On solutions of a reduced model for the dynamical evolution of contact lines Текст научной статьи по специальности «Математика»

O. Agashin, O. Korelin

GENERAL METHODS OF DIGITAL PROCESSING OF THE SPEECH SIGNAL IN CASE OF ISOLATED WORDS RECOGNITION ROBLEM WITH DSP APPLICATION

Nizhny Novgorod state technical university n.a. R.E. Alexeev

Purpose Development and deployment the real-time speech recognition systems for controlling industrial or home electronics. Providing fast and secure solution based on energy effective and independent of any operating system hardware with small form factor.

Approach A theoretical framework is proposed combine distinct techniques and methods preferred for speech recognition. Optimization of finding end points and decision making algorithms provides the most efficient solution for interactive control of small systems.

Research limitations/implications The present study provides a starting-point for further research on the speech recognition problem. For example, Hidden Markov model, wavelets and neuron network could be involved for making more precise prediction.

Value Besides, new software library providing API for processing of digital signals developed during the research. It supplies the base functionality necessary for building a simple real-time recognition system coupled with DSP-based hardware.

Key words: isolated words, speech recognition, digital signal processing, search endpoints, hardware based solution.

МЕХАНИКА ЖИДКОСТИ, ГАЗА И ПЛАЗМЫ

D.E. Pelinovsky1,2, A.R. Giniyatullin1, Y.A. Panfilova1

ON SOLUTIONS OF A REDUCED MODEL FOR THE DYNAMICAL EVOLUTION

OF CONTACT LINES

Nizhny Novgorod state technical university n.a R.E. Alexeev, Nizhny Novgorod, Russia1 McMaster University, Hamilton, Ontario, Canada

Purpose: The goal of this study is to solve the linear advection--diffusion equation with a variable speed on a semiinfinite line. The variable speed is determined by an additional condition at the boundary, which models the dynamics of a contact line of a hydrodynamic flow at 180°contact angle.

Approach: The investigation is carried out by an application of Laplace transform in spatial coordinate. Properties of Green's function for the fourth-order diffusion equation are used in analysis of implicit solutions of the linear advection-diffusion equation.

Findings: We prove local existence of solutions of the initial-value problem associated with the set of over-determining boundary conditions in the form of the fractional power series in time variable. We also analyze the explicit solutions in the case of a constant speed to show that the inhomogeneous boundary condition induces change of convexity of the flow at the contact line in a finite time.

Key words: linear advection-diffusion equation, variable speed, contact line, Laplace transform, Green's function.

1. Introduction

Contact lines are defined by the intersection of the rigid and free boundaries of the flow. Flows with the contact line at 180° contact angle were discussed in [2, 4], where corresponding solutions of the Navier-Stokes equations were shown to have no physical meanings. Recently, a different approach based on the lubrication approximation and thin film equations was developed by Benilov & Vynnycky [1].

As a particularly simple model for the flow shown on Fig. 1, the authors of [1] derived the linear advection-diffusion equation for the free boundary h(x,t) of the flow:

dh dAh Tr, ,dh ^

— + —r = V(t)—, -x > 0, t > 0. (1)

dt dx dx

The contact line is fixed at x = 0 in the reference frame moving with the velocity -V(t) and is defined by the boundary conditions h(0,t) = 1 and hx(0,t) = 0. The flux conservation is expressed by

1 3

the boundary condition hxxx(0,t) = -— (set a = 3 in equations (5.12)-(5.13) in [1]).

We assume that h, hx, hxx ^ 0 as x^-ro: in fact, any constant value of h at infinity is allowed thanks to the invariance of the linear advection-diffusion equation (1) with respect to the shift and scaling transformations. With three boundary conditions at x = 0 and the decay conditions as x ^ ro, the initial-value problem for equation (1) is over-determined and the third (over-determining) boundary condition at x = 0 is used to find the dependence of V on t.

We shall consider the initial-value problem with the initial data h(x,0) = h0(x) for a suitable function h0. In particular, we assume that the profile h0(x) decays monotonically to zero as x^-ro

© Pelinovsky D.E., Giniyatullin A.R., Panfilova Y.A., 2012.

and that 0 is a non-degenerate maximum of h0 such that h0(0) = 1, h0(0) = 0, and h0(O) < 0, see

Fig. 1. If the solution h(x,t) losses monotonicity in x during the dynamical evolution, for instance, due to the value of hxx(0,t) crossing 0 from the negative side, then we say that the flow becomes non-physical for further times and the model breaks. Simultaneously, this may mean that the velocity V(t) blows up, as it is defined for sufficiently strong solutions of the advection-diffusion equation (1) by the pointwise equation:

hxoxx^O = V(t) М0Д

(2)

which follows by differentiation of (1) in x and setting x^ 0.

The main claim of [1] based on numerical computations of the reduced equation (1) as well as more complicated thin-film equations is that for any suitable h0, there is a finite positive time t0 such that V(t) ^ -ro and hxx(0,t) ^ 0- as t f t0. Moreover, it is claimed that V(t) behaves near the blowup time as the logarithmic function of t, e.g.

V(t) ~ C1 log t0 -1) + C2 as t Î to,

(3)

where C\, C2 are positive constants.

Fig. 1. Schematic picture of the flow between rigid boundaries

This paper is devoted to analytical studies of solutions of the advection-diffusion equation (1)

and the effects coming from the inhomogeneous boundary condition hxacx (0, t ) = -1 associated with

the flux conservation. In particular, we rewrite the evolution equation for the variable u = hx in the form

Ut + Uxxxx = V(t) Ux, X > 0, t > 0, subject to the boundary conditions at the contact line

(4)

u(0, t) = 0, uxx (0, t) = -1, Uxxx (0, t) = 0, t > 0,

(5)

where the boundary conditions uxxx(0,t) = hxxxx(0,t) = 0 follows from the boundary conditions h(0,t) = 1 and hx(0,t) = 0 as well as the advection-diffusion equation (1) as x ^ 0.

To simplify the problem, we shall also consider the model for given constant V(t) = V0 and drop the third over-determining boundary conditions at the contact line:

U + uxxxx = V0Ux, x > 0 t > 0

1 (6) u(0, t) = 0,u„(0, t) = -t > 0.

Both problems (4)-(5) and (6) are considered under the initial condition u(x,0) = u0(x) with

u0(0) = 0, u'0(0) < 0 ,and u0(0) = -1 as well as the decay condition u, ux, uxx^ 0 as x ^ ro.

Using Laplace transform in spatial coordinate and Green's function for the fourth-order diffusion equation, we derive an explicit solution of the boundary-value problem (6). In the case V0 = 0,

we show that the inhomogeneous boundary condition hxxx (0, t) = uxx (0, t) = -1 leads to the secular

growth of the boundary value hxx(0,t) = ux(0,t) to positive infinity as t ^ ro. As a result, even if hxx(0,t) < 0 initially, the convexity of the solution h(x,t) at the boundary x = 0 is lost in a finite time. In the case V0 < 0, we show that no secular growth is observed but the convexity of the solution at the boundary is still lost in a finite time. Applying the same method, we prove local existence of solutions of the original boundary-value problem (4)-(5) in the form of fractional power series in time variable t. This prepares us to tackle the original conjecture on the finite-time blow-up in the dynamical evolution of the reduced model (1), which is still left opened for forthcoming studies.

The remainder of this paper is organized as follows. Section 2 reports explicit solutions of the boundary-value problem (6) for V0 = 0 and V0 ^ 0. Section 3 gives the local existence result for the boundary-value problem (4)-(5). Appendix A reviews properties of Green's function for the fourth-order diffusion equation.

2. Solution for V(t) = V

Because the coefficient V(t) changes in time variable t in the framework of the original advec-tion-diffusion equation (1), the Laplace transform in time t is not a useful method for this problem. On the other hand, the boundary-value problem (1) is formulated on half-line, and hence we can use Laplace transform in space variable x:

TO

U(p, t) = J e~pxu(x, t)dx, p > 0. (7)

0

We shall develop this method to solve the boundary-value problem (6). The explicit solution of this problem will help us to analyze the effects of the inhomogeneous boundary condition

uxx (0, t) = -1 and the constant advection term V(t) = V0 on the temporal dynamics of the advection-

diffusion equation with the fourth-order diffusion. Let us denote the boundary values:

P(t) = ux(0,t), Y(t) = Uxxx(0,t). (8)

Using Laplace transform (7), we rewrite an evolution problem associated with the advection-diffusion equation (6):

Ut + pAu - V0pU = y(t) — p + p2P(t), t > 0,

1

(9)

U (p,0) = Uo(p),

where U0(p) is the Laplace transform of u0(x) = u(x,0). By using the variation of parameters, we obtain

U (p, t) = U0 (p)e~tp 4 +tVop + J e-t -s) p 4+(t -s)Vo p (r(s) -1 p + p2fi{s))ds. (10)

o 2

Using the inverse Laplace transform, we write this solution in the form:

1 c+iw

u(x,t) = ^ J epX-tp4+tVop(f e-pU(y)dy)dp

c-iw

* c+iw ( t / 1 Л ^

+ í e\ Íe~{t~s)pA+{t~S)VaP( -1 p + p2Pis) d dp,

c-iw V 0 V 2 ' /

(11)

where Re(c) > 0 so that the singularities of the integrand in the complex p-plane remain to the left of the contour of integration.

If t > 0 is finite, u0 e l}(R+), and P,y e IL^ (R+), Fubini's Theorem implies that the integration in p and in y, s can be interchanged. Let us introduce Green's function Gt(x) for the fourth-order diffusion equation (see Appendix A):

-| C+iw +w w

Gt (x) = — J epx-tp dp = — J e~tk4+lkxdk = - J e'tk' cos(kx)dk.

2/U . 2k /-,

c-iw -w 0

Using Green's function, we can rewrite the solution (11) in the implicit form:

w ^ t

u(x,t) = JGt(x + Vot - y)uo(y)dy -1JG't-S (x + Vo(t - s))ds

0 2 o

t

+ J [Gt_ (x + Vo (t - s))r(s) + G"-s (x + Vo (t - s))0(sj№-

(12)

0

The solution is said to be in the implicit form, because the functions P(t) and y(t) determined by the boundary conditions (8) are not specified yet.

We verify that lim u(x, t) = 0, no matter what P and y are, as long as they are bounded

x ^x

function of t. Indeed, by the Lebesgue's Dominated Convergence Theorem, we have

x

JGt(x + V0t -y)u0(y)dy ^ 0 as x ^ ro

0

if u0 g l}(R+), because Gt(x) ^ 0 as x ^ ro. On the other hand, the other three convolution integrals are bounded if P,y g LXoc (R+) and t > 0 is finite, because Gt, G't and G" have integrable singularities at t = 0. By the same Lebesgue's Dominated Convergence Theorem, these three integrals decay to zero as x ^ ro.

It follows from this construction that the only way to determine the functions P(t) and y(t) in the solution (12) is to use the boundary conditions at x = 0, e.g. the boundary conditions u(0,t) = 0 and ux(0,t) = P(t). In what follows, this step is performed separately for the cases of V0 = 0 and V0 ± 0.

2.1. Case V0 = 0

We rewrite the solution (12) for V0 = 0:

u( x, t) = x Gt (x - y)u0 (y)dy -1J G't-s (x)ds + J [Gt - s (x)y(s) + G- s (x)0(s)1ds- (13)

0 2 0 o

Using (A.3) and (A.4) for Green's function Gt(x) and the boundary condition u(0,t) = 0, we evaluate this expression at x = 0 and obtain an integral equation for p and y:

-—rf-1 í r(S]ld ds +—rí3]ds = ]üt(-y)u0(y)dy. (14)

4n ^4 J0 (t - s)1/4 4n ^4 Jo (t - s)3/4 o * 0

To use the boundary condition ux(0,t) = P(t), we shall recall from equation (A.5) that the function G'"(x) behaves like O(tA) for any x > 0 and hence is not integrable in t at t = 0. Therefore,

we have to be careful to differentiate the solution in the above convolution form. The last term of the solution (13) can be computed by using the Fourier transform:

t 1 „ , ft ,4.. A

v(x, t):=J G"t-s (x)P(s)ds = — J (ik)2elkxI J e"k (t-s) P(s)ds

0 2n -x, v0 J

Differentiating this expression in x and integrating by parts in s, we obtain

\dk.

1 I t 11 e i I t Л I \

(x, t) = — J (ik)3 eikx I J e-k4(t-s)P(s)ds dp = — J—I J—(ep(s)ds dp 2ж 1 ~ , 2m k I ^ ds

-œ V 0 У -œ V 0 У

1 +<» ikx I t 1

— J — P(t)-P(0)e k4t - Je k4(t-s)p'(s)ds dp (15)

2m ~k V о У

t

1 r

- P(t) - P(0)Ht (x) - J Ht-s (x)p'(s)ds,

0

where

1 +TO e~tk4 +7kx 1 TO e~tk4 cin(kv) X

Ht(*):=-^ J ^-dk = -J--dk = JGt(y)dy. (16)

2t7 -to k t o k o

Here we note that all integrals are evaluated in the principal value sense, because the half-residue at k = 0 is canceled out in the resulting expression (15). Also we note that the decay of vx(x,t) to zero as x ^ ro is satisfied because of the symmetry and normalization of Gt in (A.6). We can now use the boundary condition ux(0,t) = P(t) to obtain the exact value for P(t):

P(t) = 2J G't(-y)Uo(y)dy - jGt- (0)ds = 2J G(-yK(y)dy + 1Ё/4) t1/4. (17) 0 0 0 T After P(t) is found uniquely from (17), y(t) is found uniquely from the integral equation (14). This computation completes the construction of the exact solution of the boundary-value problem (6) for V0 = 0 (see also [5] for other solutions of this fourth-order diffusion equation). Now we turn to the analysis of the solution thus obtained.

Theorem 1. Consider the advection-diffusion equation (6) for V0 = 0 with the initial data

u0 e l}(R+ ). Then, there exists a solution u e L (R+ x R+) of the evolution problem in the explicit form (13), where P,y e Loc (R+ ) are defined by (14) and (17) and lim P(t) = +to .

t —^to

Proof. The convolution integral in the explicit expression (17) can be analyzed from the representation (A.5) for Green's function Gt. If u0 e L}(R+), then

J G't(-y)uo( y)dy

Therefore, P e LJoc (R+ ) and P(t) ~ tm as t ^ œ due to the second term in (17). Now, the integral equation (14) for y(t) with a weakly singular kernel is well defined and solutions exist with y e L™oc(R+ ). Similarly, the solution u e L°(R+ x R+ ) is well defined by (13).

œ

0

Remark 1 One can show that there is no singularity of the solution for P(t) as t^ 0 so that /3(0) = u'0 (0) by continuity. Also, one can show that the solution of the integral equation (14) for

y(t) exists in the closed form: y(t) = 2jGy)uo(y)dy.

Coming back to the original question, if u0(0) = 0, u0(0) < 0, and u0(0) = -1, then there is a

finite value of t0 e (0,ro) such that ux(0,t) > 0 for all t > t0, that is, h(x,t) loses monotonicity at the boundary x = 0 in a finite time t0 (recall that u = hx). This dynamical phenomenon occurs because of the inhomogeneous boundary conditions uxx (0, t) = -1 even in the absence of the advection term in the fourth-order diffusion equation (6).

2.2. Case 0

We have the solution in the implicit form (12) and we need to derive integral equations on the unknown functions P(t) and y(t). One integral equation follows again from the boundary condition u(0,t) = 0:

t x ^ t

-{Gt-s (V0(t - s))r(s) + Gls (V0(t - s))/(s)]ds =j Gt (V0t - y)u0( y)dy -1 j GL (V0(t - s))ds. (18)

2-е

To find another integral equation from the boundary condition ux(0,t) = P(t), we have to use the technique explained in Section 2.1 and to compute the derivative of the solution (12) in x:

^ j i (x, t) = J G; (x + V0t - y)u0(y)dy - - J G"t_s (x + V0(t - s))ds

0 2 0 t-

+ J G-s (x + Vo(t - s))y(s)ds + - P(t) - P(0)Ht (x + Vot) (19)

02

tt

- J Ht-s (x + Vo (t - s))0;(s)ds + Vo J Gt-s (x + Vo (t - s))0(s)ds.

00

We can now use the boundary condition ux(0,t) = P(t) to obtain another integral equation for p and y:

t t

i+2B(0)HI (Vt )+:

3(t) + 23(0)Ht Vt) + 2 j Ht-s (V (t - s))3' (s)ds - 2V0 j Gt-s (V (t - s))/(s)ds

0 0 (20)

t x t v y

- 2 j G't-s (V (t - s))y(s)ds = 2 j G't Vt - y)u0 (y)dy - j G"t_s V (t - s))ds.

0 0 0

The system of integral equations (18) and (20) completes the solution (12) for the case V0 ^ 0. Because of the original motivation to study behavior of the flow on Fig. 1 for large negative V(t), see equation (3), we shall analyze the obtained solution for V0 < 0 only.

Theorem 2. Consider the advection-diffusion equation (6) for Vo < 0 with the initial data

u0 e l}(R+ ). Then, there exists a solution u e (R+ x R+) of the evolution problem in the explicit form (12), where P, y e I (R+) are defined by (18) and (20) with

o

o

o

i \vt\1/3

lim Pit) = —■—ryj lim y(t) = . (21)

t 2\V | t 2

Proof. Similarly to the proof of Theorem 1, it is easy to show from the integral equations (18) and (20) that if u0 e LX(R+), then P,P',y e L^oc(R+). We shall now compute the limit of P(t) and y(t) as t ^ ro:

Pot := lim Pit), y, := lim y(t). (22)

t ^OT t ^OT

To deal with the first integral equation (18), we first notice the explicit computation by using the Fourier transform:

(i Л 1 +ГО7'Л 0-t(k4~ikvo)\

t - ^го It 1 1 ^^

f(t):=JGt-s(Vo(t-s))ds = — J(ik)[ Je-s(kUkVo)ds dk = — J

2^ k3 - V0

dk,

-OT v 0 J

where the integrals in s and k can be interchanged by Fubini's Theorem and the integration is performed in the principal value sense. We can now explicitly compute the limit as t ^ ro by using Lebesgue's Dominated Convergence Theorem:

i- 1 +ot i ai - V 7 dk 1

lim f it) = — J —-dk = —0 J —-- =-rrr.

t2t_OT k3 - iV0 t 0 k6 + V02 3\V|

This computation gives the last term of the integral equation (18) as t ^ ro. To deal with the first term on the right-hand side of (18), we write

го i го / го Л го

JGt(Vot -y)Uo(y)dy = — J|Je-t(k4-ikVo)-i4(y)dy dk = Je-t(k-lkVo)Ua(k)dk,

where

0 2^-»V 0

1 ot _

uo(k):=^je 'X(X^. 2t o

By Lebesgue's Dominated Convergence Theorem, this integral converges to zero as t ^ ro as long

as u0 e L1(R+ ).

To deal with the second term on the left-hand side of the integral equation (18), we rewrite it in the form

J G;_s (V0(t - s))P(s)ds = — OT (ik)2 f J Pit - s)e~s(k4 -kVo)ds\k.

O 2T -OT v O J

Since P e LLOToc (R+) with the assumed limit in (22), we apply Lebesgue's Dominated Convergence Theorem and compute the integral in the principal value sense:

t — R +ot fr _ R ot b Ab _/?

lim J Gls (V0(t - s))P(s)ds = _P J -Ji-dk = _AOTj_kd^ = —P^.

t^OT0 tA 0V ' 2t -OT k3 -iV0 t 0k6 + V02 3|V|1/3

The first term on the left-hand side of the integral equation (12) is more tricky. First, we rewrite it in the form,

JGt_s(V0(t-s))y(s)ds = — J fJy(t-s)e~s(k4-lkVo) dS\dk. o 2^-^v о у

However, if y e Lfoc (R+) with the assumed limit in (22), application of Lebesgue's Dominated Convergence Theorem yields the integral in k with a simple pole at k = 0:

t y <x dk

lim | Gt_s (Vo(t - s))y(s)ds = I .

t^o 2m k(k _iV0)

The integral is no longer understood in the principal value sense. Instead, we return back to the treatment of the inverse Laplace transform in (11) with Re(c) > 0, use transformation p = ik, and shift the contour of integration in k below the pole at k = 0. As a result, computations of the integral above are completed with the half-residue term at the simple pole at k = 0 and the principal value integral:

t

lim J Gt_s (Vo(t _ s))y(s)ds = ^ tо 2ж

f ж » k2dk Л

J

_2y00

Vo| k6 + Vo2 J 3|Vo|.

Combining all computations together, we obtain the following linear equation on and y« from the integral equation (18) in the limit t ^

2y» 3cc _ 1 (23)

Vol VT 2|Vo|2/3'

To deal with the second integral equation (20), we use the Fourier transform again to write

1 <? e

Ht (Vot) = -

and

„ -t (k 4 -ikVo)

Ht (Vot) = J ---dk

2Ж1 -от k

t 1 +<*> 1 f t ,, 4 T/ ^ A

IHt-S(V0(t_s))P'(s)ds = — I 1 \p\t_s)e~s(k o)ds Idk, o 2m k vo y

where the integrals are understood in the principal value sense. If ft, ft, y e L^c with the assumed

limits (22), Lebesgue's Dominated Convergence Theorem implies that

t

Ht (Vot), I Ht_s (Vo(t _ s))ft(s)ds ^ o as t ^rc.

o

Similar to the previous computations, we prove that

J G (Vot - y)Uo( y)dy = o

o

r -1

im J G- (Vo(t - s))ds

oV " 3|V |1/3

o 3Vo|

t

im J Gt- (Vo(t - s))y(s)ds

J о т/ 2/3

o 3Vo

00

and

Д f dk _ Д

lim fG-,(V0(t-s))fl(s)ds = ДМ-

t^f J 17T J 1

-V,) V

where the last integral is computed in the principal value sense because equations (19) and (20) are derived in the principal value sense.

Combining all computations together, we have obtained the following linear equation on p„ and y» from the integral equation (20) in the limit t ^

2v 1

^-T-fe = rW. (24)

N2/3 Vd1/3

Solving the linear system (23) and (24), we obtain (21) and the theorem is proved.

Coming back to the original question, if u0(0) = 0, w0 (0) < 0, and u0 (0) = -1, then there is a finite value of t0e(0,») such that ux(0,t) > 0 for all t > t0. Therefore, like in the case V0 = 0, the function h(x,t) loses monotonicity at x = 0 in a finite time t0 (where u = hx) with the only difference that ux(0,t) remains finite and positive as t ^ We conclude that the presence of the advection term with V0 < 0 in the fourth-order diffusion equation (6) does not prevent the loss of monotonicity in x in a finite time but still stabilizes the solution globally as t ^ In both cases V0 = 0 and V0 < 0,

the monotonicity of h in x is lost because of the inhomogeneous boundary condition hxx (0, t) = -1.

3. Solution of the original problem

We shall now use Laplace transform (7) to obtain the implicit solution to the advection-diffusion equation (4) with a variable speed V(t). Let us denote

t

W (t) = \V (s)ds 0

and obtain the Laplace transform solution in the form:

U(p,t) = U0(p)e-tp4+W(t)p +|e-(t-s)P4+(W(t)-W(s))pI -1 p + p2p(s) Ids. (25)

0 V 2 J

Compared with the solution (10), we have set y(t) = 0 because of the third boundary condition in (5). Using the inverse Laplace transform and recalling the definition of Green's function Gt(x) (see Appendix A), we obtain the analogue of the implicit solution (12):

f Л t

i(x, t) = f Gt (x + W(t) - y)u0 (y)dy - - f G-s (x + W(t) - W(s))ds

0 2 0 t

+ f G"t_s ( x + W (t ) - W (s)W(s)ds.

0

Now we have two unknowns P and W and two integral equations from the boundary conditions u(0,t) = 0 and ux(0,t) = P(t).

From the boundary condition u(0,t) = 0, we obtain the integral equation:

t œ 1 t

-1 Gt_, (W(t) - W(s)W(s)ds = I Gt (W(t) - y)u0 (y)dy - -1 Gt_, (W(t) - W(s))ds. (27) 0 0 2 0

To find another integral equation from the boundary condition ux(0,t) = P(t), we differentiate

the solution (26) in x:

œ .. t

ux (X, t) = 1 Gt (X + W(t) - y)Uo (y)dy - -1 Gt- (X + W(t) - W(s))ds

0 2 0

1 r

+ - ß(t) - ß(0)Ht (x + Ж(t)) - JHt s (x + Ж(t) - W(s))ß'(s)ds (28)

^/ЧУ / ЧУ t\ \ ' ' J t-S

2 0

+ V (t )1 Gt-s ( X + W (t ) - W (s))P(s)ds.

0

From the boundary condition ux(0,t) = P(t), we obtain another integral equation:

t t

fi(f ) + 2,0(0)Ht (W(t)) + 21 Ht-s (W(t) - W(s))fi'(s)ds - 2V(t)1 G^ (W(t) - W(s))fi(s)ds

0 0 (29)

œ t v '

= 21 g; (W (t ) - y)u0 ( y)dy -1 G-s (W (t ) - W (s))ds.

0

We shall prove that the system of two integral equations (27) and (29) determines uniquely the function P(t) and V(t) locally for t > 0. The following theorem gives the result in the form of the fractional power series in t.

Theorem 3. Assume that uo e Cx (R+) such that

uo(o) = o, uo(o) = -1, <(o) = o. (30)

Then, there exists a formal solution (V, fi) of the system of two integral equations (27) and (29) in the form of the fractional power series:

X I A X I A

ft(t) = + 2 ft4tn/4, V(t) = Vo +s Vn4tn/4, (31)

n=4 n=1

where fto = u'o(o), Vo = Uo4)(o)/uo(o), and {ftn/4, V(n_3)/4}^=4 are uniquely determined.

Proof. We substitute the series representations (31) to each term of the integral equations (27) and (29). It follows from (31) that

1 1 œ

J V (s)ds = Vot3/4

t 0 n=1

œ л

3/4 + V-1- v,t(n+3)/4

л n/4

n= n + 4

and

1 ÎV, W тл 3/4 ^ 4 T. t(n+4)/4 -(t -r)(n+4)/4

it,r :=37 JV(s)ds = V>r3/4 + £ — V

t,T ■ 1/4 I W^ ' o* ^ n/4 1/4

^ tJr n=1 n + 4 r

Using the representation (A.5) of the Green function with g e C" (R), we obtain for the three terms of the integral equation (27):

•g" & J, rg " (it ,r )(t _r)n

JGls(W(t)-W(s))ß(s)ds = ßoJ^^dr + ^ßn/4Jg ^)3/4 ) dT

0 0 T n=4 o T

œ 1 t œ t ff/e \/y_r\n/4

= 4ß0g"(0)t1/4 +]T 1 g (k+2)(0)J%dr + ]Tßn/4 J g (it ,T )3t4-T) dr,

0

f f J 1 J

f Gt (W (t) - y)Uo(y)dy = f g (z - at )Uo(t1/4 z)dz = £ - u^»"/4 f g ( z - a )

- n\

(z - a )zndz

= t-4u,

J 1 J J 1 f

(0)S-(-а )k f g(k )( z) zdz + S -, u0n)(O)tn/4 f g ( z - at ) zndz,

k =0 k! 0 n=2 П! 0

and

t tg'(£ ) f i ,, „ t r

J G't_ s (W (t ) - W(s))ds = J iri dr= S -- g(k+1)(0) JIT

0 T

k=1 k!

dT.

0T

At the first powers of t1/4, we obtain a system of linear algebraic equations on the coefficients of the fractional power series (31):

1/4

2/4

3/4

4/4

5/4

4Д g 40) = u0 (0)J g ( z ) zdz,

1 f

0 = - <(0)f g ( z) z 2dz,

0 J

0 = — u

3!

0 = — u 4!

- u0"(0)J g ( z) z 3dz, !0

ff

- u04)(0)f g ( z ) z 4 dz - u0 (0)V f g '( z) zdz,

Д4/4 g "(0)f

"irW - (1 - x)

4/ 4

3 /4

dx = — u 5!

ff - u05)(0)f g ( z ) z 5dz - - <(0)V f g o( z) z 2dz

4 u0(0)V-/4f g0(z)zdz - 2g0'(0)V0,

5 .....O 5

and so on.

Using the explicit values for the integrals (A.9)-(A.13) and the initial conditions (30), we

obtain po = uO(O), V0 = m04)(O}/ Uo(O), and the linear equation

f 1

(5Ь 1

u0(0)Vi/4 + Sg"(0)|^4/4 + u05)(0) + -V0 J = 0. Similarly, we expand all terms of the second integral equation (29):

(32)

f Gt-s (W (t) - W (s))P(s)ds = Д f dT + ]S Дп / 4 f

•g (It ,T )(t -T)

n/4

.1/4

-dT

n=4 0

t ffk

4 p g (0)t3/4+s - g ^ )(0)Ji1t dT+s^Pn/4 f

3 k=2 k! 0 T n=4 0

•g(It,t )(t - T)

1/4

n/4

-dT,

J J j 1 J

f GO (W (t ) - yK( y)dy = f g ( z - at )u0(tl/4 z)dz = £ - u0n+-)(0)tn/4 f g ( z - at ) zndz

0 0 n=0 0 J 1 J J 1 J

= u0(0)S-(-a )k f g(k )( z)dz + S - u0n+-)(0)tn/4 f g ( z - at ) zndz,

k=0 k! 0 n=1 n! 0

j GO- s (W (t ) - W (s))ds = f

^"ItK J 1 ,k+2Wt Ik

-dT = S -1 g(k+2)(0)J^rdT,

-sv w ■■ K-JJ— j 3/4 ^ ,,6 v /J ^3/4

0 0 T k=0 k! 0 T

0

0

n

0

0

0

0

0

0

0

0

0

t

at » 1 ,,, , , Ht (W(t)) = I g(z)dz = S —— g(k)(o)atk+1, o k=o(k +1)!

and

IHt_s(W(t)_W(s))ft(s)ds = S ftn/4 S g(k)(o)Iitkr+1(t_r)(n_4)/4dr.

o n=4 k=o (k +1)! o

At the first powers of t1/4, we obtain a system of linear algebraic equations on the coefficients of the fractional power series (31):

x

fto = 2uo(o)j g ( z )dz,

t0/4

0

œ

t1/4: 0 = 2<(0)J g ( z) zdz - 4 g "(0),

0

œ

t2/4: 0 = <(0)Jg(z)z2dz,

0

0

,3/4

œœ

t3'4 : 2ß0g(0)V, - 8ß0g(0)V> = -w04)(0)Jg(z)z3dz - 2u0(0)V Jg'(z)dz,

0

8 8 1 œ œ

t4/4 : ß4/4 + 8ß0g(0)Vi4 -8ß0g(0)V-/4 = -u05)(0)Jg(z)z4dz - 2u'0(0)V, Jg'(z)zdz 5 3 12 0 0

œ

8

"o(0)V1/4 J g'(z)dz

u,

0

and so on. Again, using the explicit values for the integrals (A9)-(A.13) and the initial conditions (30), we obtain J30 = u0(0), V0 = u04)(0)/u'0(0), and the linear equation

8 1

- - u0(0) g (0)V-/4 + (J4/4 + u05)(0)+-V0) = 0.

The system of linear equations (32) and (33) has a unique solution

(33)

V1/4 = 0, ß4/4 =-u05)(0) -1V0, (34)

provided that

f g(0)gff (0)=rf 4X4}=£

which is true. Note that the constraint Vo = uo4)(o)/u'o(o) also follows from the pointwise equation (2) obtained for sufficiently smooth solutions. Similarly, the second equation (34) follows from the advection-diffusion equation (4) after one derivative in x and the limit x ^ 0 and t ^ 0.

It remains to prove that the system of linear equations obtained from the system of integral equations (27) and (29) can be solved at each order of t(n+11>/4 and tn/4, respectively, for n > 4. From the previous computations, we can deduce that the first integral equation at t(n+1)/4 gives a linear equation on variables (pn/4, V(n-3)/4) of the power series (31):

0

1(1 " 4 .

-Рп / 4 g " (0) Г 3/4 dx +-- w0(0)F(„_3)/4 j g'(z) zdz = •••, (35)

0 x n +1 ( ) 0

where the dots on the right-hand side denote the terms expressed through derivatives of u0(x) at x = 0 and the previous terms of the power series (31). Similarly, the second integral equation at tn/4 gives another linear equation on variables ( pn/4,V(n-3)/4):

-

Pn/4 - - J0g(0)V(„-3)/4 = - . (36)

The system of linear equations (35) and (36) is non-degenerate if

C^g«}^^ (37)

The coefficients {Cn} are computed numerically for n > 1 (see Fig. 2). The sequence is monoton-ically increasing. It approaches closely to 1 at n = 8, where C8 ~ 0.96, and n = 9, where C9 ~ 1.04. Therefore, Cn ^ 1 for all n > 1 so that the linear system is non-degenerate and a unique solution for (Pn/4, V(n-3)/4) exists for any n > 4.

Fig. 2. Numerical approximations of Cn defined by (37)

In the present time, we cannot prove yet that the system of integral equations (27) and (29) leads to a finite-time blow-up, according to the conjecture in [1]. Nevertheless, numerical computations show that the blow-up holds for a generic set of initial data. Fig. 3 shows the behavior of functions P(t) and V(t) near the blow-up time. It follows from this figure that P(t) = hxx(0,t) ^ 0 at the

1/3

same time as V(t) ^ with P(t) V(t) ^ C0, where C0 > 0 is a numerical constant. In other

1/3

words, we conclude with the conjecture that P(t) ~ V(t)- as V(t) ^ in a finite time t0e (0,«).

j—|—I I 11

Fig. 3. Numerical computations of P(t) and V(t) for the advection-diffusion equation (1). Acknowledgement. Authors received funding from the Federal Target Program "Research and scientific-pedagogical cadres of Innovative Russia" for 2009-2013". We thank E.S. Benilov for enlightening discussions and for preparing Fig. 3 based on numerical codes in [1].

Appendix. Green's function

Let us define the fundamental solution of the fourth-order diffusion equation:

ht + = 0 x e R, t > 0,

h |i=0 = S(x), x e R,

(A.1)

where 5 is a standard Dirac delta-function in the distribution sense.The fundamental solution is usually referred to as Green's function and we shall denote it by

h(x, t) = Gt (x), x e R, t e R. Using the Fourier transform in x, we can obtain the explicit expression for Green's function:

1 ад 1 ад

Gt (x) = — f e-tk+ikxdk = - f e-&4 cos(kx)dk. 2k j k{

(A.2)

In particular, we have Gt(-x) = Gt(x) for all x e R and

Gt (0) = 1 ад e-4 dk

К 0

4kT

(A3)

G"(o) = _1 Ik 2e~tk 4 dk = _^T, (A.4)

m o 4mt

where r is the standard Gamma function.The Green's function can be represented in the self-similar form by

Gt(x) = g(-£■ \ g(z) = 1Je~_4 cos(kz)dk, (A.5)

t Vt y m o

where g e L2(R) n Lx(R). Therefore, Gt decays to zero as t^« in any Lp norm forp > 2. In particular, \Gt(x)| < ||g|r /t1/4, |G"(x) < ||g"|r /11/2, and so on, for any x e R.

By the stationary phase method (see, e.g., Chapter 5 in [3]), g(z) and all derivatives of g(z) decay to zero as |z| faster than any algebraic powers. This gives the decay of Gt(x) and any x-derivative of Gt(x) as |x| ^ « for any fixed t > 0. Although Gt and g are not L1 functions, they satisfy the normalization conditions:

IG (x)dx =I g(z)dz = 1, t > o. (A.6)

R R

The even function g: R ^ R satisfies the ordinary differential equation

dz4 dz

4 df = g + zd-g, z 6 R, (A.7)

subject to the initial values

1 ( 1 ^ 1 ( 3 ^

g (0) = — Г- g " (0) = 0, g "(0) = -—г-l g " (0) = 0, (A.8)

4 I 4ж \ 4 I

4ж

and the decay behavior as |z| It is clear from the differential equation that g e C'(R) satisfies a number of integral constraints:

x

I zg( z)dz = _4 g " (o), (A.9)

o

x -

I z2 g (z)dz = o, (A.10)

0

x 1

1 z3 g ( z)dz = _8g (o), (A.11)

0

x

1 z 4 g ( z )dz = _12, (A.12)

0

and so on.

X г

J z5 g ( z )dz = 164! g " (0),

(A.13)