An analysis of the Avalos-Triggiani problem for the linear Oskolkov system of non-zero order and a system of wave equations

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The Avalos-Triggiani problem for a system of wave equations and a linear Oskolkov system of non-zero order is investigated. The mathematical model contains a linear Oskolkov system describing the flow of an incompressible viscoelastic Kelvin-Voigt fluid of non-zero order, and a wave vector equation corresponding to some structure immersed in the fluid. Based on the method proposed by the authors of this problem, the existence of a unique solution to the Avalos-Triggiani problem for the indicated systems is proved.

Avalos-triggiani problem, incompressible viscoelastic fluid, linear oskolkov system

Короткий адрес: https://sciup.org/147242593

IDR: 147242593   |   DOI: 10.14529/mmp230407

Текст краткого сообщения An analysis of the Avalos-Triggiani problem for the linear Oskolkov system of non-zero order and a system of wave equations

Let Q be a bounded domain in R n ,n = 2, 3, with sufficiently smooth boundary dQ. Let u = col(u 1 , u 2 ,..., u n ) be a n— dimensional velocity vector n = 2, 3 , the scalar function p be a pressure, and the vector w = col(w 1 .w 2 ,..., w n ) be a vector of displacement of a body, which occupies the domain Q s , and is immersed in a fluid occupying the domain Q f . Therefore, Q = Q s U Q f , Q s П Q f = dQ s = r s is the common boundary of Q s , and Q f . Let us denote the outer boundary of Q f by r f (see Fig).

Our goal is to investigate the Avalos–Triggiani problem [1, 2] for the case when the fluid in Q f is an incompressible viscoelastic Kelvin-Voigt fluid of the nonzero-order [3]. The considered mathematical model is determined by the system

K

(1 kV 2 )u t ^ V 2 u ^ e l V 2 w l + V p = 0    V (t, x) G (0, T] x Q f = Q Tf ,   (1)

l=1

∂wl —— = u + ai W1,   ai G R-, ∂t в1 G R+,    l = 1, K,    V(t,x) G QTf, (2) V • u = 0, V(t,x) G QTf, (3) wtt — V2w + w = 0 V(t, x) G (0, T] x Qs = Qts (4) with the boundary value conditions ulrf = 0,       V(t,x) G (0,T] x rf = rTf,                      (5)

w i | r f

= 0,        V (t, x) G r Tf ,

(6)

u = W t ,

V (t,x) G (0,T] x r s = r Ts ,

(7)

-      = pv      v (t,x) e r Ts                          (8)

and the initial value condition

(w(0, ), W t (0, ), w i (0, ),..., w k (0, ),u(0, )) = (w o , w i , w io ,..., w k o ,u o ) e H , (9) where H = (H 1 (Q s )) n x (L 2 (Q s )) n x H 1 x ... x H K x H f and H l = (L 2 (Q s )) n , l = TT" K, H f = { f e (L 2 (Q f )) n : V- f = 0 in Q f and [f v] | r f = 0 } .

Fig. Physical model

In system (1), the parameters κ and µ characterize the elastic and viscous properties of the fluid, respectively, the parameters e l , l = 1, K determine the time of pressure retardation (delay), v is a unit normal vector. In the case of K = 0, к = 0 , problem (1) -(8) was investigated in [1,2], and for K = 0, к = 0 in [4], [5]. The case of K = 0, к = 0 is investigated for the first time.

1. Reduction to the Cauchy Problem

Following [1,2], we assume that p(t) satisfies the following elliptic problem:

Ap = 0 in QTf, ∂u ∂w p = dV •v - dV •v on Гт',                        (10)

∂p

— = Au v   on   r Tf .

∂ν

Then the pressure p can be represented as follows:

p(t) = D s { (ddVt) v - dwVt) v) r } + N f ((Au(t) v) r Tf )    in    Q Tf ;

where the Dirichlet map D s is defined by the relations

f Ah = 0 in Qf, h = Ds(g) О < h= ∂h g on rs, =0 on rf, ∂ν and the Neumann map Nf is defined by the relations

f Ah = 0

in

Q f ,

h = N f (g ) ^ s

h =0 ∂h

on

r s ,

< dV = g

on

r f

Then original system (1) – (4), which describes the interaction of the fluid and the body immersed in the fluid, takes the form

K

(1 - KV2)ut - ^V2u - ^ в1 V2wi - G1W - G2U = 0   V(t,x) G ^Tf,(11)

l =1

dW = u + aiwi,   ai G R-,   в1 G R+,   l = 1?X

V- u = 0, wtt - V2w + w = 0 V(t, x) G QTs(14)

with the boundary value conditions u|rf = 0, V(t,x) G Гт,, wl Ipf = 0       V(t,x) G rTf, u = Wt,       V(t,x) G rTs, where

G i w = V{D s { ( 5^ • v) r }} in   Q t, ,

G 2 u = -V{ D s { ( ddtl v) r^ } + N , ((^u(t) v) F Tf ) }    in    П т, .

Let us rewrite the problem (11) – (17), in which pressure is excluded, in the form of an abstract Cauchy problem:

Lv) = Mv, v(0) = v 0 , where the operators L and M are defined by the matrices

I

O

O

O

... O

O

O

I

O

O

... O

O

O

O

I

O

... O

O

L :=

O .

O .

O .

I

.

... O

..

O .

.

.

O

.

.

O

.

.

O

.

.

O

..

..

... I

.

.

O

O

O

O

O

... O

A κ

/

O

I

O

O

... O

O

M : =

A -

O

O .

IO O O

.

O α 1 O

.

O O α 2

.

... O

... O

... O

..

O I I

.

.

Here v = col(w, w t , w 1 , is clear out of the conte

.      ..      .     .      .         .

.                       .                   .                        .                         ..                                   .

O   O   O    O   . . .  α K       I

G i    O e i A в 2 А ... в к A vA + G 2 )

w 2 ,..., w K , u), A K = 1 kV 2 , I is a unit op xt. We study the problem (18) using the results

erator. Its domain obtained in [6–9].

Lemma 1. Let к E R , ^ E R + , the operators L and M be linear continuous operators from G to H ( L,M E L( G , H ) ), then there exists L - 1 E L( H ). Here the space G = (H 2 (^ s )) n x (H 2 (Q s )) n x9 1 x ... x9 K x9 f , where 9 l = (H 2M )) n , l = TT K 9 f is closure according to the norm of the space (H 2 (fi s )) n that is the space of infinitely differentiable solenoidal functions such that (15) – (17) are fulfilled.

Theorem 1. For any к E R, ^ E R + and v 0 E G , there is the unique solution to the problem (18) v E C ^ ((0,T ], G )

In conclusion, we note that we intend to develop our research in the direction indicated in [10–12].

Acknowledgements. The work was carried out within the framework of solving problems for the development of the laboratory of Differential Equations and Mathematical Physics of Yaroslav-the-Wise Novgorod State University. The authors express their gratitude to Professor G.A. Sviridyuk for his attention to the work and discussion of the results.

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