SECULAR TERMS FOR THE KINETIC MCKEAN MODEL Текст научной статьи по специальности «Математика»

DIFFERENTIAL EQUATIONS AND

CONTROL PROCESSES N. 1, 2023 Electronic Journal, reg. N &C77-39410 at 15.04.2010 ISSN 1817-2172

http://diffjournMl.spbu.ru/ e-mail: [email protected]

Nonlinear partial differential equations Ordinary differential equations

Secular terms for the kinetic McKean model

Dukhnovsky S. A.

Moscow State University of Civil Engineering, Moscow, Russian Federation

e-mail: [email protected]

Abstract. In this article, we investigate the kinetic McKean model. The perturbed solution of the Cauchy problem is sought in the form of Fourier series. The Fourier coefficients for the zero and nonzero modes are written out, respectively. The original system is reduced to an infinite system of differential equations. An approximation for the systems is constructed. Under certain assumptions, we find secular terms (non-integrable part). This, in turn, will allow us to prove for the first time the exponential stabilization of the solution in the future.

Keywords: kinetic model, Fourier series, Knudsen parameter, secular terms 1 Introduction

We consider the well-known kinetic McKean model [2, 5, 9]:

dtu + dxu = -(w2 — uw), x e R,t> 0, (1)

£

dtw — dxw = — (w2 — uw), (2)

£

u(x, 0) = u0,w(x, 0) = w0, (3)

where u0(x) = u0(x + 2n), w0(x) = w0(x + 2n) is periodic functions. This system describes a monatomic gas with two groups of particles with corresponding densities u = u(x,t),w = w(x,t) . The first group moves with the speed c = 1, the other c = -1, the parameter £ corresponds to the Knudsen number in the kinetic theory of gases. The McKean system is a non-integrable system, i.e. the Painleve test is inapplicable.

The physical description of the Boltzmann equation is described in a fundamental article [1, 8]. The asymptotic stability of kinetic systems of Carleman, Godunov-Sultangazin and Broadwell for periodic initial data were studied in the works [7, 11, 12, 13, 14]. The proofs of the theoretical results were confirmed numerically in the works [17, 18]. The exact solutions of the systems are presented in [2, 3, 4, 5, 9, 10, 15, 16]. The secularity condition for the kinetic Carleman system was found in [6]. In this work, approaches and methods (see [6, 11, 13]) will be applied for our system as well as for the above systems. The McKean system has been largely unexplored. We will single out the non-dissipative part of the solution and reduce the problem of the existence of a global solution to a nonlinear equation in the Hilbert space. This will later allow us to prove for the first time the exponential stabilization of the solution.

2 Fourier solution for the McKean system

We study the Cauchy problem for small perturbations of the equilibrium state w2 = uewe, ue,we > 0 of the system (1)-(3). Let be

u = ue + wlJ2e2u, w = we + wlJ2e2w. (4)

Then

dtu + dxu — we1(uu — u) = ewlJ2(w2 — uw), x G r, t > 0, (5)

£

dt w — dxw + we-(w — u) = — ewlJ2(w2 — uw), (6)

£

u |t=o= u0, wu |t=o= wu0. (7)

For periodic solutions with zero means

u(t, x) = uo(t) + uk(t)eikx, w(t, x) = wo(t) + ^ wk(t)eikx,

kGZo kGZ o

zo = {k G z, k = 0},

we introduce weight spaces L2,Y(R+; H), H with norms:

»00

si^(m+;h) = / i «o(t) i2 dt+

Jo

»00

+ i k i2"i «k(t) i2 dt, iii S |(=oi||H,=| «0 i2 + E i k |2" | «? |2 .

k€Z0 k€Z0

Here y > 0, a = const.

Theorem 1 For any a > 2 and w2 = uewe > 0 there exist ^0,q £ (0,1) such that for periodic initial data (u0 ,W°) with zero averages satisfying the inequality

ll|U°||k + ll|W°||k < s2q,

there exists a global solution (u,W) £ (R+; Ha) to Cauchy problem (5)-(7), where 7 = e^0 > 0.

Hence, the local equilibrium principle with an exponential stabilization to the equilibrium state holds.

Theorem 2 Let a > 2 and let the condition of Theorem 1 be fulfilled. Then

,2e = uewC)

a positive equilibrium state (ue = const > 0,we = const > 0,w;f = uewe) is

exponentially stable:

IIIu(x,t) -Ueiik < ci(|||S0|||hct + iiiw^ik)e"27lt,

i i i w(x,t) - weiik < c^|||S|||hct + | | | w0| | | u^je~2llt, where c1,c2 > 0, y1 > 7 > 0.

These theorems will be proven in a future publication for the first time. We assume that the average

1 />2n 1 />2n

«0 = — S(x)dx = w° = — w°(x)dx = 0. 0 2 Wo ( ) 0 2n J0 ( )

Let us rewrite the system (5)-(7) in terms of Fourier coefficients for k = 0

dt«k + ik«k = -(dtwk - ikwk), (8) dtwk - ikwk + 1 we(wk - «k) = -ewe1/2 ^ (wkiwk2 - «klwki), (9)

ki+k2=k,k,ki,k2€Z

«k it=0= «k,wk |t=0 = w°. (10)

and the zero mode k = 0

d d fi-w

= - dtW°' (11)

—Wo + ^we(w0 - u0) = -EW^2 (wkiWk2 - U^W^), (12)

ki+k2=0,k1GZ,k2GZ

uo |t=o= 0, wo |t=o= 0. (13)

Solving the equation (8), we find its solution

Uk = -Wk + (uk + wo)e-ikt + 2ik / eik(s-t)Wkds. (14)

Jo

For k = 0 we have

Uo = -Wo.

We rewrite out the sum for k = 0

y^ (Wki Wk2 - UkiWki ) =

ki+k2 = k,ki€Z,k2GZ

= WoWk - UoWk + WkWo - UkWo + ^ (Wki Wk2 - Uki Wki) =

ki +k2=k,kGZo,ki€Zo,k2GZo

r t

¿k + Wk

o

d Wk2 - (Uki + ikit

ki+k2 = k,kGZo,ki€Zo,k2GZo

ft

= 4WoWk - Wo ((uk + wo)e-ikt + 2iky eik(s-t)Wkds) + + ^ (2wkiWk2 - Wk^(uoi + w°°i)e-ikit+

EZo,k2GZo

+2ik^ eiki(s-t)Wkids)). (15)

Substituting (14), (15) to (9), we have an infinite system of ordinary differential equations (ODEs)

d 1 1 f1

—wk - ikwk + 2we-wk - 2ikwe- eik(s-t)wkds = (16)

dt £ £ J o

= we1/21 dk e-ikt+ £

+£We1/2(Ik(w) - 2Bk(w, w)) - £w1/2Tkadd(w), Wk |t=o= wo, k g zo = {k g z, k = 0}.

Here Tkadd(w) is the perturbation operator of the base system d 1 1 f1

—wk - ikwk + 2we- wk - 2ikwe- eik(s-t)wkds = (17)

dt £ £ J o

= we1/2-dke—ikt + £w1/2 (lk(w) — 2Bk(w, w)) , wk |t=o= wo, k g zo = {k g z, k = 0},

where

dk = we1/2(uk + wo),Tkadd(w) = wo(4wk — (uk + wo)e—ikt — 2ik i eik(s—t)wkds)

lk (w) = £ (u°k! + wox )e—ikltwk2,

'I + wS1 )e—iklt"

ki+k2=k,kGZo,kiGZo,k2 GZo

ft

Bk(w,w) = ^ w°^w°i — ik^ eikl(s t)wkids).

ki+k2=k,kGZo,kiGZo,k2GZo o

Let's make a replacement for the transition to zero initial data

i

wk

= woe(ik—2we!)t + yk,yk g l2,y(r;h). (18)

In what follows, we will consider the system (16) without the perturbation operator. Substituting (18) to (17) and taking into account that

ik / eik(s—t^woe(ik—2weè)t + yk) ds = wkik (e(ik—2weè)t — e—ikt).

2(ik - We 1)

then we have for yk an infinite system of ordinary differential equations (ODEs)

d 1 1 ft

Tk(yk) = -77yk - ikyk + 2W^ yk - 2ikWe- / eik(s-t)ykds = (19) dt £ £ J o

= We1/21Dke-ikt + fk(t)e-2we^ + £We1/2(Lk(y) - 2B(y,y)), yk |t=o= 0, k g zo = {k g z,k = 0},

where

d°=wi/2(«°+wo )—wo,

ik — we e

Bk(y, y) = ^ yk^yki — ike y eiki(s—t)ykids

ki+k2=k,kGZo,kiGZo,k2 GZo °

fk(t) = -T^ 1w0e" + ewy2(fL(t) - 2fB(t)),

- Wee

fL(t)= E « + w0, )e-k"w02 e'k2i,

Ci + wki)g

ki+k2=k,kGZo,kiGZo,k2 6Zo

fkB(t) = ^ w02e^wS. e(iki-2we1 * - -kiwkl ^ x

ki+k2=k,kGZo,ki€Zo,k2£Zo

_ -T1w0i 'ki^ " 2(-Ti - wej)

X I e(iki-2wei)t _ g-ikit

Lk (y) = E (u°ki + w0i )e-ikityk2 +

ki+k2=k,kGZo,ki€Zo,k2 6Zo

w2e(ik2-2w^)tfyki - -Ti /'

+ E (w02e(ik2-2w^»(yki - iki / eiki(s-t)ykids) +

ki+k2=k,k£Zo,ki6Zo,k2£Zo

0

3 Equation for zero mode

From the system (12) for the zero mode, we obtain dtw0 + 21 wew0 = -ew^2 (2w0w0 + E (w°iw°2 - u°Iw°2 )), (20)

ki+k2=0,ki£Zo,k2 6Zo

w0 |t=0= 0. (21)

Rewrite (20) as the Riccati equation

-dw0 + 21 wew0 = swj1/2^2w0w0 - /0(w) + 2B0(w, w)^ , (22)

dt s V /

w0 |t=0= 0, (23)

where

l0(w)= E (u°I + w0i )e-ikit w°2 ,

ki+k2=0,ki£Zo,k2 6Zo

B0(w,w)= E w°2 (w°1 - iki J eiki(s-t)wkids).

ki+k2=0,ki£Zo,k26Zo 0

Let be

w° = w0e(ik-2we 1 )t + yk ,yk £ L2,7 (r).

Here

2

llyllL(R+) = / e2Yt | y |2 dt. o

Then we have d1

-yo + 2^Weyo = -£we1/2(2yoyo - fo(t)e-2we11 - 1o(y) + 2Bo(y,y))

dt o £

yo |t=o= 0,

where

Bo(y, y) = E yk^yki - ik^ eiki(s-t)ykids),

ki+k2=o,kiGZo,k2GZo o

fo(t) = foL(t) - 2foB(t),

foL(t)= E (uki + Woi )e-'kiH eik2t,

t 1 = ' 1 U'ij

ki + k2=o,kiGZo ,k2GZo

/oB (t)= E < «'*(«

"B (t) = W eik2Hwo e(iki"2w i» - , 1^ ,, x

ki+k2 =o,kiGZo,k2GZo

3(iki-2Wei)t _ e-ikit

ik1Wki

ki^ " 2(ik1 - We 1)

x e

Lo(y)= E (uki + woi )e-ikityk2 +

ki+k2=o,kiGZo,k2GZo

+ E (wk2e(ik2-2wei)t (yki - ik1 ft eiki(s-t)ykids) +

ki+k2=o,kiGZo,k2GZo o

+yk2 (<e(!ti"2w-i)t - ^ 1) (e(,ti"2w"•)t - e_'M)))

4 Finite approximation

To construct an approximation solution of the Cauchy problem (5)-(7), a finite approximation of the infinite system (19) is introduced for m g n:

Tk(y<"") = - ifcykm) + 2We1yf - 2ikWe1 [' e'k<'" ^'"ds = (24)

dt £ £ 0

= We1/21Dkm)e-'kt + /km)(t)e-2weit + £we1/2(Lkm)(y(m)) - 2Bkm)(y(m),y(m))),

ykm) |t=o= 0, k g zo, | k |< m,

Here

ikw

Dkm) = w1/2 («Î + w0 ) - w0,

ik we g

Bkm)(y(m), y(m)) = £ yk2 (». - ik1 [ e':kl('-t)ykids),

ki +k2=k,|ki i<m,ik2i<m

ikwe 1 o ikt , _„„1/2/' /L fk (t) = m-fTwke + £w1 ( fk,m(t) - 2/k,r

ik — w1 £ V ' '

__cB

'k (6) = w 1" wke + £we \Jk,m(t) - 2 f k,m

ik - we1 £ V ' '

ZkLm(t)= £ («°ki + wki )e-ikitwk2 eik21,

kl+k2=k,ikli<m,ik2i<m

B (t) = V" w0 e^ ( w0 e(iki-2wei, 1°i ,, x

/Bm(t) = £ wk2 eik2t(

kl+k2=k,ikli<m,ik2i<m

_ ik1w0i 'kl^ " 2(ik1 - we1 )

X | e(ikl-2wei)t _ e-iklt

¿k",(y(m,)= e (u°i + w0i ^+

ki+k2=k,|ki|<m,|k2|<m

+ E (w02e(ik2-2we 1)t (yki - -ki J' eikl(s-t)ykids) +

ki+k2=k,|ki|<m,|k2|<m 0

+yk2 (w°xe(iki-2we 1 )t - -Tiw0i (e(iki-2we1 )t - e-ikit V 2(-ki we £) ^

The solution of the system (24) will be sought in the form

y(m) = Q(m)T —i ( g—ikt) + T-i(z (m)) z (m) I = 0 yk = Qk Tk (e )+ Tk (zk ), zk |t=0=0,

Q(m) £ hm), z(m) £ l2,Y(R+; hm)), where z(m) = (4m), | k |< m,k = 0). Then

zkm) = (wi/2 jDkm) - Qkm))e-ikt + fkm) (t)e-2we 11+ (25)

+swei/2 (Lkm) (Qkm)Tk-i(e-ikt) + Tk-i(zkm))) --2Bkm) (Qkm)Tk-i(e-ikt) + Tk-i(zkm)), Qkm)Tk-i(e-ikt) + Tk-i(4m);

In the variables (zkm),Qkm)), the system (25) under the secularity condition

we1/21 Dkm) - Qkm) = 0, | k |=1,...,m, (26)

£

will be written as:

4"" = /°"'»(t)e-2w if + £'w,1/2 (-L°m) (QrVle-'^) + T,-1^"»)) — —2Bkm) (q°'")T°—1(e—ikt) + T°—1 (z°m» ), Q^1 (e—ikt) + T°—1 (z°m» ))).

We get the system in the Hilbert space L2,Y (r+; h-"»). For zero mode we set yo = zo. In this case

zo = —£w1/^t e2wei(s-t) (2zozo — /o(t)e-2w^ — lo(y) + 2Bo(y, y))ds. (27) Here is no secularity condition for the zero mode.

5 Local equilibrium

We will find a solution to the secularity condition from the principle of local equilibrium. Taking into account (14) and (18), we have

c t

ukm) = -ykm) - woe('k-2wei)t + (uk + wo)e-'kt + 2ik / e'k(s-t)ykds+

ik

ik — we 1

- e

= _ y(m) + e(iJ —2We 1 )t WelWj +

k ik — we 1

ee

t ,0 We1 „„0\„-iki

+2ik / eik(s—i)yjds + (uj--^wj)e—iki, (28)

JO ik — We 1

We separate the non-integrable part using (24)

ik / eik(s—i)Qjm)Tk-1(e—iks)ds - —iki + Rj, (29)

jo 2we

2w V^T°—1(e—ikt) — ikT—1(e—ikt) + 2we-T°—1(e—g L^(r; h<"'»). Applying the formula (29), we get

r t

.q,",?—1 (e-ikt) — T^^fc / e^^^z«"»^ (30)

o

1 „., U„o

u(m) - -(

+ (u0 — , Wee . w0 — -Qkm))e—iki + e(ik—2we 1 )iTWeIW

+W^(ddtTk—1(e—iki) — ikTk—1(e—iki) + 2we Jt—1 (e—

0

If

= w(u° - ik-i )'|k|< m, (31)

e£

then we have ukm) ^ 0, when t ^ to. For the second component in ha ), we have

(m)

Qkm)Tk-1(e-ikt) + Tk-1(zkm)) + w0e(ik-2wl)t ^ 0,t ^ to.

w° - qk '(e ""') + ')+ w°e- —^ ^ 0,t ^

Thus, under the condition (31), we have the local equilibrium.

6 Conclusion

The one-dimensional McKean system was investigated. Secular terms were found that do not belong to our space L2,Y. As a result, we obtain a nonlinear equation in the Hilbert space. In what follows, we obtain priori estimates for one, an existence theorem for a solution using the fixed point theorem. We will also prove the weak convergence of the approximative solution to the weak solution and just the classical solution. From here, the exponential stabilization of the solution to a positive equilibrium state will follow (see theorems 1, 2).

References

[1] Godunov S. K., Sultangazin U. M. On discrete models of the kinetic Boltz-mann equation. Russian Mathematical Surveys, 1971; 26(3): 1-56.

[2] Euler N., Steeb W.-H. Painleve test and discrete Boltzmann equations. Australian Journal of Physics, 1989; (42): 1-10.

[3] Tchier F., Inc M. and Yusuf A. Symmetry analysis, exact solutions and numerical approximations for the space-time Carleman equation in nonlinear dynamical systems. The European Physical Journal Plus, 2019; 134(250): 1-18.

[4] Dukhnovskii S. A. Solutions of the Carleman system via the Painleve expansion. Vladikavkaz Math. J., 2020; 22(4): 58-67. (In Russ/

[5] Dukhnovsky S. On solutions of the kinetic McKean system. Bul. Acad. Stiinie Repub. Mold. Mat., 2020; 94(3): 3-11.

[6] Vasil'eva O. A., Dukhnovskiy S. A. Secularity condition for the kinetic Carleman system. Vestnik MGSU, 2015; No. 7, 33-40.

[7] Dukhnovskii S. A. Asymptotic stability of equilibrium states for Carleman and Godunov-Sultangazin systems of equation. Moscow University Mathematics Bulletin, 2019; 74(6): 246-248.

[8] Vedenyapin V., Sinitsyn A., Dulov E. Kinetic Boltzmann, Vlasov and related equations. Amsterdam, Elsevier, 2011, xiii+304 pp.

[9] Dukhnovsky S. A. The tanh-function method and the (G'/G)-expansion method for the kinetic McKean system. Differential equations and control processes, 2021; No.2, 87-100.

[10] Lindblom O., Euler N. Solutions of discrete-velocity Boltzmann equations via Bateman and Riccati equations. Theoretical and Mathematical Physics, 2002; 131(2): 595-608.

[11] Radkevich E. V. On the large-time behavior of solutions to the Cauchy problem for a 2-dimensional discrete kinetic equation. Journal of Mathematical Sciences, 2014; 202(5): 735-768.

[12] Dukhnovskii S. A. On the rate of stabilization of solutions to the Cauchy problem for the Godunov-Sultangazin system with periodic initial data. J. Math. Sci., 2021; (259): 349-375.

[13] Vasil'eva O. A., Dukhnovskii S. A., Radkevich E. V. On the nature of local equilibrium in the Carleman and Godunov-Sultangazin equations. Journal of Mathematical Sciences, 2018; 235(4): 392-454.

[14] Radkevich, E. V. On discrete kinetic equations. Doklady mathematics, 2012; 86(3): 809-813.

[15] Ilyin, O. V. Symmetries, the current function, and exact solutions for Broadwell's two-dimensional stationary kinetic model. Theoretical and Mathematical Physics, 2014; 179(3): 679-688.

[16] Ilyin, O. V. Existence and stability analysis for the Carleman kinetic system. Computational Mathematics and Mathematical Physics, 2007; 47(12): 1990-2001.

[17] Vasil'eva O. The investigation of evolution of the harmonic perturbation the stationary solution of the boundary value problem for a system of the Carleman equations. Matec web of conferences, 2017; (117), 00174.

[18] Vasil'eva O.A. Numerical solution of the Godunov-Sultangazin system of equations. Periodic Case. Vestnik MGSU, 2016; No. 4: 27-35. (In Russ.)