\_{0i} \left( \overrightarrow{\xi} \right) \left| \det \hat{A} \right| \frac{1}{\left( 2 \sqrt{\pi \nu t} \right)^n} \exp \left[ -\frac{\left( \overrightarrow{\chi} - \overrightarrow{\xi} - t \overrightarrow{u\_0} \left( \overrightarrow{\xi} \right) \right)^2}{4 \nu t} \right] \tag{12}$$

As distinct from (11), the average solution (12) of Eq. (10) is already arbitrarily smooth on any unbounded time interval and not only providing the positiveness of the determinant of the matrix A^ .

If, on the other side, we neglect the viscosity forces when B ! ðÞ¼ t 0 in (11), the smooth solution (11) is defined, as was already noted, only under the condition det A^ > 0 [22–26] (see Appendix). This condition corresponds to a bounded time interval 0 ≤ t < t0, where the minimum limiting time t<sup>0</sup> of existence of the solution can be determined from the solution to the following nth-order algebraic equation (and successive minimization of the expression obtained, which depends on the spatial coordinates, with respect to these coordinates):

$$\begin{aligned} \det \hat{A}(t) &= \mathbf{1} + t \frac{d u\_{01}(\mathbf{x}\_{1})}{d \mathbf{x}\_{1}} = \mathbf{0}, \boldsymbol{n} = \mathbf{1} \\\\ \det \hat{A}(t) &= \mathbf{1} + t d \dot{v} \vec{u\_{0}} + t^{2} \det \hat{U}\_{012} = \mathbf{0}, \boldsymbol{n} = \mathbf{2} \end{aligned} \tag{13}$$

$$\det \hat{A}(t) = \mathbf{1} + t d \dot{v} \vec{u\_{0}} + t^{2} \left( \det \hat{U}\_{012} + \det \hat{U}\_{013} + \det \hat{U}\_{023} \right) + t^{3} \det \hat{U}\_{02} = \mathbf{0}, \boldsymbol{n} = \mathbf{3} \tag{14}$$

where det <sup>U</sup>^ <sup>0</sup> is the determinant of the 3 � 3 matrix <sup>U</sup>0nm <sup>¼</sup> <sup>∂</sup>u0<sup>n</sup> <sup>∂</sup>xm , and det <sup>U</sup>^ <sup>012</sup> <sup>¼</sup> <sup>∂</sup>u<sup>01</sup> ∂x<sup>1</sup> ∂u<sup>02</sup> <sup>∂</sup>x<sup>2</sup> � <sup>∂</sup>u<sup>01</sup> ∂x<sup>2</sup> ∂u<sup>02</sup> <sup>∂</sup>x<sup>1</sup> is the determinant of a similar matrix in the twodimensional case for the variables ð Þ <sup>x</sup>1; <sup>x</sup><sup>2</sup> . In this case det <sup>U</sup>^ <sup>013</sup>, det <sup>U</sup>^ <sup>023</sup> are the determinants of the matrices in the two-dimensional case for the variables ð Þ x1; x<sup>3</sup> and ð Þ x2; x<sup>3</sup> , respectively.

high-frequency EMF waves. The present new VCR theory gives this possibility due to the transformation of a longitudinal Bose-condensed plasmon into transverse VCR photon, during the scattering of a plasmon on the relativistic electron [14, 37]. Moreover in this new VCR theory, the VCR phenomenon has the same nature as

Fundamental turbulence problem was unsolved during many years by virtue of

The importance of this problem is determined by stability and predictability problems in all fields of science where solutions and methods of hydrodynamics are used. In this connection in 2000, the problem of the existence of smooth timedependent hydrodynamic solutions was stated as one of the seven Millennium Prize Problems (MPPs) by the Clay Institute of Mathematics [27]. MPPs relate only to incompressible flows "since it is well known that the behavior of compressible flows

Here we show that even for a compressible case, it is possible to obtain exact analytical, time-dependent, smooth-at-all-time solutions of Hopf equation (4) (which gives also new class solution also for vortex typ. 2D and 3D Euler equation) when any viscosity of super threshold friction is taken into account [22–26]. With the aim to introduce effective volume viscosity (in addition to external friction in (4)), let us consider the n-dimensional Hopf equation (4) in the moving with velocity Við Þt coordinate system, where Við Þt is a random Gaussian delta-

Við Þ<sup>t</sup> Vjð Þ<sup>τ</sup> � � <sup>¼</sup> <sup>2</sup>νδijδð Þ <sup>t</sup> � <sup>τ</sup>

In (9) δij is the Kronecker delta, δ is Dirac-Heaviside delta function, and the coefficient ν characterizes the action of the viscosity forces. In the general case, the coefficient ν can be a function of time when describing the effective turbulent viscosity, but also it can coincide with the constant kinematic viscosity coefficient when the random velocity field considered corresponds to molecular fluctuations. We will restrict our attention to the consideration of the case of constant coefficient ν in (9).

<sup>þ</sup> uj <sup>þ</sup> Vjð Þ<sup>t</sup> � � <sup>∂</sup>ui

As shown in Appendix, in the case of an arbitrary dimensionality of the space (n = 1, 2, 3, etc.), Eq. (10) has the following exact solution (see also [22–26]):

> � x ! <sup>þ</sup> <sup>B</sup> !

dt1Við Þ <sup>t</sup><sup>1</sup> , <sup>A</sup>^ � Anm <sup>¼</sup> <sup>δ</sup>nm <sup>þ</sup> <sup>t</sup> <sup>∂</sup>u0<sup>n</sup>

∂xj

� � � �!

ðÞþt tu<sup>0</sup>

is an arbitrary smooth initial velocity field. The solution (11)

! ξ

Thus, the initial equation (4) (for the case μ ¼ 0) takes the form:

∂ui ∂t

h i Við Þ<sup>t</sup> <sup>¼</sup> <sup>0</sup> (9)

¼ 0 (10)

<sup>∂</sup>ξ<sup>m</sup> , det <sup>A</sup>^ is the determinant of the

det A^ (11)

the absence of analytical, time-dependent, smooth-at-all-time solutions of the nonlinear hydrodynamic equations. A few exact solutions are known in hydrodynamics, but none of these solutions is time-dependent and defined in unbounded

for numerous physical systems where dissipative instability is realized when corresponding excitations in a medium become energetically favorable at some

super threshold conditions [12, 32–36].

is abominable" [41].

ui x !; t � �

where BiðÞ¼ <sup>t</sup> <sup>Ð</sup><sup>t</sup>

matrix A^ , and u0<sup>i</sup> x

58

¼ ð dn ξu0<sup>i</sup> ξ � �! δ ξ!

0

!� �

1.2 Exact solution of hydrodynamic equations

Nonlinear Optics ‐ Novel Results in Theory and Applications

space or in space with periodic boundary conditions [38–40].

correlated in-time velocity field for which the relations hold:

In the two-dimensional case, the condition in the form of Eq. (13) exactly coincides with the collapse condition obtained in [42] in connection with the problem of propagation of a flame front investigated on the basis of the Kuramoto-Sivashinsky Eq. (3). In this case for exact coincidence, it is necessary to replace <sup>t</sup> ! b tðÞ¼ Us exp <sup>γ</sup><sup>0</sup> ð Þ ð Þ�<sup>t</sup> <sup>1</sup> γ0 in (13).

dI dt � �

when div u<sup>0</sup>

with div u<sup>0</sup>

(where x

B !

formula (16):

the Euler variables.

(when <sup>t</sup> <sup>¼</sup> 0) to <sup>τ</sup> <sup>¼</sup> <sup>1</sup>

of the inequality

condition (13).

61

! <sup>¼</sup> <sup>x</sup>

! t; a ! � � <sup>¼</sup> <sup>a</sup>

t¼0

DOI: http://dx.doi.org/10.5772/intechopen.80813

! ¼ 0 in (13) when n = 2.

¼ � <sup>ð</sup> d3

mum time of implementation of the collapse <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup> ffiffiffiffiffiffiffi

! <sup>þ</sup>tu<sup>0</sup> ! a

matrix of the first derivatives of the velocity <sup>U</sup>^ im <sup>¼</sup> <sup>∂</sup>ui

U^ im a !; t

> <sup>∂</sup>u xð Þ ; <sup>t</sup> ∂x � �

x div u<sup>0</sup>

! det <sup>2</sup>

Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory…

In fact, in accordance with this criterion proposed in [45], the collapse of the solution is not possible in the case of the initial divergence-free velocity field, i.e.,

pressible flow, the initial condition corresponded just to the initial velocity field

On the basis of the solution (11), using (13) and the Lagrangian variables a

� � <sup>¼</sup> <sup>U</sup>^ <sup>0</sup>ik <sup>a</sup>

In this case the expression (16) exactly coincides with the formula (30) given in [45] for the Lagrangian time evolution of the matrix of the first derivatives of the velocity which must satisfy the three-dimensional Hopf equation (10) (when

ðÞ¼ t 0 in (10)). In particular, in the one-dimensional case when n = 1, in the Lagrangian representation from (11) and (13), we obtain a particular case of the

x¼x að Þ ;t

The solution (17) also coincides with the formula (14) in [45] and describes the catastrophic process of collapse of a simple wave in a finite time t<sup>0</sup> whose estimate is given above on the basis of the solution to Eq. (13) in the case n ¼ 1 with the use of

Let us take into account only the external friction. For this purpose it is necessary to consider the case with μ > 0 in Eq. (4). In this case we can also obtain the

[22, 23]). The new time variable τ now varies within the finite limits from τ ¼ 0

μ > 1 t0

for given initial conditions, the quantity det A^ > 0 for all times since the necessary and sufficient condition of implementation of the singularity (13) will be not satisfied because the change t ! τð Þt must also be carried out in the

where a is the coordinate of a fluid particle at the initial time t ¼ 0.

exact solution from the expression (11) (for the case when in (11) B

in them the time variable <sup>t</sup> by the variable <sup>τ</sup> <sup>¼</sup> <sup>1</sup>� exp ð Þ �t<sup>μ</sup>

¼

du0ð Þ a da <sup>1</sup> <sup>þ</sup> <sup>t</sup> du0ð Þ <sup>a</sup> da

<sup>μ</sup> (as t ! ∞). This leads to the fact that in the case of fulfillment

! ¼ 0. However, in this case the violation of criterion (15) does not exclude the possibility of the collapse of the solution by virtue of the fact that the criterion (15) does not determine the necessary condition of implementation of the collapse. Actually, in the example considered above (in determination of the mini-

<sup>U</sup>^ <sup>0</sup> <sup>&</sup>gt; <sup>0</sup>; I <sup>¼</sup>

L1L<sup>2</sup> p

!� �), we can represent the expression for the

!� �A�<sup>1</sup> km a !; t

<sup>∂</sup>xm in the form:

ð d3 x det <sup>2</sup>

U^ (15)

!

(17)

<sup>2</sup><sup>a</sup> ) for two-dimensional com-

� � (16)

!

<sup>μ</sup> (see (31) in Appendix and

¼ 0) changing

(18)

In the one-dimensional case, when n = 1, from Eq. (13) we can obtain the minimum time of appearance of the singularity <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup> max du<sup>01</sup> <sup>x</sup>ð Þ<sup>1</sup> dx1 � � � � � � > 0. In particular,

for the initial distribution <sup>u</sup>01ð Þ¼ <sup>x</sup><sup>1</sup> <sup>a</sup> exp � <sup>x</sup><sup>2</sup> 1 L2 � �, a <sup>&</sup>gt; 0, it follows that <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>L</sup> a ffiffi e 2 p obtained for the value <sup>x</sup><sup>1</sup> <sup>¼</sup> <sup>x</sup>1max <sup>¼</sup> <sup>L</sup>ffiffi 2 <sup>p</sup> . In this case the singularity itself can be implemented only for positive values of the coordinate x<sup>1</sup> > 0 when Eq. (13) has a positive solution for time.

This means that the singularity (collapse) of the smooth solution can never occur when the initial velocity field is nonzero only for negative values of the spatial coordinate x<sup>1</sup> < 0.

Similarly, we can also determine the vortex wave burst time t<sup>0</sup> for n > 1. For (13) in the two-dimensional case (when the initial velocity field is divergence-free) for the initial stream function in the form <sup>ψ</sup>0ð Þ¼ <sup>x</sup>1; <sup>x</sup><sup>2</sup> <sup>a</sup> ffiffiffiffiffiffiffiffiffiffi L1L<sup>2</sup> <sup>p</sup> exp � <sup>x</sup><sup>2</sup> 1 L2 1 � x2 2 L2 2 � �, a <sup>&</sup>gt; 0, we obtain that the minimum time of existence of the smooth solution is equal to <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup> ffiffiffiffiffiffiffi L1L<sup>2</sup> p <sup>2</sup><sup>a</sup> .

In the example considered, this minimum time of existence of the smooth solution is implemented for the spatial variables corresponding to points on the ellipse <sup>x</sup><sup>2</sup> 1 L2 þ x2 2 L2 ¼ 1.

1 2 In accordance with (13), the necessary condition of implementation of the singularity is the condition of existence of a real positive solution to a quadratic (when n = 2) or cubic (when n = 3) equation for the time variable t. For example, in the case of two-dimensional flow with the initial divergence-free velocity field div u<sup>0</sup> ! ¼ 0, in accordance with (13), the necessary and sufficient condition of implementation of the singularity (collapse) of the solution in finite time is the condition:

$$\det U\_{012} < 0 \tag{14}$$

For the example considered above from (14), there follows the inequality x2 1 L2 1 þ x2 1 L2 2 > <sup>1</sup> 2 . When this inequality is satisfied, for n = 2 there exists a real positive solution to the quadratic equation in (13) for which the minimum collapse time <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup> ffiffiffiffiffiffiffi L1L<sup>2</sup> p <sup>2</sup><sup>a</sup> > 0 given above is obtained.

On the contrary, if the initial velocity field is defined in the form of a finite function which is nonzero only in the domain <sup>x</sup><sup>2</sup> 1 L2 1 þ x2 2 L2 2 ≤ <sup>1</sup> 2 , then the inequality (14) is violated, and the development of the singularity in a finite time turns out already to be impossible, and the solution remains smooth in unbounded time even regardless of the viscosity effects.

The condition of existence of a real positive solution of Eq. (13) (e.g., see (14)) is the necessary and sufficient condition of implementation of the singularity (collapse) of the solution, as distinct from the sufficient but not necessary integral criterion which was proposed in [45] (see formula (38) in [45]) and has the form:

Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory… DOI: http://dx.doi.org/10.5772/intechopen.80813

$$\left(\frac{dI}{dt}\right)\_{t=0} = -\int d^3x \,\mathrm{div}\,\overrightarrow{u\_0}\,\,\det^2\hat{U}\_0 > 0; I = \int d^3x \,\mathrm{det}^2\hat{U} \tag{15}$$

In fact, in accordance with this criterion proposed in [45], the collapse of the solution is not possible in the case of the initial divergence-free velocity field, i.e., when div u<sup>0</sup> ! ¼ 0. However, in this case the violation of criterion (15) does not exclude the possibility of the collapse of the solution by virtue of the fact that the criterion (15) does not determine the necessary condition of implementation of the collapse. Actually, in the example considered above (in determination of the minimum time of implementation of the collapse <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup> ffiffiffiffiffiffiffi L1L<sup>2</sup> p <sup>2</sup><sup>a</sup> ) for two-dimensional compressible flow, the initial condition corresponded just to the initial velocity field with div u<sup>0</sup> ! ¼ 0 in (13) when n = 2.

On the basis of the solution (11), using (13) and the Lagrangian variables a ! (where x ! <sup>¼</sup> <sup>x</sup> ! t; a ! � � <sup>¼</sup> <sup>a</sup> ! <sup>þ</sup>tu<sup>0</sup> ! a !� �), we can represent the expression for the matrix of the first derivatives of the velocity <sup>U</sup>^ im <sup>¼</sup> <sup>∂</sup>ui <sup>∂</sup>xm in the form:

$$
\hat{U}\_{im}\left(\overrightarrow{a},t\right) = \hat{U}\_{0ik}\left(\overrightarrow{a}\right)A\_{km}^{-1}\left(\overrightarrow{a},t\right) \tag{16}
$$

In this case the expression (16) exactly coincides with the formula (30) given in [45] for the Lagrangian time evolution of the matrix of the first derivatives of the velocity which must satisfy the three-dimensional Hopf equation (10) (when B ! ðÞ¼ t 0 in (10)). In particular, in the one-dimensional case when n = 1, in the Lagrangian representation from (11) and (13), we obtain a particular case of the formula (16):

$$\left(\frac{\partial u(\mathbf{x},t)}{\partial \mathbf{x}}\right)\_{\mathbf{x}=\mathbf{x}(a,t)} = \frac{\frac{du\_0(a)}{da}}{\mathbf{1} + t\frac{du\_0(a)}{da}}\tag{17}$$

where a is the coordinate of a fluid particle at the initial time t ¼ 0.

The solution (17) also coincides with the formula (14) in [45] and describes the catastrophic process of collapse of a simple wave in a finite time t<sup>0</sup> whose estimate is given above on the basis of the solution to Eq. (13) in the case n ¼ 1 with the use of the Euler variables.

Let us take into account only the external friction. For this purpose it is necessary to consider the case with μ > 0 in Eq. (4). In this case we can also obtain the exact solution from the expression (11) (for the case when in (11) B ! ¼ 0) changing in them the time variable <sup>t</sup> by the variable <sup>τ</sup> <sup>¼</sup> <sup>1</sup>� exp ð Þ �t<sup>μ</sup> <sup>μ</sup> (see (31) in Appendix and [22, 23]). The new time variable τ now varies within the finite limits from τ ¼ 0 (when <sup>t</sup> <sup>¼</sup> 0) to <sup>τ</sup> <sup>¼</sup> <sup>1</sup> <sup>μ</sup> (as t ! ∞). This leads to the fact that in the case of fulfillment of the inequality

$$
\mu > \frac{1}{t\_0} \tag{18}
$$

for given initial conditions, the quantity det A^ > 0 for all times since the necessary and sufficient condition of implementation of the singularity (13) will be not satisfied because the change t ! τð Þt must also be carried out in the condition (13).

In the two-dimensional case, the condition in the form of Eq. (13) exactly coincides with the collapse condition obtained in [42] in connection with the problem of propagation of a flame front investigated on the basis of the Kuramoto-Sivashinsky Eq. (3). In this case for exact coincidence, it is necessary to replace

In the one-dimensional case, when n = 1, from Eq. (13) we can obtain the

2

when the initial velocity field is nonzero only for negative values of the spatial

we obtain that the minimum time of existence of the smooth solution is equal to

In the example considered, this minimum time of existence of the smooth solution is implemented for the spatial variables corresponding to points on the

In accordance with (13), the necessary condition of implementation of the singularity is the condition of existence of a real positive solution to a quadratic (when n = 2) or cubic (when n = 3) equation for the time variable t. For example, in the case of two-dimensional flow with the initial divergence-free velocity field

! ¼ 0, in accordance with (13), the necessary and sufficient condition of implementation of the singularity (collapse) of the solution in finite time is the

For the example considered above from (14), there follows the inequality

solution to the quadratic equation in (13) for which the minimum collapse time

On the contrary, if the initial velocity field is defined in the form of a finite

violated, and the development of the singularity in a finite time turns out already to be impossible, and the solution remains smooth in unbounded time even regardless

The condition of existence of a real positive solution of Eq. (13) (e.g., see (14))

is the necessary and sufficient condition of implementation of the singularity (collapse) of the solution, as distinct from the sufficient but not necessary integral criterion which was proposed in [45] (see formula (38) in [45]) and has

<sup>2</sup><sup>a</sup> > 0 given above is obtained.

function which is nonzero only in the domain <sup>x</sup><sup>2</sup>

. When this inequality is satisfied, for n = 2 there exists a real positive

1 L2 1 þ x2 2 L2 2 ≤ <sup>1</sup> 2

implemented only for positive values of the coordinate x<sup>1</sup> > 0 when Eq. (13) has a

This means that the singularity (collapse) of the smooth solution can never occur

Similarly, we can also determine the vortex wave burst time t<sup>0</sup> for n > 1. For (13) in the two-dimensional case (when the initial velocity field is divergence-free) for

1 L2 � � max du<sup>01</sup> <sup>x</sup>ð Þ<sup>1</sup> dx1 � � �

<sup>p</sup> . In this case the singularity itself can be

L1L<sup>2</sup> <sup>p</sup> exp � <sup>x</sup><sup>2</sup>

det U<sup>012</sup> < 0 (14)

� � �

, a <sup>&</sup>gt; 0, it follows that <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>L</sup>

> 0. In particular,

1 L2 1 � x2 2 L2 2

� �

, then the inequality (14) is

a ffiffi e 2 p

, a > 0,

<sup>t</sup> ! b tðÞ¼ Us exp <sup>γ</sup><sup>0</sup> ð Þ ð Þ�<sup>t</sup> <sup>1</sup>

positive solution for time.

coordinate x<sup>1</sup> < 0.

<sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup> ffiffiffiffiffiffiffi L1L<sup>2</sup> p <sup>2</sup><sup>a</sup> .

ellipse <sup>x</sup><sup>2</sup> 1 L2 1 þ x2 2 L2 2 ¼ 1.

div u<sup>0</sup>

x2 1 L2 1 þ x2 1 L2 2 > <sup>1</sup> 2

condition:

<sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>e</sup> ffiffiffiffiffiffiffi L1L<sup>2</sup> p

the form:

60

of the viscosity effects.

γ0

in (13).

Nonlinear Optics ‐ Novel Results in Theory and Applications

for the initial distribution <sup>u</sup>01ð Þ¼ <sup>x</sup><sup>1</sup> <sup>a</sup> exp � <sup>x</sup><sup>2</sup>

obtained for the value <sup>x</sup><sup>1</sup> <sup>¼</sup> <sup>x</sup>1max <sup>¼</sup> <sup>L</sup>ffiffi

minimum time of appearance of the singularity <sup>t</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>

the initial stream function in the form <sup>ψ</sup>0ð Þ¼ <sup>x</sup>1; <sup>x</sup><sup>2</sup> <sup>a</sup> ffiffiffiffiffiffiffiffiffiffi

Providing (18), the solution to the n-dimensional EH equation is smooth on an unbounded interval of time t. The corresponding analytic vortical solution to the three-dimensional Navier–Stokes equation also remains smooth for any t≥0 if the condition (18) is satisfied [22–26].

The second example, which is represented here, also gives new perspectives on the basis of the new exact solution (in the Euler variables) for n-dimensional Hopf equation because this equation is known as the possible model for weak nonlinear optic problems [46]. The importance of the new solution is connected with its Euler form in dependence from space variables, which are not represented in the solution of the Burgers-Hopf equation well known before (see [45] and

Hydrodynamic Methods and Exact Solutions in Application to the Electromagnetic Field Theory…

The Appendix presents a procedure for deriving the exact solution of the 3D

When the external friction coefficient tends to zero in Eq. (4), μ ! 0, Eq. (4)

In the unbounded space, the general Cauchy problem solution for Eq. (19) under

In (20), δ is the Dirac delta function. Using known (see farther) properties of the delta function, it is possible to express the delta function in (20) with the help of an

In (21), the matrix A^ depends only on the initial velocity field and is as follows:

To infer (21), it is necessary to use the following delta-function property that is

δ ξ! �ξ<sup>0</sup> � � !

det <sup>∂</sup>Φ<sup>k</sup> ∂ξ<sup>m</sup> � � ξ ! ¼ξ<sup>0</sup> !

are defined from the solution of the equation

¼

Φ ! ξ0 � � !

� � � �

� x ! þtu<sup>0</sup> ! ξ � � � �!

> ∂u0<sup>k</sup> ξ � �!

> > ∂ξ<sup>m</sup>

� � � �

¼ 0 (24)

¼ 0 (19)

!� � may be obtained as follows (see also in

! þt u! x

det A^ � � �

� (21)

(22)

(23)

!; t � � � � (20)

� x

The Hopf equation in the n-dimensional space (n = 1..3) is as follows:

∂ui ∂t þ ul ∂ui ∂xl

! <sup>0</sup> x

> ð dn ξu0<sup>i</sup> ξ � �! δ ξ!

Eq. (19) may be represented in an implicit form as follows:

!; t � � � � � δ ξ!

> ! ξ � �! :

δ Φ ! ξ � � � �!

!

<sup>A</sup>^ � Akm <sup>¼</sup> <sup>δ</sup>km <sup>þ</sup> <sup>t</sup>

!; t � � � � <sup>¼</sup>

identity true for the very velocity field meeting Eq. (19):

! þt u! x

A. Exact solution of n-D Hopf equation (n = 1, 2, 3)

also coincides with the Hopf equation (19).

DOI: http://dx.doi.org/10.5772/intechopen.80813

! � t u! x

arbitrary smooth initial conditions u

δ ξ!

true for any smooth function Φ

In (23), the values ξ<sup>0</sup>

63

� x

others).

Hopf equation.

[22, 23]):

ui x !; t � � <sup>¼</sup> <sup>u</sup>0<sup>i</sup> <sup>x</sup>

Note that under the formal coincidence of the parameters μ ¼ �γ<sup>0</sup> (see the Sivashinsky equation (3) in Introduction), the equality τðÞ¼ t b tð Þ takes place providing the implementation of the singularity (13) when n = 2 and in accordance with the solution of the Kuramoto-Sivashinsky equation in [42] and the regularization of this solution for all times if (18) takes place.

Moreover the example of interesting prosperity for the direct application for solution (11) (see also (12)–(18)) may be done in the connection of the results [46], where the description of light propagation in a nonlinear medium on the basis of the Burgers-Hopf equation is done.

Indeed, in [46], the model of light propagation in weak nonlinear 3D Coul-Coul's medium with small action radii of nonlocality is represented. In [46], it was stated that in the geometric optic approach, this model is integrated and described by the Veselov-Novikov equation which has a 1D reduction in the form of the Burgers-Hopf equation. The last equation is considered in connection with nonlinear geometrical optics when 1D reduction is made for the case when the refractive index has no dependence on one of the space coordinates. It is important when the property of nonlinear wave finite-time breakdown for Burgers-Hopf solutions is considered in the application to the case of nonlinear geometrical optics. These solutions are useful for modeling of dielectrics which have impurities which induced sharp variations of the refractive index. Indeed, in the points of breakdown, the curvature of the light rays obtained discontinues property as it takes place at the boundary between different media [46].

In [46], the only hodograph method is used for the Burgers-Hopf (or Hopf equation which is obtained from the Burgers' equation in the limit of zero viscosity) equation solution in this connection. Thus the direct analytical description of the 1D–3D solutions to the Hopf equation in the form (11) gives the new possibility also for the nonlinear optic problem which is considered in [46]. For example, according to this solution, it is possible to obtain the important effect of avoidance of finite-time singularities when viscosity or friction forces are taken into account (when condition (18) takes place for the case of external friction).
