CC BY
14
YAK 517.55
Successive Approximation for the Inhomogeneous Burgers Equation
Received 06.04.2017, received in revised form 08.03.2018, accepted 09.06.2018 The inhomogeneous Burgers equation is a simple form of the Navier-Stokes equations. From the analytical point of view, the inhomogeneous form is poorly studied, the complete analytical solution depending closely on the form of the nonhomogeneous term.
Keywords: Navier-Stokes equations, classical solution. DOI: 10.17516/1997-1397-2018-11-4-519-531.
Azal Mera*
University of Babylon Babylon Iraq
Vitaly A. Stepanenko^
Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041
Russia
Nikolai Tarkhanov*
Institute of Mathematics University of Potsdam Karl-Liebknecht-Str. 24/25, 14476, Potsdam
Germany
Introduction
The inhomogeneous Burgers equation is the simplest nonlinear model equation for diffusive waves in fluid dynamics. It reads
ut - vuXx + uuX = f, (0.1)
where u stands generally for the fluid velocity, x the space variable, t the time variable, v is the kinematic viscosity, or the diffusion coefficient, and f a given forcing term. The inverse R = 1/v of the diffusion coefficient is known as the Reynold number. Burgers [3] first developed this equation primarily to shed some light on turbulence described by the interaction of two opposite effects of convection and diffusion. However, turbulence is more intricate in the sense that it is both three-dimensional and statistically random in nature. Note that equation (0.1) is parabolic, if v > 0, whereas (0.1) with v = 0 is hyperbolic. More importantly, the properties of solutions to parabolic equations are significantly different from those of hyperbolic equations.
The mathematical structure of equation (0.1) includes a nonlinear convection term uu!x which makes the equation more interesting, and a viscosity term of higher order uxx which regularises the equation and produces a dissipation effect of the solution near a shock. When the viscosity
1 [email protected] ^ [email protected] © Siberian Federal University. All rights reserved
coefficient v vanishes, the Burgers equation reduces to the transport equation, which represents the inviscid Burgers equation u't + uu'x = f.
The study of equation (0.1) goes back as far as Forsyth [7] who treated an equation which converts by a change of variables into the Burgers equation. In [1] Bateman introduced the equation (0.1). He was interested in the case where v ^ 0, and in studying the movement behaviour of a viscous fluid when the viscosity tends to zero. Burgers published a study on equation (0.1) in his paper [3] devoted to turbulence phenomena. Using the transformation discovered later in [6] and independently in [8] Burgers continued his study of state aspects of what he called "nonlinear diffusion equation." The results of this work can be found in [4]. The objective of Burgers was to consider a simplified version of the incompressible Navier-Stokes equations by neglecting the pressure term. Among the most interesting applications of the Burgers equation in dimension one we mention traffic flow, growth of interfaces, and the financial mathematics, see for instance [11,18].
The nonlinear Burgers equation with f = 0 can be converted to the linear heat equation and then explicitly solved by the Hopf-Cole transformation. We look for explicit solutions to the forced Burgers equation (0.1), where f (x,t) is the forcing term in a cylinder C := I x (0,T) over a finite interval I = (a, b) of the real axis. In this work we focus on existence, uniqueness and regularity results for the inhomogeneous equation.
For f = —\w'x, equation (0.1)becomes u!t-vu'Xx +uu'x = —\w'x, which is known as the Burgers stochastic equation, where w = w(x,t) stands for the white noise. Using the transformation u = —\v'x one sees readily that this is equivalent to the equation
I n ^ / I \2
vt — vvxx — 2 (vx) = w,
which was introduced in [9] and quickly became the default model for random interface growth in physics.
In [2], the main result is the existence and uniqueness of a solution to the inhomogeneous Burgers equation in the anisotropic Sobolev space H2'1(C). This latter is defined to consist of all functions u e L2 (C) whose weak derivatives d^d?u belong to L2(C) for all nonnegative integers a and j satisfying a + 2j < 2. In our paper we develop another approach which has the advantage of being consructive and extends to more general nonlinear problems. It goes back at least as far as [15].
1. Linearisation
The change of the unknown function by
U' d
u = -2v ~Uj = -vdX loS U2 (1.1)
reduces the inhomogeneous Burgers equation to
ut — vulx + uu'x = — 2v — (
d ( U' — vux
t — vuxx + uux = — 2Vd)X\-U-
=f
in C. While being intermediate in the Hopf-Cole transformation, the latter equation is equivalent to
U — vuxx + V (x,t)U = 0, (1.2)
where V(x,t) = — J f (x,t)dx — c(t).
The most familiar boundary value problem for solutions of parabolic equation (0.1) is the first mixed problem
u(x, 0) = Uo(x), if x € I, d 3)
u(x,t) = ul(x,t), if (x,t) € dI x [0,T], ( )
where we require u0(x) = ul(x, 0) for x € dI. The inverse of transformation (1.1) is given by
U(x, t) = C(t) exp ^ — — j u(x, t)dx^J,
where C(t) is an arbitrary function of t independent of x. Problem (1.3) transforms immediately to the third mixed problem for solutions to parabolic linear equation (1.2)
U(x, 0) = C(0) exp (--l uo(x)dx\ if x € I,
_ \ 2vJ J (1.4)
UX(x,t) + —ui(x,t)U(x,t) = 0, if (x,t) € dI x [0,T],
the condition on the lateral boundary being of Robin type.
While the transformed Burgers equation given by (1.2) is linear, the coefficient V(x,t) multiplying U is in general a nonsmooth function of (x,t) € C. Let alone that the boundary conditions become harder to handle. Note that equation (1.2) is of independent interest. In [10], Kac uses the equation to evaluate the Paley-Wiener integral, in which application the coefficient V(x, t) depends solely on x. To this end he reduces (1.2) to a Fredholm equation of the second kind which is solved by the Laplace transform. The solution of the integral equation is built in the form of a series expansion over eigenfunction of the equation. In order to establish the convergence of the series Kac first requires the boundedness of V from above, but then he gets rid of this restriction.
In our work we go to treat the inhomogeneous Burgers equation (0.1) directly by reducing it to a nonlinear Fredholm equation of the second kind. This approach can actually be specified within the general framework of nonlinear Fredholm operators in Banach spaces. It leads to existence, uniqueness and regularity theorems provided that the nonlinear term is dominated by the principal linear part of the problem.
2. Reduction to an integral equation
Denote by x,t) the standard fundamental solution of convolution type to the heat equation
u!t = vu'Xx in R x R, i.e.,
^(x,t) = \ 6XP( ivt)' if f>
0, if t < 0.
Using the properties of convolution we get (dt — vd2x)(^ * u) = S * u = u for all distributions u of compact support in R x R. As is known, the operator := ^ * u maps distributions of compact support in R x R continuously into distributions in R x R.
Rewrite equation (0.1) in the form u't — vu'Xx = f — N(u) in the cylinder C, where N(u) := uu'x. Assume that u € H2'1(C) and f € L2(C). By Theorem 12 of [16] it follows that u € C(C), and so N(u) € L2(C). Hence, both sides of the equation belong to L2(C). On multiplying both sides of the equation by the characteristic function xc of C we obtain an equality of distributions of compact support in R x R, namely
Xc (u't — vu'Xx) = (dt — vdX) (xcu) + [xc, dt — vd2x] u = xcf — XcN(u),
where the bracket [xc, dt — vdx] stands for the commutator of the operator of multiplication by Xc and the heat operator. Applying the fundamental solution & yields
Xcu + & (xcN(u)) = & (xc f) — & ([xc, dt — vd2x] u) (2.1)
on all of R x R. An easy manipulation of the Stokes formula shows that the distribution [xc, dt — vd^]u is supported on the boundary of C. To wit,
{[Xc,dt — vdx] u,g) = — ugdx + v (udxg — dxug) dt =
■Jdc Jdc
,-b t_T ,T _b
= / ug |t=0 dx + v (udxg — dxug) \Xx=adt ■J a J 0
holds for all smooth functions g with compact support in R x R. We have thus proved the following lemma.
Lemma 2.1. For any function u e H2'1(C) satisfying the inhomogeneous Burgers equation, we get
u — Ps(dx' u) + Vv (N (u)) = Pi(uo) + Pd(ui) + Vv (f),
where
Pi(uo) = ^(x — -,t)uodx', Ps (v) = v '^(x — -,t — ■)vl\X,=dt',
1 t , o t x =a
Pd(ui) = —v dx'^(x ■ ,t — ■)ui\XX'^adt', Pv(f) = I / ^(x ■ ,t — ■)fdx'dt'. Jo Ji Jo
If I = (—&>, to), then both Ps(dx'u) and Pd(ul) vanish and the initial value problem for the inhomogeneous Burgers equation reduces to the nonlinear integral equation of Volterra type
u(x,t)+ [ ( ^(x — ■,t — ■)N(u)dx'dt' = Pi(uo)+ Pv(f) (2.2)
JI Jo
for (x,t) e I x (0,T). This equation can be solved by successive approximation, see for instance [10]. In the case of finite intervals I one has to substitute for ^ the Green function of the first mixed problem for the heat equation in the half-strip I x (0, to).
3. The main theorem
For 1 < p < to and an integer s > 0, we denote by Lp(I) and Hs(I) the usual spaces of Lebesgue and Sobolev, respectively. If B is a Banach space, we write Lp((0,T), B) for the space of all measurable functions u : (0,T) ^ B with the property that
/0
\\u\\b dt)
i/p
d
< CO.
As usual, for p = to one substitutes the essential supremum of \\u\\B on (0, T) for the integral on the right-hand side.
We study the first mixed problem for a class of semilinear parabolic equations which includes, in particular, the Burgers equation in the cylinder C. More precisely, consider
u't — v (t)u'x + e(t)uu'x + c(x,t)u'x = f in C,
u(x, 0) = uo(x), if x e I, (3.1)
u
(x,t) = 0, if (x,t) e dI x (0,T),
where f G L2(C) and u0 G H(I) are given functions. The coefficients v(t) and e(t) are assumed to take on their values in bounded intervals of R>0 away from zero. The coefficient c(x,t) is required to be bounded in some sense in all of C. More precisely, we assume that both \c(x,t)\ and \c'x(x,t) \ are bounded by a constant R> 0 for all (x,t) G C.
Theorem 3.1. Suppose that f G L2(C), u0 G Hq(I), and the coefficients v, £ and c satisfy the above conditions. Then problem (3.1) admits a unique solution u G H 21(C).
The proof of Theorem 3.1 is based on the Galerkin method. As usual we introduce an approximate solution by reduction to finite-dimensional spaces. By the Galerkin method we establish the existence of an approximation solution, using an existence theorem for a system of ordinary differential equations. We approximate the equation of problem (3.1) by a simpler equation. On using a compactness argument we then make a passage to the limit, thus obtaining the desired solution to (3.1).
4. Galerkin method
On multiplying the equation u't — vu'Xx + £uu'x + cu!x = f by a test function g G H (I) and integrating by parts from a to b we get
/• b /• b /• b /• b /• b
/ u'tgdx + v (t) u'xg'xdx + £(t) uu'x gdx + c(x,t)u'xgdx = fgdx (4.1)
•J a J a J a a a
for almost all t G (0,T). This is a weak formulation of the differential equation of (3.1). Any function u G HX(C) satisfying (4.1) and the conditions of (3.1) is called a weak solution to (3.1).
To prove the existence of a weak solution of problem (3.1), we choose an orthonormal basis (ej )j=1 2 in L2 (I) consisting of the eigenfunctions of —d2x for the Dirichlet problem
—d2xej = Xj ej in I,
ej = 0 on dI
in I. An easy calculation shows that
\ ( jn \2 , , I 2 . /. x - a\
= ï- , ej (x) = \ -- sin [jn--
j \b - a) j ' Mb - a \ b - aJ
for j = 1, 2,.... Each function u G L2(C) can be decomposed into the Fourier series
œ
u(x,t) = ^2 Ck(t)ek(x), k=i
where ck = (u,ek )l2(i) and the series converges in the L2(I) -norm for almost all t G (0,T). Given any n = 1,2,..., we look for an approximate solution un to problem (3.1) of the form
n
un(x,t) = £ ck(t)ek(x) (4.2)
k=i
for (x,t) G C. The coefficients ck (t) depend on n, however, we do not display this explicitly by abuse of notation. Since the system (ek )k=12 is an orthonormal basis of L2 (I), it follows that this system is an orthogonal basis of H^I). More precisely, the norm of ek in H1 (I) just amounts
to V1 + Xk and one easily checks that (u, ek)Hi(I) = (1 + Xk)(u, ek)l2(i) for all k = 1, 2,.... By the above, we assume u0 e H^(I), and so
uo(x) = ^2 co,kek(x) k = 1
on I, where c0,k = (u0,ek)l2(i) and the series converges in the H 1(I) -norm for almost all t e (0,T). We require each approximate solution un to satisfy the system
^n
/•b fb fb fb
/ dtunCjdx + v(t) dxundxCjdx + z(t) undxunejdx + c(x,t)dxunejdx =
J a J a J a J a
b
= fej dx, (4.3)
a n
un(•, 0) = co,kek k=i
for all j = 1,... ,n and almost all t G (0, T).
Remark 4.1. By the very construction, the sequence of initial data un(•, 0) converges to u0 in the Sobolev space Hq (I).
The simple part of Galerkin method consists in establishing that system (4.3) possesses a solution.
Lemma 4.2. For each n = 1, 2,..., system (4-3) admits a unique solution un of the form (4-2). Proof. Since e1,... ,en is an orthonormal system in L2(I), we get
/b n rb dtunejdx ^^ ck(t) ekejdx = cj(t)
k=i
for all j = 1,..., n. On the other hand, from the equality —d^ek = Xkek it follows that
b n b v(t) dxundxej dx = v(tck(t)\k ek ej dx = v (t)cj (t)Xj.
k=i
Summing up we see that system (4.3) is equivalent to an initial value problem for the coefficients
c1(t),..., cn(t). To wit,
cj + ^j v(t)cj = fj (t) aj,k (t)ck - aj,k,i (t)ck ci in (0,T),
k = 1 k = 1,...,n (4.4)
cj(0) = c0,j
for j = 1,... ,n, where
bb
j,k (t) = c(x,t)dxek ej dx, aj,k,i(t) = e(t) ek dxeiej dx ab a
fj (t) = f f
a
r e jdx.
The left-hand sides of equations (4.4) constitute a system of n uncoupled linear ordinary differential equations. The right-hand sides are well-defined quadratic functions of c1(t),..., cn(t) whose coefficients are integrable functions of t e (0,T), for f e L2(C) and ej, c(x,t) are regular. A familiar argument shows that the initial value problem (4.4) has a unique maximal solution defined on some interval [0, Tn] with Tn < T. If Tn < T, then \\un(-,t)\\Hi(I) must tend to as t ^ Tn. The a priori estimates we shall establish later show that this does not happen, and therefore Tn = T, cf. [17, p. 192]. □
5. A priori estimate
In the sequel we use the letters C1; C2, etc. to designate diverse constants. They need not be the same in numerious applications unless otherwise stated. As mentioned, we assume
^ 5 t) 5 ^ (5.1)
£! 5 £(t) 5 £2 V 7
for all t e [0, T], where Vj and £j are positive constants.
Lemma 5.1. There is a positive constant C\ independent of n, such that for all t e [0, T] we get
r t
uj
\\un(-,t)\\l*{I) + V! f \\dxun(-,s)\\l2{I)d,s < C1. Jo
Proof. Multiplying (4.3) by cj (t) and summing up for j = 1,... ,n one obtains
r-b b -i b
1 d !-b !-b 1 !-b !-b
2 dt u^dx + v(t) (dxun)2dx — - dxc(x,t)u2ndx = fundx.
a a a a
Indeed, because of the boundary conditions one gets
t-b r b
£(t) J u2ndxundx = £(t) J 3 dx(unfdx = 0
and an integration by parts yields
fb 1 fb
c(x,t)undxun dx =-- dxc(x,t)u2ndx.
J a 2 J a
Then, on integrating in t over [0,t] and using estimates (5.1) one concludes readily that 1 ^
1 r
- \un(-,t)\2L2{I) + V1 J \\dxun(-,s)\\2L2{I)dS <
1 R ft ft
< -\\un(0)\L2(I) + 2 J \\un(-,s)\\l2 (I)ds + J0 \\f (■,s)\L2(I)\un(-,s)\L2(I)ds
for all n = 1, 2,....
Using the Poincare inequality
(b — a)2
\\un\\L2{I) < --- \\dxun\\L2{I)
along with the elementary inequality
\rs\ < -r2 + --s2 (5.2)
for e = 2 V1 we get
( b — a) 2
\\un (^,t)\\2L2 (I) + V1 \\dxun(^s)\\2L2{I)ds <
o
< \\un(; 0)\\2L2(I ) + R 0 \\uri (■, s)\\2L2{I)ds + 0\\f (■, s)\\2L2(I)ds.
The second term on the right-hand side of this inequality is obviously dominated by
R Jo ) + V1 J \\dxun^,s' )\\2L2{i )dS')dS
for all t € [0,Tj. Since the sequence (un{:, 0))n=12 converges in Hq(I) to u0 (cf. Remark 4.1) and f € L2(C), there is a positive constant C independent of n, such that
(b - a)
2
\\unt, 0)\2l2{i) \\f \\Ue) < C
whence
\\un {•ML (I) + V1 J0 \\dxun{^ S)\2L2{i)ds <
< C + RJ (\\un^,s)\\2L2(i) + V1J \\dxur^,s')\\2L2(i)ds')ds.
By the Gronwall inequality,
\\un{•, t)\\L2(I) + V1 \\d un{•, s)\2L2(I)ds < C exp(Rt)
holds for all t € [0, Tj. On choosing C1 = C exp {RT) we establish the desired estimate. □
The Poincare inequality shows that our next assertion actually strengthens Lemma 5.1. Lemma 5.2. There is a positive constant C2 independent of n, such that for all t € [0, Tj we get
\dxun{^,t)\2L2(I) + v1 J \\dXun^,s)\\2L2 (I)ds < C2.
Proof. The proof of this a priori estimate is much the same as that of Lemma 5.1. □
Lemma 5.3. There is a positive constant C3 independent of n, such that for all t € [0, Tj we get
\\dtun\\2L2(c) < C2. Proof. For any n = 1, 2,..., consider the function
5n(x,t) = f {x,t) + v{t)dXun(x,t) - e(t)un(x,t)dxun{x,t) - c(x,t)dxun(x,t)
of {x,t) € C. To establish that dtun is bounded in L2(C) we will first show that the sequence (5n)n=1,2,... is bounded in L2(C). By the very assumption, we get f € L2(C). According to Lemmata 5.1 and 5.2, the terms cdxun and vdxun are bounded in L2(C) uniformly in n. It remains only to check that eundxun is bounded in L2(C) uniformly in n.
Lemma 5.1 shows that the norm \\un\L~([0 T] Hi(I)) is bounded uniformly in n. Then, using the embedding of ) into L^(I) yields
J \\eundxun\\2L2(i)dt < e22^ (\\un^,t)\\2L^(i)\\dxun^,t)\\2L2(i)^ dt <
< 4C2 0 (\\un{;t)\\2m(i)\\dxun{,t)\\2L2{i))dt <
< 4C2 \\un(^,t)\\2L^ ([0 ,T], Hi(I))\dxun{^,t)\\2L2(C)) dt,
>([o ;
where C is a constant independent of n. Hence it follows that Sn is bounded in L2(C) uniformly in n. This already implies that the sequence (dtun)n=1,2 ,... is bounded in L2(C). Indeed, from (4.3) we get
i'O i'O i'b
/ dtunejdx = f + v(t)6%un - e(t)undxun - c(x,t)6xun) ejdx = Snejdx Ja Ja Ja
for all j = 1,... ,n and almost all t e (0,T). On multiplying both sides by cj(t) and summing up for j = 1,... ,n we obtain
, b
\\dtUn\\2L2(iI) = SndtUndx < \\5n\\L2(i)\\dtUn\\L2(i) J a
for all n = 1, 2,.... Hence, \dtun\\L2(C) ^ \\^n\\L2(c) is bounded uniformly in n, as desired. □
6. Existence of a weak solution
Lemmata 5.1, 5.2 and 5.3 show that the Galerkin approximation un is bounded in L^([0,T],L2(I)) and L^([0,T], H2(I)), and dtUn is bounded in L2(C) uniformly in n. So, one can extract a subsequence of un (we continue to write un for this subsequence) such that un ^ u weakly in L2([0,T], H2(I)), un ^ u strongly in L2([0,T], L2(I)) (which just amounts to L2(C)) and almost everywhere in the rectangle C, and dtun ^ dtu strongly in L2(C). Obviously, the limit function u belongs to H2,1 (C).
Lemma 6.1. Under the assumptions of Theorem 3.1, problem (3.1) admits a weak solution u e H2'1(C).
Proof. Since dtun converges to dtu in the L2(C) -norm, it follows that
T /■ b t b
/ / dtung dxdt ^ / dtug dxdt Jo J a Jo J a
for all g € L2(C). On the other hand, as un converges to u both weakly in L2([0,T],H2( in the norm of L2(C), we see that undxun converges to udxu weakly in L2(C), and so
,-T ,-b ,-T ,-b
no pl pb
e(t) undxung dxdt ^ / e(t) udxugdxdt,
JO J a
l b l b / / c(x, t) dxung dxdt ^ / c(x, t) dxug dxdt
O a O a
O a O a
for all g € L2(C). We make use of these properties when passing to the limit in problem (4.3), as n ^ to. Given any fixed index j, we apply the Fubini theorem to deduce that
b b b b b / dtuejdx + v(t) dxudxejdx + z(t) udxuejdx + c(x,t)dxuejdx = fejdx (6.1) Ja Ja Ja Ja Ja
for almost all t € (0,T).
Let g be an arbitrary function of Hq(I). Since (ej)j=i,2,... is an orthogonal basis in Hq(I), the function g can be written as
g = J2 gj ej,
j=i
where gj are the Fourier coefficients of g with respect to the basis and the series converges in the H 1(I) -norm. On multiplying the equalities of (6.1) by gj, summing up for j = 1,...,N and letting N ^ x> we get (4.1) fulfilled for all g € Hq(I). By the very construction, un(■, 0) converges to u0 in the H 1(I) -norm. On the other hand, un(x, 0) converges to u(x, 0) for almost all x € I, for un ^ u almost everywhere in C and dtun ^ dtu in the L2 (C) -norm. It follows that the initial data u(■, 0) of u coincide with u0. Moreover, since each function un belongs to L2([0,T],Hq(I)) and un ^ u weakly in L2([0,T],H2(I)), the limit function u vanishes on the lateral boundary of C. Thus, u is a weak solution of problem (3.1), as desired. □
7. Uniqueness
Lemma 7.1. Under the assumptions of Theorem 3.1, the solution of problem (3.1) in H 21(C) is unique.
Proof. We first observe that any solution u G H2,!(C) of problem (3.1) is in L«([0,T],L2(I)). Indeed, it is easily seen that such a solution u satisfies
1 d fb fb 1 ''b
1 d 1 u2dx , v,(t) I (dxU)2d
1 d f f 1 fb f
— ~t u2dx + v(t) (3xu)2dx--3xc(x,t)u2dx = fudx
J a J a J a J a
fb fb 1 for almost all t G [0, T], because v23xudx = —8xu3dx = 0 and
J a J a 3
b b 1 2 1 b 2 c(x,t)(dxu)udx = c(x, t) — dxu2dx = —— 3xc(x,t)u2 dx.
J a J a — — J a
Arguing as in the proof of Lemma 5.1 we get
\\u(-,t)\\2L2{I) + \\3xu(-,s)\\2L2{I)ds <
'0
ft tu a)2 ft
< \\uo\\2L2(I) + RJ \u(-,s)\\2L2(i) ds J \\f (-,s)N L2(I )ds,
and so there is a constant C > 0 such that
r-t
u( , t) 2 2tT\ + Vi I \\OxM
\\u(-,t)\\2L2 (I ) + vi\ \\dxu(,s)\\2L2{I)ds <
0
< C + R0 (\\u(., s)\\lHI) + V! 0 \\dxu(s')\\lHI)ds')ds. By the Gronwall lemma,
\\u(-,t)\\2L2(I ) + Vil \\dxu(-,s)\\2L2(I)ds < C exp(Rt)
0
holds for all t G [0, T]. This shows that u G L<([0, T],L2(I)) whenever f G L2(C) and uo G L2(I).
We now assume that u\,u2 G H2'!(C) are two solutions of problem (3.1). Put u = u\ — u2. It is clear that u G L<([0,T], L2(I)). The equations satisfied by u! and u2 lead to
b
/ (dtug + v(t)dxJudxJg + e(t)udxuig + e(t)u2dxug + c(x, t)8xug) dx = 0
a
for all g £ Hq(I). On choosing g = u(-,t) as a test function, for any fixed t £ [0, T], we deduce
lo (
that
2 dt
2dt ^(I) + v(t)\\dxu\\2L2{I) t-b
dxu
i'b fb i'b = —e(t)j u2dxu\dx — e(t) u2udxudx — c(x,t)udxudx. (7.1)
•J a J a J a
b b
An integration by parts yields / u2dxu1dx = —2 udxuu1dx, and so (7.1) becomes
aa
1 d fb 1 fb
2 dt WuW^I) + v(t)\dxu\2L2(I) = £(t) (2ux — u2)udxudx +2 dxc(x,t)u2dx.
aa
By inequality (5.2) with e = 2v\, we get b
s(t) (2ui — u)udxudx ^
a
£2 2 < J— (2 \\ul\\L^([0,T],L2(I)) + \\u2\\L~{[0,T],L2{I))) \\u\\h(I) + vi \\dxu\\2L2 (I) ■
4v\
Furthermore,
/ dxc(x,t)u2dx ^ R\u\Ls(I)
a
for all t £ [0, T]. Hence it follows that there is a positive constant C with the property that
2d \\u\\h(I) < C \\u\\2LHI), and so the Gronwall lemma implies u = 0, as desired. □
8. Boundary value problems in a bounded domain
The study of boundary value problems for the heat equation was initiated by the familiar paper [13]. In [12], Kondrat'ev developed a general theory of boundary value problems for linear parabolic equations in bounded domains of R" x Rt. The focus of [12] is on the asymptotics of solutions at characteristic points of the boundary.
Let G be the domain in Rx x Rt consisting of all (x,t) £ R x R with the property that t £ (0,T) and gi(t) <x < g2(t), where g\ and g2 are functions on [0, T] which are continuously differentiable in the open interval (0,T) and satisfy Qi(t) < g2(t) for all t £ [0,T]. One looks for a solution to the first mixed problem for the Burgers equation in G, that is
u't — vu'xx + uu'x = f in G,
u(x, 0) = u0(x), if x £ (ei(0),g2(0)), (8.1)
u(x,t) = 0, if (x,t) £ d(g1(t),g2(t)) x (0,T),
where f £ L2(C) and u0 £ H1(g1(0), g2(0)) are given functions.
Using the results obtained in the foregoing sections, we look for conditions on the functions g1 and g2 which guarantee that problem (8.1) admits a unique solution u £ H2,1 (G). In order to solve problem (8.1), we apply the method which was used, e.g., in [14] and [5]. This method consists in establishing that the problem admits a unique solution when the domain G is transformed into a rectangle by means of a suitable change of variables preserving the anisotropic Sobolev space H2,1(G).
Theorem 8.1. If f £ L2(C), u0 £ H^(Q1(0), q2(0)) and \g'2(t) — g[(t)\ < C for all t £ (0,T), then problem (8.1) has precisely one solution in H2,1 (G).
The proof of this assertion invokes an appropriate change of variables which allows one to use Theorem 3.1.
Proof. Namely, consider the mapping h : G ^ R x R given by
= x - tpi{t) = t
V2 (t) - <fl(t)'
for (x,t) £ G. This mapping transforms G into the rectangle C = (0,1) x (0,T). Setting u(x,t) = v(y, s) and f (x,t) = g(y, s), we rewrite (8.1) equivalently in the form
' v „ . 1 / v'(s)y + v'i(s) , . r
vs vyy +--— vvy--—-vy = g in C,
s (v(s))2 yy v(s) y v(s) y (82)
v(■, 0) = v0 on I, (82)
v = 0 on dI x (0,T),
where I = (0,1), p = p2 — and v0(y) = u0(p(0)y + ^1(0)). It is easily verified that the change of variables (y,s) = h(x,t) preserves the spaces HQ, H2,1 and L2. Moreover, the conditions on the coefficients of (8.2) imposed in Section 3. are fulfilled. Therefore, we are now in the setting of problem (3.1), and so the desired result follows from Theorem 3.1. □
Theorem 8.1 can be generalised to the case where p1 and p2 are Lipschitz continuous functions on [0,T] instead of C 1(0,T). On the other hand, the question arises what happens if p1(0) = p2(0). In this latter case (p1(0),0) is an isolated singular point at the boundary of G. The study of problem (8.1) near this point would require analysis in domains with conical or more generally cuspidal points at the boundary.
The first author gratefully acknowledges the financial support of the Ministry of High Education of Iraq.
References
[1] H.Bateman, Some recent researches on the motion of fluid, Mon. Weather Rev., 43(1915), 163-170.
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Последовательное приближение для неоднородного уравнения Бюргерса
Азал Мера
Университет Вавилона Вавилон Ирак
Виталий А. Степаненко
Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041
Россия
Николай Тарханов
Потсдамский университет Карл-Либкнехт-Штр., 24/25, Потсдам, 14476
Германия
Неоднородное уравнение Бюргерса представляет собой простой вид уравнений Навье-Стокса. С аналитической точки зрения неоднородная форма плохо изучена, а полное аналитическое решение тесно зависит от формы неоднородного члена.
Ключевые слова: уравнения Навье-Стокса, классическое решение.
