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Section 4. Mathematics
Turdiev Ulugbek, Senior teacher, Tashkent University of Information Technologies Karshi branch named after Muhammad al-Khwarizmi Tashkent, Uzbekistan E-mail: [email protected]
THE CAUCHY PROBLEM FOR A ONE-DIMENSIONAL SYSTEM OF BURGERS-TYPE EQUATIONS ARISING IN TWO-SPEED HYDRODYNAMICS
Abstract. The Cauchy problem for a one-dimensional system of Burgers-type equations arising in two-speed hydrodynamics is considered. The formula for solving the Cauchy problem in the class of functions of finite smoothness is obtained. It is shown that with the disappearance of the kinetic coefficient of friction, which is responsible for energy dissipation, the formulas proceed to the well-known solution of the Cauchy problem for the one-dimensional Burgers equation.
Keywords. Two-speed hydrodynamics, Burgers-type system, Florin-Hopf-Cole transformation.
Introduction
In recent decades, mathematicians have become increasingly interested in problems related to the behavior of solutions of systems of partial differential equations, with a small parameter with higher derivatives, and taking into account kinetic parameters. These problems arose from physical applications, mainly from modern hydrodynamics (compressible multiphase fluids with low viscosities). An analogy of the Burgers equation arises, for example, in the study of a weakly nonlinear one-dimensional acoustic wave moving at a linear speed of sound. In this case, the nonlinear in terms of the terms in the system of equations of Burgers type are obtained from the dependence of the speeds of sound on the amplitude of the sound wave, and the terms with the second derivative and the velocity difference represent the attenuation of sound waves associated with energy dissipation. In other words, these terms ensure the continuity of solutions and represent dissipative processes associated with the production of entropy. These members, in turn, provide non-reversal of waves [1]. The considered system is a special case of the system of equations of two-speed hydrodynamics in the one-dimensional case [2-6].
A one-dimensional analogue of the Navier-Stokes equations for compressible fluids can be considered a system of Burgers-type equations, which is a system of nonlinear convection-diffusion equations
ut + uux = vux-b (u-u)) (l)
ut + uuxx = vuxx + b(u - u), (2)
where u and u can be considered as the speeds of subsystems with dimensions [x ] / [t], that make up a two-speed continuum with corresponding partial densities p and p, p = p + p is the total density of the continuum, b = — b, b - is the friction coefficient with the dimension 1/ [t] ,which is an analogue of the Darcy coefficient for porous media. Positive constants v and v play the role of the kinematic viscosities of subsystems with the dimension [x] / [t].
The system of equations of two-speed hydrodynamics and the system of equations of Bugers type have much in common. For example, the adjective term corresponding to the dependence of sound on the amplitude of sound waves and linear viscosities v, v, friction coefficient b [1] in the right-hand side, responsible for the attenuation of sound waves. As for the properties of the solutions, they are completely different. With the Burgers equation system with vanishing coefficients v, v, b both strong (shock waves) and weak discontinuities are formed, while solutions of the two-speed hydrodynamics system do not possess such features. However, the scope of applicability of this system is by no means limited to the examples given; such systems arise in many problems, which is what determines its meaning.
The Cauchy problem for a system of equations of Burgers type
Consider for system (l), (2) in r [0,T ] =
= {(t, x ) :0 < t < T, x e aCauchy problem with the following initial data
Section 4. Mathematics
( = u ( x ), u = u ( x ) x e R.
t = 0 ov >> t=0 cA >
(3)
We will be interested in solutions of the Cauchy problem have a look
¥ t=o = ¥o (x)>
for a system of equations of the Burgers type (1), (2), in con- | ,x ) = Jgv (x&t) (Ç)dÇ + J Jgv (x,o,t -t) (^(Ç^dÇdT -trast to [7; 8] in the class of finite smoothness, namely u - o—
U t œ
—JJgt (x,Ç,t - t t )(vln|(£, t )-Vhnyfa t ))dÇdr (6)
v 0-»
c t c
f (t,x ) = f Gv (x,, (W)d + i f G (x,-) (f)f (W^to +
-co 0 -co
a t c
+ -ffgv(x,w,t-r)f(w,t)(vln^(W.T)-vinf(w,t))dwdt (7)
(t, c)2 u (t,x)e C1,2 (r[0 T]) is the class — functions onee continuously differentiable in s and twice contiguously differ-entiable in c.
The formula for solving the Cauchy problem for a Burgers-type equation system
Convenient to make ehe replacement variables Florin-Hopf-Co-e
$ = Exp
y = Exp
—— I udx 2v
--\udx
2v
0 -œ
Where
(x-I)2
4at
there is a fundamental solution to the one-dimensional heat
In this case, the functions u and v are expressed in terms equation.
Further suppose, as in [9], that the Cauchy data u0 (c), u0 (c) satisfy the following conditions
u(z)ez = a(c2) Juo(z)ez=o(c2), (8)
of the functions $ and y by the formulas
- fa -
u = -2v—, u = -2vL-cL. 4 ¥
In terms of the function $ and y the system of dynamic equations (l) and (2) takes the form
T1 = (vIT)c - -^).
x 4 v ¥
=f v + b (ln^).
m )c \ y )x v yv
From here, after integrating over c, we get b
fat = --fa(v lnfa-v lny) + C (t U, (4)
v b
Y, = vyc^ — yz(vln^-vlny) + C2(t)y, (5)
u
Where Cl (t) and C2 (t) are arbitrary functions of time. Solutions of the Cauchy problem for system (4), (5) with
For large |x|.
Introduced the following functions
(x _ y )2 y F (u, x, y ,t) = A-LL +
2t
, (x-y)
Ju(t ,n)dn.
(9)
data
=o=4>o (x )
F (u, c, y,t,T) = -+ J u (T,n)-n. (10)
2 (t - A)
Note that the following equalities are true.
y
F(u0,c,y,t)- F(u0,c,y,t) = J [u0 (n)- u0 (n)]-n.
y
F (u, c, y,t ,t)-F (u, c, y,t,T) = J u (t,v)-u (T,n)]-n. For any c,y,t,t .
Differentiating both sides of equality (6) and (7) with respect to the variable c. after simple transformations we get
1 œ 1 < b tœ x <1 1 ^ ^
(¡)x (t,x )=-— ju0 (1)GV (x,1,t )Exp[-—ju0 (n)dri] d<+ — JJ Gv (x,<,t-r^Exp -—jv(i:,n)dri]j [u{r,r])-u{r,T])
-<X:
0-<x> t œ
àï\d< dx,
1 ^ 1 b t ™ c e ^^ y/x (t,c )=-— ju0 (eG {c,e,t )Ecp[-—ju0 (n—]e%-—U c-Gy (c,e,t-t )Ecp -—Jv (t,n)en]j [(t,n)-u (r,q) en-e-t,
v -rn v 0 v 0 t _ v 0 0 _
From here, taking into account (9), (10) and from the definition of the fundamental solution of the thermal conductivity operator, we obtain
1 c — £
-f2 (u,uc,%,t,r)e%eT, (11)
1 x E 1 1 t
fa (t, x )=- —T= i ^^ Exp[-—F (uo,x ,E,t ,)]dE- — if 2W4nvt _Jœ t 2v ' 2v o
1 p x -p 1 1 i Vx (t,xr—^ f-Exp[-—F(<,x,e,t,)]d%-— \ f
2W4^vt - t ?v v ' 2v J J
2v ' "' " 2v 0L,]4nv(t -t) t The following notation is used in formulas (11) and (12):
G2 (u,ux,Ç,t,r)dÇdr. (12)
F2 (u,v,ç,r,t,x) = {Cj (t) + b[Fj (u x,t,t) - Fl (vx,t,tExp --1 Fl (ux,t,t)
2v
G2 (u,v,£,r,t,x) = {C2 (t) -b[F (ux,t,t,%) - F1 (vx,t,t)]Exp F1 (ux,t,t,%)
Theorem. Let u0 (x), u0 (x), e C2(R) dW^ (R) and sat- Then to solve the Cauchy problem (l) - (3) Spread For-
isfy relation (10), where W2, (R) is the Sobolev space. mulas
i ——— Exp[—— F(u0x,—,t,)]d—+\ i . ^ x—— F2 (u,ux—,t,r)d%dr J-» t 2v v ' ' JoJJ(t-t) t-t v ' ' u (x ,t) =--- \j~ -,
i Exp[--F (u0x ,—,t ,)]d— + \ i --^ F2 (u,uix,—,t,t~)d—dt
2v JoJ -»J(t-t) '
J x—0Exp[—— F(v0x—,t,)]d— + \ J . ^ x——G2 (u,u x—,t,t)d—dt J-» t 2v v ' ' joj-» j(t-t) t-t v ' ' u(x,t) =--- \f -'
J Exp[-—F((x—,t,)]d—+J) G2 (u,ux—,t,t)d—dt
The investigation. Let u0 (x), u0 (x) satisfy the conditions Then for solving the Cauchy problem (l) - (3) the fol-
of the theorem. lowing formulas are valid
f^^Exp^Fiu^t,]) jX^1 + -T [uM — JZçp {u,ùx,^,t,r)d^dT ^
u
j Exp[—— F (u0x ,Ç,t ,)) j Exp[—— F (u0x ,Ç,t,))
j —oo 2v ' ~œ 2v
r Exp[~—F(uax,Ç,t,)]dÇ fT 4 1 + T\u(x,t)-G2(u,ùx,Ç,t,r)dT (x,t) = ^_^ 2v 1 j °J'T__:_. (14)
f ^Exp[-2vF (u0x ,)) f^Exp[-2vF ((x ,))
Comment. When the friction coefficient b disappears well-known solution of the Cauchy problem for the Burgers
(in the absence of energy dissipation due to the friction coef- equation [9].
ficient and Ck (t) = 0(k = 1,2) solution (13), (14) goes to the
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