CC BY
6
Research article
UDC 517.928.1:517.958:519.633
PACS 02.30.Jr, 02.30.Mv, 11.10.Jj, 05.20.Dd, 05.40.Jc,
DOI: 10.22363/2658-4670-2021-29-2-126-145
The asymptotic solution of a singularly perturbed Cauchy problem for Fokker—Planck equation
Mohamed A. Bouatta1, Sergey A. Vasilyev1, Sergey I. Vinitsky1,2
1 Peoples' Friendship University of Russia (RUDN University) 6, Miklukho-Maklaya St., Moscow, 117198, Russian Federation 2 Joint Institute for Nuclear Research 6, Joliot-Curie St., Dubna, Moscow Region, 1^1980, Russian Federation
The asymptotic method is a very attractive area of applied mathematics. There are many modern research directions which use a small parameter such as statistical mechanics, chemical reaction theory and so on. The application of the Fokker-Planck equation (FPE) with a small parameter is the most popular because this equation is the parabolic partial differential equations and the solutions of FPE give the probability density function.
In this paper we investigate the singularly perturbed Cauchy problem for symmetric linear system of parabolic partial differential equations with a small parameter. We assume that this system is the Tikhonov non-homogeneous system with constant coefficients. The paper aims to consider this Cauchy problem, apply the asymptotic method and construct expansions of the solutions in the form of two-type decomposition. This decomposition has regular and border-layer parts. The main result of this paper is a justification of an asymptotic expansion for the solutions of this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.
Key words and phrases: asymptotic analysis, singularly perturbed differential equation, Cauchy problem, Fokker-Planck equation
It is well known that the differential operator, which is applied in the theory of measure, has such form:
The solution of the equation L* ^ = 0 is Borel measures on an open set O G Rd and there is the relation
(received: March 31, 2021; accepted: May 25, 2021)
1. Introduction
L = atj dx, dx, + btd,
l^ij^d, deN.
© Bouatta M.A., Vasilyev S.A., Vinitsky S.I., 2021
© ®
This work is licensed under a Creative Commons Attribution 4.0 International License
http://creativecommons.org/licenses/by/4.0/
If the measure ^ has a density p, then p is conjugate solution of the equation
dx,dx a%'Jp(x) — dx b%p(x) = 0, x e n.
Similarly, we can consider parabolic operators in the form
P = df —dT dT alJ + dT b%,
o Us ^ Us j Us ^ '
and there are appropriate parabolic equations P*^ = 0 for finding measures ^ on x [0,7]. The equations for the study of density have the form of Fokker-Planck equation (FPE)
dtp(x, t) — dx dx a^(x, t)p(x, t) + dx,b%(x, t)p(x, t) = 0.
FPE equation uses for analysis a macroscopic process but for a small subsystem.
We can formulate the singularly perturbed Cauchy problem for FPE in the form:
edtp(x, t, e) — dx dx (x, t)p(x, t, e) + dx,bl(x, t)p(x, t, e) = 0,
p(x,0,e) = p0 (x), xen, Vpo(x)eCZ° (n),
where £ > 0 is a small parameter.
If we assume £ = 0, we can get a degenerate Cauchy problem in the following form:
dXi dx (x, t)p(x, t) — dXi b% (x, t)p(x, t) = 0,
p(x,0)=po (x), xen, Vpo (x)ecg° (n),
where solutions p(x, t) are solutions of the degenerate problem and p may differ from solutions p(x,t) significantly.
A large number of methods have been developed for the analytical and numerical study of FPE solutions [1]-[8]. Hyung Ju Hwang and Jinoh Kim [9], [10] study the initial-boundary value problem for the Vlasov-Poisson-Fokker-Planck equations in an interval with absorbing boundary conditions. They introduce the Deep Neural Network (DNN) approximated solutions to the kinetic Fokker-Planck equation in a bounded interval and study the large-time asymptotic behavior of the solutions and other physically relevant macroscopic quantities. Shu-Nan Li and Bing-Yang Cao [11] obtained solutions based on the fractional Fokker-Planck equation (FFPE) with a generic time- and length-dependence of an "effective thermal conductivity" (fteff), namely, KeSLa with L being the system length. They formulate the effective thermal conductivity in terms of entropy generation, which does not rely on the local-equilibrium hypothesis. Hrishikesh Patel and Bernie D. Shizgal [12] compare the Kappa distribution of space plasmas modelled with a particular Fokker-Planck equation for a two component system with the linear Fokker-Planck equation that has been used to study the Student ¿-distribution. Lucas Philip and Bernie D. Shizgal [13] consider the one-dimensional bistable Fokker-Planck equation with specific drift and diffusion coefficients so as to model protein folding. Yunfei Su and Lei Yao
[14] study the hydrodynamic limit for the inhomogeneous incompressible Fokker-Planck equations.
The development of the asymptotic analysis of singularly perturbed differential equations and systems of differential equations was made by A. N. Tikhonov [15], M. I. Vishik and L. A. Lyusternik [16], A. B. Vasil'eva [17], S. A. Lomov [18], V. A. Trenogin [19], J. L. Lions [20] and other researchers during the second half of the 20th century. There is a large number of recent works. O. Hawamdeh and A. Perjan [21] study an asymptotic expansions for linear symmetric hyperbolic systems with small parameter. Using the boundary layer functions method of Lyusternik-Vishik, A. Perjan [22] obtains the asymptotic expansions of the solutions to the Cauchy problem for the linear symmetric hyperbolic system as the small parameter e ^ 0. A. N. Gorban [23] investigates a model reduction in chemical dynamics with slow invariant manifolds and singular perturbations. Bor-Yann Chen, Liy-ing Wu and Junming Hong [24] consider singular limits of reaction diffusion equations and geometric flows with discontinuous velocity.
In this paper we apply the results of the paper [21] and investigate the Cauchy problem for the singularly perturbed Tikhonov-type symmetric system of non-homogeneous constant coefficients linear parabolic partial differential equations (LpPDE system) with a small parameter. We use the asymptotic method for this Cauchy problem and construct expansions of solutions in the form of decomposition, which has regular and border-layer parts. The main result of this paper is a proof of a justification theorem of an asymptotic expansion for this Cauchy problem. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations.
2. Singularly perturbed Cauchy problem for LPPDE
system
We consider the following singularly perturbed Cauchy problem (Pe),
P€u(x,t,e) = f(x,t), x£ Rd, t^0, (1)
u(x,0,e) = u0(x), x e Rd, (2)
where £ > 0 is a small parameter.
Thus, Pe = P0+eP1 is a parabolic operator, where Pi = Aidt + Bi(dx) + Di,
P=1 P,q=1
Bzp = 0>7t)ns,t=i, Ctpq = (csvtq)ns,t=i, A = (dlt)7Z,t=i are real constants of
symmetric n x n matrices and bzJtt > 0, c^tq > 0, dzst > 0 (Vs,t = 1,... ,n), O 1, u(x,0,e) : Rd x [0,to) x (0,to) ^ Rn, f(x,t) : Rd x [0, to) ^ Rn, f(x,t)eC1,
A) = \Im 0) , Ax =(0 0 | , O^m^n, \0 0j \0 In_m/
where Ik is an identity matrix and
^ = + eA1, B(dx)=BQ (dx ) + eB1 (dx), D = Do + eDx, Lt (9X ) = Bi (dx) + Di, i = 0,1, dx = (d/dXi ,...,d/dXd).
The special forms of matrices A0 and A1 determine the natural representations of matrices Bi, Di by blocks in the forms:
^ (9X )=(%1 f/) BJ2 f* )V A ^), г = 0,l,
\Bh(9X) b%3(dx)) \d;2 dz3J
where B1A(dx), Di e (R), Bz2(dx), Dz2 e M™^(R), Bi3(dx),
Di3 e M(R), and * means transposition, and
d d
Bzj (dx) = ^ Bp1 dXp - ^ Cp>qdXpdXq, i = 0,1, J = 1,2,3,
P=1 P, <1=1
R01 = (h0p) _ r01 = (r0pq) _
^p (ust )s,t=1;m, pq ( st )s,t=1,m,
p02 = (h0P) _ _ r02 = (r0P1) _ _
p (st .! s=1,m,t=m+1,n, pq ( st .! s=1,m,t=m+1,n,
f>03 = (h0P) _ r02 = (r0P1) _
p (st )s,t=m+1,n, pq ( st )s,t=m+1,n.
The aim of our work is to construct the asymptotic solution u(e,x,t) for (P£) with a small parameter e ^ 0.
Thus, the investigation of the solution u(e,x,t) depends on the structure of the operator P£. The norm, which determines the convergence of the perturbed system solution, is also very important.
We denote the usual Sobolev spaces by Hs with the scalar product in the form:
(%v)s = i (i+eymmdt
JRd
where s e R, u(^) = F[u] (£ e Rd) and F-1 [u] are the direct and the inverse Fourier transforms of the function u in S'. Let HJ = (Hs )d be a notation of the Hilbert space, which is associated with the scalar product
d
(f 1, f2 )s,d = "^j(f1j,f2j )s , fi = (fi1 ,.",fid ), i = 1,2,
3=1
and with the norm || • \\sd, which is generated by this scalar product.
Let D'((a,b),X) be a space of vectorial distributions on (a,b) with values in Banach space X. We can set
W k'p (a, b; X) = {ue D' ((a, b);X); uij) e LP (a, b;X), j = 0,1,..., k},
for k e N* and 1 < p < to, where is the distributional derivative of order j and W0,p (a, b; X) = IP (a, b; X) for k = 0. We denote operator Li:j(dx) in the form:
Li j(dx) = Bij(dx) + Dip
and
F = col(f,g), Uo = col(u0 ,Ui),
where f,u0 e Mmx1 (R), g,Ul e Mi™-™)*1 (R). We assume that
H1: Bip, Cipq, Di, i = 0,1, p,q = 1,d are real symmetric matrices;
H2: (D C, Or- > (D03 V,V)R~-™ > Q0\V\2, with Q0 > 0; for all (e Rn and
v e R^- m.
Thus, the operator (P£) is a symmetric parabolic system (H1) and the operator (P0) is an elliptic-parabolic system in case: det B03 ^ 0 and B02 = 0.
3. Formal asymptotic expansions of the singularly perturbed Cauchy problem (Pe)
We construct the formal asymptotic expansions of the solutions u(e, x, t) for the Cauchy problem (P£) on the positive powers of the small parameter e in this section.
We can use the following asymptotic expansion of the solution u(e,x,t) for the problem (P£) according to the method of Lyusternik-Vishik [16]:
u(e,x,t) = V(x,t,e) + Z(x,r) = '¿jsk(Vk(x,t) + Zk(x,r)) + RN(e,x,t), (3)
=0
where r = t/e, and Z(x,r) = Z0(x,r) + ••• + eNZN(x,r) is the boundary layer function, which describes the singular behavior of the solution u(e, x, t) within a neighborhood of the set {(x,0), x e Rd}, which is the boundary layer.
The function V(x,t) = V0(x,t) + ••■ + eNVN(x,t) is the regular part of expansion (3).
We assume that the function Z(x,r) is small for large r, i.e. Z ^ 0 as t —y to). There is the solutions behavior u(e,x,t)^-u(0,x,t) of the singularly perturbed Cauchy problem (Pf), when e — 0 within the boundary layer, then the function Z(x,t) has to be reduced for the discrepancy elimination of the solutions u(e, x, 0) and m(0, x, 0).
We can substitute expansion (3) into (1) formally and identify the coefficients of the same powers of e, which contain the same variables.
Then we can get the following equations:
P0Vk = Fk(x,t), xe Rd, t>0, (4)
where
F0 = f(x,t), Fk = -P1 Vk_1, k=1,...,N,
^0 dTZk = Fk (x,r), k = 0,1,...,N, (5)
A1 (L0 zN + L1 ZN_1 + 8t Zn ) = 0, xe Rd, t > 0, F0 =0, F1 = —L0Z0 -A1 8tZ0,
Fk = —L(0Zk_1 — L1 Zk_2 — A1 &tZk_1, k = 2,..., N,
(P0 + eP1 )Rn = F(x,t,e), xe Rd, t > 0, (6)
F = -£n+1 (P1 VN + L1ZN) — eN A0 (L0 ZN + LxZN_x )
We can substitute (3) into initial condition (2)
RN(e,x,0) = 0, xe Rd, (7)
V0(x,0) + Z0(x,0) = U0(x), xe Rd, (8)
Vk(x,0) + Zk(x,0) = 0,xe Rd, k=l,...,N. (9)
We can use the following notation for convenience
Zk = ( , Vk = ( , Fk = , Fk = , (10)
where ,vk,fk,FkA e M^(R), Yk,wk,gk,Fk2 e MR).
We can use (5), (8), and (9) for Xk and Yk so that
&TXk = Fk1, Xk ^ 0, T^+w, (11)
and
dTYk + Lo3 Yk = Fk2 (x, r), xe Rd, t > 0,
Y0(x,0)=u1 (x)-w0(x,0), xe Rd, (12)
Yk (x,0) = -wk (x, 0), k = l,...,N, xe Rd,
where
F01 = 0, F11 = -^oi Xo — L02 Yo , Fki = -L{)iXk_i — L02Yk_i — LuXk_2 — L12Yk_2, k = 2,..., N, F02 = -L*o2^o), Fk2 = -L*o2Xk — Li3Yk_i — L\2Xk_i, k = l,..., N
Lij(o = Btj(o+m3, 1 = 0,1, j = 1,2,3.
Similarly, we can obtain the problems for vk and wk from (4) and (8), (9).
dt vk + Loi vk + l02 wk = fk (x,t), k = 0,l,...,N, l02vk + L03wk = 9k(x,t), x£ Rd, k = 0,1,...,N, t > 0, ^ Vo(x, 0) = Uoo(x) - Xo(x, 0), vk(x,0) = -Xk(x,0), k=1,...,N, x£ Rd.
Thus, we have the problems for determining the functions Xk, Fk, vk, wk and Rn .
4. Justifying asymptotic expansions of the singularly perturbed Cauchy problem (Pe
We investigate the validity of the expansion (3) in the following sections. We can consider the problem (13) in the next form
| dtv + L0iv + L02W = f(x, t), L*02v + L03w = g(x,t), xe Rd, t>0, (14)
v(x, 0) = h(x), x e Rd,
d d
Loj = Boj(dx) + Doj = ^ B°o0dXp — ^ Cp{dx^dx^ + D(jj, j = 1,2,3.
P=1 0,Q=1
We use the following problem for the solvability and regularity justifications of the problem (14)
'dt m + (D01+i\î\èoi mm + (D02+AmAOMO = m, t), (DO2 + i\C\B O2 (0)m + (Dos + AmXi (0)M0 = g(t, t),
M,o) = h(o,
(15)
%(o = itB0?&/\£\)-^\ i &m2),
0=1 0,0=1
where i = 0,1, j = 1,2,3, £ e Rd. We prove the following lemmas.
Lemma 1. The matrix D03 + ¿\Ç\£>03($) is invertible for Ç e Rd under the assumptions (H1), (H2) and the function ^ ^ (D03 + ï\Ç\^03(Ç))-1 is bounded on Rd.
Proof. We can use the method of the simultaneous reduction of two matrices to the diagonal form for proving this lemma and we assume that
D*03 = D03 and D03 = (d°st )s>t=n=m^., d°st >0 (s,t = n- m,n). There is an orthogonal matrix T1 G Mn-m (R), T-T1 = In_m, which
T1D03Tx = Al = diag(X1,..., Xn-m),
where Xk >0, k = 1,... ,n — m are the eigenvalues of matrix D03. We can use the transformation of the matrix B03 (£) in the form:
C(0 = A- TIB03 (i)Ti A-1.
As the matrix C(£) is a real symmetric, then there exists an orthogonal matrix T2 ({) G M(Rn-m), such that
T*C(0T2 = A(0 = diag(^i(£),..., (0),
where (£),..., p>n-m(0 are real eigenvalues of matrix C(^). Thus, we have the transformations of this type:
T * (OD03 T(0 = In_m, T * (0B03(0T(0 = A(0, (16)
where T($) = T1A^1 T2(£). We can use (16) so that
D03 + i\t\B03 (0 = T 0-1 (0(In-m + №A(.0)T-1(0. It means that the matrix D03 + i\£\B03(£) is invertible and we have
(D03 + №B03(0)-1 = T(0A1(0(In-m — i\Z№))T*(Z), (17)
where
A1 (0 = diag((l + \^\2v!)-1,..., (1 + \i\2yl-m)-1 )■ The orthogonality of the matrix T2 (£) implies the boundedness of the function C^T(C) on Rd.
The boundedness of the matrix (D03 + i\£\B03(^))-1 follows from (17). Lemma 1 is proved. □
We can obtain the solution of the problem (15) from Lemma 1
p(t,t) + K(0v(tt) = H(tt),
M,0) = h(0,
where
w&t) = (D03+z\c\B03(or1 m,t) — (d*02+z\c\b*02(orn,t)), (19)
K(0 = D01 + №¿01 (0 — (D02 + №¿02 (0)(D03+
+ №B03 (0)-1 (D*02 + mB02(O), (20)
H(t,t) = m,t) - (D02 + №B02m.Dos + №Bos(or'g&t). Lemma 2. The matrix K(£) can be represented in the form
K(0 = Ko(0 + im-1 (8 + |^|2k2(0, C e Rd, (21)
under the assumptions (HI), (H2), where the functions £ ^ Kj(£), j = 0,1,2 are bounded on Rd and K1,K2 are real symmetric for K2 > 0.
Proof. Let us substitute (17) into (20). We can obtain the representation (21), where
Ko (0 = Go1 - G02 T T 0 G*o2 - |^|2 {G^ TA1AT 0 ^ + bo2 TKX AT 0 G^),
Ky (0 = boi + Go2TAi AT0G*o2 - Go2TAiT0b0^-
- bo2TAi T0 G*o2 - |?|2 bo2 TAi AT0 b0^,
K2(0=bo2 TAi T 0b*o2.
Accordingly, Kj (£), j = 0,1,2 are bounded on Rd and K'i = Ki, K'2 = K2. It remains to prove that K2 > 0. Let us denote the eigenvalues of the real symmetric matrix A as Xj(A), j = 1,... ,m, where Xi ^ X2 ^ ■■■ ^ Xm. We can use Ostrowski's theorem so that
Xj(K2(£)) = Xj(b^TAiT0b*o2) = 0jXj(Ai) > 0,
where 0 < Xi(bo2TT0b*o2) < 0j < Xm(bo2TT0b*o2)■ It means that K2 > 0. Therefore, Lemma 2 is proved. □
We can prove the following proposition.
Proposition 1. Let the assumptions (Hi), (H2) be fulfilled and I e N0. If the conditions h e H^+2l+i, F = col(f,g) e Wl'i(0,T;H^+2) are true, then there exists a unique strong solution V = col(v, w) e W(0, T; Hsa) of the problem (14) and
\\V\\wl,^(o,T;H§) < G(T) (\\h\\s+2l+i,m + \\F\\wl,1{o,T;H^+2)) . (22) Proof. Consider the Cauchy problem
I m+K&m = 0, (23)
v(0) = h, 0<t<T,
in the Hilbert space H = [f = (fx,...,fm); (1 + |^|2)§fk(0 e L2(Rd), k = 1,... ,m], equipped with the scalar product (f,g)H = JRd (1 +
\?\2)s(f,9)R™ d£. We can use the representation (21) and demonstrate that the operator —K(£) • H ^ H satisfies the conditions
Re(—Kf, f)H < u(f, f)H, Re(—K* f, f)H < u(f, f)H, f G H,
where u = sup^eRd \\K0(£)llRm^.Rm + $ with a positive parameter 5 > 0. This means that the operator —(K + ul) is maximal dissipative on H.
The Cauchy problem (23) generates a C0 semigroup of operators {T(t), t > 0} on H [21]. Thus, we have the next estimation \v(-,t)\H < ewt\\h\\H for any hGH, i.e. \\f(t)\\ < eojt, where
Jt №(', t)\2H < —(K0V(;t), v(;t))H — (v(;t), KfJv(;t))H < 2u\v(-,t)fH.
Using Parseval's equality, we can get that the Cauchy problem (F[Kv] = K($)v)
ftv(t) + KV(t)=°, (24)
X0) = V0, 0<t<T,
where the operators {T(t),t > 0} on Hsm generates the semigroup C0, where v(-,t) = T(t)v0, \\T(t)\\ < e°jt. Thus, we can solve the Cauchy problem
jtz(t) + (K + uI)z(t) = f(ty<*, (25)
X0) = V0, 0<t<T, where the semigroup C0 has the representation in the form T0(t) = T(t)e-wt.
Hence, there exists a unique mild solution of this problem z G C([0, T];Hsm) for every y() G Hsm, fG L1 (0,T;Hsm) [21], and
z(t) = T0(t)V0 + f T0(t — s)f(s)e°JSds, 0
\Z\c([0,T];H?n) ^ W^Ws,™ + \f\L1(0,T;H;^n) .
Moreover, if the next y0 G H,fr+21, f G Wl'1(0,T;H^l) and I G № are true, then z is a strong solution of the problem (25), z G W(0,T; H^) and
\z\wl,^(0,T;H^) ^ C(T)(\y0\s+2l>m + \ f\wl,1(0,T;H^)).
We can note that the solution y of the Cauchy problem
| y(t) + Ky(t) = f, (26)
_y(0) = V0, 0<t<T,
and the solution z of the problem (25) are connected with the equality
y(t) = e-^ z(t).
Consequently, we have the same for y0, f and I G N * so that
WWW ) <C(T)(\\y0 W
Using (18), the last estimation and the boundedness of the matrix (G03 + ¿|<C|6(0)-1, we can obtain the next estimation
\\v\\wl'^{0,T;H^) ^
< C(T)(\\h\\
+ WfWwmp^-^^) + MwM^T^-J). (27)
We can get the estimation from (19) and (27) in the form:
\\w\\wl'^(0,T;H^) ^
< C(T) (Wh\\s+2l+1,m + \\f\\wl,1(0,T;H;;r+1) + W9\\wl,1{0,T;H;:+2m)) . (28)
Thus, the estimations (27) and (28) imply the estimation (23). Proposition 1 is proved. □
Let us consider the next Cauchy problem
(dTY + L03Y = F(X,T),XG Rd, r > 0, ^
\y(x,0) = y0(x), xG Rd.
Proposition 2. Let the assumptions (H1), (H2) be fulfilled and I G N*. If
the conditions y0 G Hsn-lm, F G W^ (0, to; Hsa-m) are true, then there exists
a unique strong solution Y G W^ (0, to; Hsa-m) of the problem (29) and the inequality is satisfied for this solution
\\9lTY(;r)\\s>n-m < Ce-^(\\y0Ws+Wa-m+
1-1 pT
+ ^WdvTF(;0)\\s+l-v-i>n-m +1 eq°eWdlTF(;9)Ws,n-m M). (30)
v=0 J0
Proof. The operator -L03(dx) is a dissipative under the assumptions (H1), (H2) and it generates the C0 semigroup of the contractions S(t) on Hsa-m. Thus, there exists a unique mild solution Y G C([0, to); Hsa-m) of the Cauchy problem (29). Hence, we can obtain the estimation \\S'(r)W < e-q°T, r > 0, which with the next equality
Y(;r) = S(t)V0 + [ S(6)F(; T-e)de
0
gives the estimation (30) in the case I = 0. We can obtain the estimation (30) by differentiating to r the equation (29) in the case 1^1. Proposition 2 is proved. □
Using these propositions, we can determine the functions Vk and Zk. Hence, it follows from (11) for k = 0 that X0 = 0. We can find the main regular term V0 = co\(v0 ,w0) of the expansion (3) from (13) and Proposition 1. Instantly, we have the following:
w0(x, 0) = F-1[(G03 + z|^03(0)-1 m, 0) - (G*02 + A№r2(№o(0)]. Lemma 1 and the Parseval equality permit us to obtain the next estimation
\\w0 (', 0)\\s,n.-m ^ C(\\g(-, 0)|s,n-m + K || s+1,m) ^
<C(\U0 \U,. + \\*,M)L,n ). (31)
Proposition 2 permits us to define the function Y0 as a solution of Cauchy problem (12). Moreover, we can obtain the next inequality from (30) and (31)
\dlrY0(;T)\s>n-m < Ce-^(\U0\\s+m,ra + \F(; 0)\\s+u). (32)
Thus, we can find the main singular term Z0 = col(0, F0) of the expansion (3).
Let us obtain the next terms of this expansion. Let us suppose that the terms V0,..., Vk-1 and Z0,..., Zk-1 are already found. We can obtain the terms Vk and Zk and show that the next estimations
\Vk\wl,^(0,T;H^) ^ C(T)(\U(0\s+2l+3k+1,n +
s+2l+3k-2,n
), (33)
and
HZk(;T)\a>n ^Ce-^(1 + Tk)(\^0\ s+l+k+1,n
s+l+k,n
) (34)
are true, if we suppose that such estimations are true for previous terms. We can note that the estimations (33), (34) for V0 and Z0 follow from (22) and
(32). 0 0
At first, if we solve the problem (11), we can get
/TO
Fk1 (;d)dd,
where the integral exists due to the estimation (34) for Zk-1. Using (34) for Zk-1 and for Zk-2, we obtain the next estimation:
\dlTXk(■,r)\\a,m = \dlT-1 Fk1 (■,r)\\a,m <
< C(\dlT-1 Zk-1 (;r)\s+hn + \dlT-1 Zk-2(■,r)\s+lin) <
< Ce-q°T(1 + Tk-1 )(\U0\s+l+k,n + \F(;0)\s+l+k-hn), (35)
for 1^1. Similarly, we can get the estimation (35) in the case I = 0.
Using Proposition 1 and vk(•, 0) = —Xk(•, 0), we can solve the problem (13) and find the functions Vk.
Using the next estimation
\\vk \\W ) <c(T)(\xk M)\\
s+2l+1,rri + Wk-1 \\w i,~(0,T;H°+3) ),
and also (22), (33) for Vk-1 and (35) for Xk, we can find the estimation (33) for Vk.
Instantly, we can obtain the next equality
wk(x,0) = F-1 [(D03 + i№03(®)-1 (gk(t,0) - (D02 + iltlB02(Z))xk(C,0))] and establish the estimation
W^k (-,0)W <C(\\9k (",0)W s+1,m) ^
< C(\Xk-1 (-,0)Ws+1,m + \\xk(-,0)Ws+1,m + \\wk-1 (-,0)\\s+1,n-m) <
<C(\\ U0 \\ s+k+1,n
+ \\ F(;0) \\
s+k,n ). (36)
Using (34) for Zk-1 and (35) for Xk, we can obtain the next unequality
WdlTFk2(;T)\\ Sin-m < C(\\ dlTXk(-,T)\\ s+1,m + WdlTZk-1(-,r)\\ a+1,n) <
< Ce-q°T(1 + rfc-1 )(\\U0Ws+i+k+1,n + \\F(; 0)\\a+z+fc,n). (37)
We can find the next estimation from (30), (36) and (37)
\\9lrYk(;T)\s;n-m < Ce-q°T(\\wk(;0)\\s+l!n-m+
+ J1 \№Fk2(-, 0)\s+i-v-1,n-m + [ eq°d\dlTFk2(;9)\Sin-rn M) < v=0 J0
< Ce-q°T(1 + r)(\U0\s+l+k+1,n + \F(-, 0)\\a++fc,n). (38)
The estimations (35) and (38) imply the estimation (34) for Zk. We can prove the main result of our work.
Theorem 1. Let us suppose that B and G satisfy conditions (H1), (H2) and 0 < I < N + 1. If the conditions U0 G Hn+2l+3(N+1), F G Wl+1,1 (0,T; H^+2l+3(N+1^) are true, then there exists a unique strong solution U G Wl,^(0,T; Hsn) of the problem (Pe). The expansion (3) is true for this solution, where Vk and Zk are determined by problems (13), (11), (12) respectively and they satisfy the estimations (33), (34). The estimation
\RN1 \wl.™(0,T;H*m) +£l/2\RN2\wl.™(00,T;H*n_m) ^G(T)eN+1 1 (39)
is true with C(T) depending on ^ \\U~0 L+2Z+3(Ar+1),ra, \F\wl+i1(0,T ; H^+21+3(N+1'>) and q0 for the remainder term R^ = col(RN1 ,R^2). In particular, if we assume N = 0, then there is the next estimation
WU-V0 - Z0\c{[o,T\;H'n) < C(T)e1'4.
Proof. Using the properties of the C0 semigroup of operators, we can obtain the solvability of the problem (P£). Indeed, the operator -(B(dx) + D) is closed and dissipative on Hsa. This operator generates the C0 semigroup of contractions on Hsn, which solves the problem (P£). Moreover, the conditions
u0 e Hsn+l, f e wl>1(0,T;H*), d%f(■ ,0) e Hsn+l-v-1, v = o,...,i-i, 1^1
imply the regularity of the solution U e W(0,T;Hsn). Using the method from [21], we can prove the estimation (39). Furthermore, all constants depend on the norms, which are indicated in the Theorem 1, and they are represented by C(T). Let us denote the next relations Rj = dltRN, Ru = dltRNi, i = 1,2. We can find that (BRl, Rl)sn is a pure imaginary value from the condition (H1). Consequently, we can get the next equation
j^AR^ ■ ,t),Rl( ■ ,t))s , n + 2(GRl (■ ,t),Rl (■ ,t))s , n = 2Re(dlt F( ■ ,t),Rl (■ ,t))s , n.
Using the assumption (H2), we can get the next inequality
d
¿t(ARl (■,t), Rl (,t))s,n + 2% (R12 (■,t), R12 (,t))s,n-m ^
< 2\(d\F(,t),Rl(;t))s>n\. (40)
The estimations (33) and (34) yield the next estimation
mF(■ ,t),Rl(■ ,t))s>n\ < eN+1 \(Pi(dltVN(■ ,t)) + e-1 Li(d'TZN(■ ,r)), (41) Ri( ■ ,t))s,n \ + eN-l\(Lo (dlT ZN (■ ,r)) + Li (dlT ZN-i (■ ,t)),
A-o Rl ( ■ ,t))s,n \ <
< C(T)(eN-1 K(t)\\Rli(■ ,t)\\Sim + (eN+1 + K(t)eN+1-1 )\Rl(■ ,t))ltn),
where 0 ^t ^T,t = t/e and K,(t) = e-q°t/£(1 + (t/e)N). Integrating (40) by t and using (41), we can get the next inequality
\\Rn (■ ,t))Hm + e\\Rl2( ■ ,t))\\ + 2qo I \\Ri2 (■ ,0)1
0
< \\Rn(■ Mlm + e\\Rl2(■ ,0))\\ln-m + C(T)(eN-1 i k(0)\R1i(■ ,0)\s;m d0+
0
+ I (eN+1 + K(d)eN-l+1 )\RjX;e)\Sin dd), O^t^T, (42)
0
We can note that
Ri(;0) = J2 (-A-1 (B(dx) + D))l-v-1 A-1 dXF(;0), I > 1,
v=00
and according to (7), R0(-,0) = 0.
Therefore, using the equality A-1 A0 = A0 and (34), (35), we can find the next estimation
\\^-1 d^tF( -, 0)\\s,n <
<eN+1 \\(^-1 P1 d?VN)(-, 0)\\s,n + eN+1-»\(A-1L1 d»ZN)(- ,0)\\a,n+ + eN-v\A0(L0dvTZN + hdvTZN-1)(-, 0)\\s,n <
< C(T)(eN + eN-v) < C(T)eN-v,
where 0<e<1,0^v^N.
Thus, we can obtain the next inequalities
\Ri(-,0)Wa,n < £ \\^-1 (B(dx) + D))1 -V-1 A-1 d,XF(-,0)\^ <
=0
-1
< C(T) ^ £-(l-v-1) - £N-v < C(T)eN-1+1. (43)
=0
If the conditions l<N+1,0^t^T,e^0 are true, we can obtain the estimations
<mRi1 (-,0)\\a,mdd< [f\(d)dd+ [f\(d)\Rl1 (-,e)fs>mdd<
00
<C(T)e+ [ k(0)\\ru(-,0)W2s,m dd, (44)
0
and
C(T) [ (eN+1 + K,(9)eN-l+1 )\Rl(-,d)\atn d6 <
0
< C(T)eN-l+1 + q0 [ \Rl2(-, 0)\2,n-m d0+ 0
+ C(T) [ (eN+1 + K(d)eN-l+1 )\\Rl1(-,e)\2Sim dd. (45)
0
Using the next inequality
\R-11 (-,t))W2.rn + £\Rl2 (-,t))W2 + Q0 I \\Rl2 (-,0)\\
0
<C(T)(eN-l+1 + I (eN+1 + K,(9)eN-1 )\\Rl1(-,e)\2s>m d0),0^t^T,
0
and the estimations (43), (44), (45), we can find the inequality (42).
[
Using Gronwall's lemma and the last inequality, we can get the estimations
\\R11 ( • ,t)Hm < C(T)eN-l+1(46)
and
e\\Ri2( • ,t)Hn-m + Qo I \\Rl2 (• ,0)\2an-m de^C(T)eN-l+\ O^t^T. (47)
Jq
Using (43) and (47), we can obtain the estimation
\Rl2 (.^,t)\'s,n-m ^
^ \Rl2 0)\\s,n-m + 2 I \Rl2 (,Q)\s,n-m\R(l+1)2 (^,0)\s,n-m dO <
rt \ 1/22
2
Ws'ri-m dd\ X
^C(T)e2(N-l+1) + 2 (J \\Rl2(v (f \\R{1+1)2 (;0)\\2,n-m dè) / < C(T)eN-l+1/2, O^t^T. (48)
The estimates (46) and (48) imply the estimate (39). Therefore, Theorem 1 is proved. □
Thus, we justify asymptotic expansions of the singularly perturbed Cauchy problem (Pf ).
5. Conclusions
In this paper we investigate the Cauchy problem for the singularly perturbed Tikhonov-type symmetric system of Fokker-Planck equations. This system consists of non-homogeneous constant coefficients linear parabolic partial differential equations with a small parameter. For these singularly perturbed Cauchy problems a method for constructing asymptotic solutions is proposed. We use the asymptotic method for this Cauchy problem and construct expansions of solutions in the form of decomposition, which has regular and border-layer parts. The asymptotic solutions in the form of regular and boundary-layer parts are obtained and the question of asymptotic solutions behavior when e ^ 0 is investigated. The main result of our work is a justification of an asymptotic expansion for this Cauchy problem. We prove the justification theorem for the asymptotic solutions. Our method can be applied in a wide variety of cases for singularly perturbed Cauchy problems of Fokker-Planck equations. The Fokker-Planck equation is connected with the Chapman-Kolmogorov equation for the transition probability function of a Markov process.
Our results give the approach to investigate the fast-changing processes in liquids and gases, plasma, solid state theory, magnetic, hydrodynamics, radiophysics, telecommunication technology, chemistry, biology, finance and
so on. An extension of Fokker-Planck equations with a small parameter to model non-Markovian processes is also possible.
Acknowledgments
This paper has been supported by the RUDN University Strategic Academic Leadership Program and funded by RFBR according to the research projects No. 18-07-00567.
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For citation:
M. A. Bouatta, S. A. Vasilyev, S. I. Vinitsky, The asymptotic solution of a singularly perturbed Cauchy problem for Fokker-Planck equation, Discrete and Continuous Models and Applied Computational Science 29 (2) (2021) 126-145. DOI: 10.22363/2658-4670-2021-29-2-126-145.
Information about the authors:
Bouatta, Mohamed A. — PhD's degree student of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: [email protected], phone: +7(495)9522823, ORCID: https://orcid.org/0000-0002-5477-8710) Vasilyev, Sergey A. — Candidate of Physical and Mathematical Sciences, Assistant professor of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: vasilyev\[email protected], phone: +7(495)9522823, ORCID: https://orcid.org/0000-0003-1562-0256, ResearcherID: 5806-2016, Scopus Author ID: 56694334800)
Vinitsky, Sergey I. — Leading researcher of Bogolyubov Laboratory of Theoretical Physics of Joint Institute for Nuclear Research, Professor of Department of Applied Probability and Informatics of Peoples' Friendship University of Russia (RUDN University) (e-mail: [email protected], phone: +7(49621)65912, ORCID: https://orcid.org/0000-0003-3078-0047, ResearcherID: B-7719-2016, Scopus Author ID: 7003380373)
УДК 517.928.1:517.958:519.633
PACS 02.30.Jr, 02.30.Mv, 11.10.Jj, 05.20.Dd, 05.40.Jc,
DOI: 10.22363/2658-4670-2021-29-2-126-145
Асимптотическое решение сингулярно возмущённой задачи Коши для уравнения Фоккера—Планка
М. А. Буатта1, С. А. Васильев1, С. И. Виницкий1,2
1 Российский университет дружбы народов ул. Миклухо-Маклая, д. 6, Москва, 117198, Россия 2 Объединённый институт ядерных исследований ул. Жолио-Кюри, д. 6, Дубна, Московская область, 141980, Россия
Асимптотические методы — очень важная область прикладной математики. Существует множество современных направлений исследований, в которых используется малый параметр, например статистическая механика, теория химических реакций и др. Использование уравнения Фоккера—Планка с малым параметром очень востребовано, поскольку это уравнение является параболическим дифференциальным уравнением в частных производных, а решения этого уравнения дают функцию плотности вероятности.
В работе исследуется сингулярно возмущённая задача Коши для симметричной линейной системы параболических дифференциальных уравнений в частных производных с малым параметром. Мы предполагаем, что эта система является неоднородной системой тихоновского типа с постоянными коэффициентами. Цель исследования — рассмотреть эту задачу Коши, применить асимптотический метод и построить асимптотические разложения решений в виде двухкомпонент-ного ряда. Таким образом, это разложение имеет регулярную и погранслойную части. Основным результатом данной работы является обоснование асимптотического разложения для решений этой задачи Коши. Наш метод может быть применён для широкого круга сингулярно возмущённых задач Коши для уравнений Фоккера—Планка.
Ключевые слова: асимптотический анализ, сингулярно возмущённое дифференциальное уравнение, задача Коши, уравнение Фоккера—Планка
