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ISSN 2074-1871 Уфимский математический журнал. Том 12. № 2 (2020). С. 107-117.
NONPOTENTIALITY OF SOBOLEV SYSTEM AND CONSTRUCTION OF SEMIBOUNDED FUNCTIONAL
V.M. SAVCHIN, P.T. TRINH
Abstract. Works by S.L. Sobolev on small-amplitude oscillations of a rotating fluid in 1940's stimulated a great interest to such problems. After the publications of his works, I.G. Petrovskv emphasized the importance of studying general differential equations and systems not resolved with respect to the higher-order time derivative. In this connection, it is natural to study the issue on the existence of their variational formulations. It can be considered as the inverse problem of the calculus of variations. The main goal of this work is to study this problem for the Sobolev system. A key object is the criterion of potentiality. On this base, we prove a nonpotentialitv for the operator of a boundary value problem for the Sobolev system of partial differential equations with respect to the classical bilinear form. We show that this system does not admit a matrix variational multiplier of the given form. Thus, the equations of the Sobolev system cannot be deduced from a classical Hamilton principle. We pose the question that whether there exists a functional semibounded on solutions of the given boundary value problem. Then we propose an algorithm for a constructive determining such functional. The main advantage of the constructed functional action is applications of direct variational methods.
Keywords: Nonpotential operators, Sobolev system, semibounded functional. Mathematics Subject Classification: 35M12, 35Q35, 47G40.
1. Introduction We consider the following Sobolev system of partial differential equations [1]
* = IT +1? = F '
* = w+ = F •
Zt3, \ 9u3 dp o (1.1)
N(u■p) = IT + = ^
~4. , du1 d u2 du3 .
N'(u p) = a? + sJ2 + au; =
(x, t) = (x\x2,x3, t) eQt = n x (0,T),
u u2 u3 u
Q C R3 is bounded by the smooth surface dQ, F\ i = 1,4, are given continuous functions on
Qt.
V.M. Savchin, P.T. Trinh, Nonpotentiality of the Sobolev system and the construction of
a semibounded functional.
©Savchin V.M., Trinh P.T. 2019.
The publication has been prepared with the support of the "RUDN University Program 5-100". Submitted November 10, 2019.
Denoting F = (F 1,F2, F3, F4), N = (N 1,N2, N3, N4),N = N - F, we let
D(N) = {(u, p):ul eC1(QT), PeC1(Q); u%=0 = u^x1, x2, x3),
. . 1 o Q __\ )
ul\ t=T = u\(x,x2,x3), i=l, 3, p\dQ = ^,
where u}(x) e C(Q), i = l, 3, j = 0, l, are given functions, Q = Q U dQ, QT = Q x [0,T]. Denoting bv fc the unit vector (0, 0, l), we represent system (1,1) in form [1]:
NUu,p) = ^- - [u x k]i + ^ = F\ i = 1, 3, K ' dt dx%
ÑHu,P) . ^ ^ = F4
i=1
(1.3)
This system describes small oscillations of a rotating fluid. In [1], there was proved the existence of a solution of (1,1) in a Hilbert space H as well as its continued dependence on the initial data. The Cauchv problem in an unbounded space was solved in an explicit form.
The work of S.L, Sobolev was continued by P.A Aleksandrvan, T.I, Zelenyan, V.N, Maslennikova, and others, see [2] and the references therein. The Sobolev system in the case of two space variables was studied in [3]. By means of the Fourier transform, the solution of the Cauchv problem was obtained in the form of convolutions with kernels having locally integrable properties. The asymptotic behavior of this solution for large values of time was studied.
The problem of existence of variational formulation, Hamilton principle for (1.1), (1.2), was not been studied before. In modern interpretation [4], it can be considered as an inverse problem of the calculus of variations (IPCV), The main aim of this paper is to study the existence of solutions of IPCV for problem (1.1), (1.2).
2. Nonpotentiality of Sobolev system
Let U, V be normed linear spaces over the field of real numbers R, U C V; and 0y be the zero element in U and V respectively; D(N) C U ^ R(N) C V be an arbitrary twice Gateaux differentiable operator with the domain D(N) and the range R(N).
We denote by N the first Gateaux derivative of N at the point u E D(N) defined by the formula [5]
N'h = —N(u + £h)L=o = 5N(u, h).
ae
The mapping &(u; •, •): V xU ^ R linear in each argument and depending on the parameter u E U is called a local bilinear form.
The derivative &u(h; v, g) is defined as
K(h; v, g) = — $(u + eh; v, g)l£=o.
A function $ is called a nonlocal bilinear form if it is independent of the parameter u, that is, $(u; •, •) = (•, •). Then $u(h; v, g) = 0.
We say that (•, •) : V x U ^ R is a non-degenerate nonlocal bilinear form if
1) the condition (v, g) = 0 for all g E U implies that v = 0y;
2) the condition (v, g) = 0 for all f E V implies th at g = 0u-
Definition 2.1. The operator N : D(N) C U ^ V is said to be potential on the set D(N) with respect to a local bilinear form $(u; •, •) : V x U ^ R if there exists a functional F^ : D(Fn) = D(N) ^ R such that SFN[u, h] = $(u; N(u),h) for all u E D(N), h E D(NU). Here Fn is called the potential of the operator N.
Further, we shall make use of the following theorem.
Theorem 2.1. [6] Let N : D(N) Ç U ^ V be a Gâteaux differentiable operator on the convex .set D(N) and a local bilinear form $(u; ■, ■) : V x U ^ R be .such that for all fixed elements u G D(N ) and h,g G D(N'U ) the function <^(e) = $ (u + eh; N (u + eh), g) belongs to C 1[0,1]. Then the potentiality of the operator N on D(N) with respect to $ is equivalent to
JNAg (u) = $(u; N'u h, g) + $U(h; N (u), g) = $(u; N'ug, h) + $'u (g; N (u),h) (2.1)
for all u G D(N), g,h G D(NU). In this case
i
Fn [u] = J $(u(X);N(u(X)),u — uo) d\ + FN [uo], (2.2)
o
where u(X) =u0 + X(u — u0) and u0 is an arbitrary fixed element from D(N).
N
the local bilinear form $, In physics literature, functional (2.2) is called the action functional, or action for short.
Remark 2.1. If $ is a nonlocal bilinear form, then (2.1) becomes
(N'u h, g) = {N'ug ,h) for all u G D(N), g,h G D(N'U). (2.3)
Let us introduce a classical nonlocal bilinear form by
$1(w, g) = (v, g) = ^ vi(x, t)gi(x, t)dxdt. (2.4)
Theorem 2.2. Operator (1.1) is not potential on set (1.2) with respect to nonlocal bilinear form (2.4).
Proof. By (1.1) we find the Gateaux derivative
N '
/1 —1 0 /A
dt dx1
1 <L 0 —
1 iXt 0
dt XJ dx2
0 0 — —
dt dx3
0 I
Xdx1 dx2 dx3 /
In accordance with conditions (1.2), we have
D(NU) = {(h1,h2,h3,h4) : h GC 1(QT>¿=173, h4 GC1 (H); h*\t=o = 0,
h\t=T = 0, i = TT3, h4\dn = 0}.
Let us prove that operator (1.1) does not satisfy criterion (2.3). Denoting by h' = (h1 ,h2,h3) and g' = (g1, g2, g3), we get
$1( NU h, g)
Qt
t(
i=1 v
dh
dh — [h' x k]l +dx'
dh4\ i A dh_ .
dXX^)9 + ^ dx9 ' i=1
x .
Using the chain rule, we obtain
3
$1(NUh, g)= / J]
Qt
i=1
id gi
Dt(higi) — ti^ — [h' x k]%g% + Dx,(h4g%)
— h4 ~~ + Dxi (h'g4) — h^
dx1 y ' dx
,id g4
dx dt for all h, g G D(N'U ),
d
d
where Dt = —, Dxi = —.
dt dx%
By virtue of the Divergence theorem and the condition h e D(N'U), we have
T
0 o
[DXi (h4g1) + Dx2 (h4g2) + DX3 (h4g3)] dx1dx2dx3dt
T
h4g1 dx2dx3 + J h4g2 dx^x3 + J h4g3 dx^x2 | dt = 0. 0 Vo dO dO
Similarly, we have
T
[.DX1 (h1g4) + DX2 (h2g4) + DX3 (h3g4)] dx1dx2dx3dt
0 o
T
and
h1 g4 dx2dx3 + J h2g4dx1dx3 + J h3g4 dx^x2 | dt = 0, 0 \&o an an
3
[Dt(h1g1) + Dt(h2g2) + Dt(h3g3)} dxdt = ^(h'gdx = 0.
Qt
i=1
Applying the above results, we get
3
$1( N'uh, g) = J '"
Qt
t (
=1
dg
dgl
[h' xk]g— > -i'-h
dt
x
3
- - £ =1
dx1
x .
(2.5)
On the other hand, we have
$1( N'ug ,h)
Qt
t (
=1
c)g_ dt
dg_4 dx
't-[s' x if^ )hi +v h4
V + z
=1
dx%
d x d .
(2.6)
In (2.5), the coefficient at h4 is — ^^ and in (2.6) it is ^^ . Hence $1(NUh, g) is not
dx1 ^ dx
=1 =1
indentically equal to $1( N^g,h). Thus, criterion (2.3) is not satisfied.
□
In view of Theorem 2.2, the following question arises. Does there exist a bilinear form such that the operator N of problem (l.l), (1.2) is potential with respect to this form? The answer is given in Section 3.
Before that, let us study the existence of the matrix variational multiplier for operator (l.l).
Definition 2.2. An invertible linear operator M : D(M) C R(N) ^ V is called a variational multiplier for the operator N : D(N) CU ^ V if the operator N = MN is potential on the set D(N) with respect to the given bilinear form.
Theorem 2.3. There is no matrix variational multiplier M = [m,ij(x, t)}'4j=1 for operator N (l.l). '
Proof. Suppose that there exists a matrix variational multiplier
M = {mtj(x, t)}4j=1
4
for (1.1). Then the operator N(u) = MN(u) is potential with respect to classical bilinear form (2.4).
Denoting m-ij = mij(x, t) we get
4 4
mirN'U hgldxdt
(nu h,9 )=f EE
QT i=1 ^
Qt
4 3
£ E
i=1 r=1
dhr ,lr dh4\ i dhr i ~dt - [hxk] + 8^)gmir + d^4
dxdt.
Using the chain rule, we obtain
43
'i I Nuh,9S
Qt
*1 9) = /£ E
=1 =1
Dt(hrgimir) - hrgi^m- - hrd-mir - [h' x k]rgm at at
+ Dxr (h4gimir) - h4g'^mX^ - h4
+ Dxr(hrgimi4) - hrg'^m4 - hr^Q^mi4
dx dt.
Since h,g E D(N'U), we get:
» 4 3 „ 4 3
/ Y,^Dt(hrgimir)dxdt = ^^(h
=1 =1 =1 =1
Qt n
Ymir )\= dx = 0,
and
E [Dxi(h4glmn) + Dx2 (h4#im»2) + D^(h4g^3)] dxdt
Qt
=1
cT 4
I h g%mi\ dx dx + J h gimi2 dx dx + J h gimi3 dx dx [on an an
= 0,
and
E [Dxi (h1^'x'4) + Dx2 (h2gimi4) + Dx3 (h3g^4)] dxdt
Qt
=1
T 4
h gimi4 dx dx + J h gimi4 dx dx + J h g% dx dx mi4 [on an an
Applying the above results, we obtain:
43
'1 I Nu,h,9
= 0.
(NUh,g) = [ EE
QT i=1 r=1
- hrgidW - hr%m- - [h' x kXgimv,
a m
- h4gi tr - h4^—mir - hrgi"''"t4 - hr^—mi4
a mi4 a
dxr dx
r„.i _ hr _
r ~dxr
dxr
-/{£ h [¿(
Qt 1 r=1 l i=1 v
—mir + —mi4 + gQ1,
+ [-^rm4r + m 4 + g4Q1,4r
+ h4
3 r 3 /
z E(
r=1 L i=1 ^
(
dg% dxr
djf dt
dg 4
dxr
mir + gl
* If) ++(
dg4 4 dm4r
m4r + g
dx
dxr
x ,
where
Q1,
' dmi1 dmi4
+
dt dx1
dmi2 dmi4
+
dt dx2
dmi3 dmi4
+
< dt dx3
mi2
H1
r = l,
r = 2, i = M.
= 3,
On the other hand, we have C 4 4
$1 (^NUg, h) = ^ ^ mirN'Ugh dxdt
Qt = r=1
A 3
Qt
Qt
1=1 r=1
^r
dg'
hM ^ — b'x k] + + v^
;dgr
x
ir-^ (r / , dg4 dg1 \
— [9 xk] mn + + )
r=1 i=1 ^ '
+ h^ ^ ( ~j^m4r — [g' x k]%m4i + m4r + — m4^ } dxdt.
r=1
dxr
dxr
Hence,
$1 (iVUh, g)— $1 (jVUy ,h)
Qt
33
£ h'\ E(
r=1 L i=1 V
^% , , , dg1
— (mir + mri) + mi4 dt dxr
dg4 dgi • \
+ ~dx[mri + "9x^4 + S^QMr — b x k]*mri j
+ ( %44m4r + +
4r
+ h
3 r 3 / .
£ e(!
r=1 L i=1 V
x x ' -^—m4r + -—mir + -—m44
dt dxr dxr
(2.7)
$ Cf4 ■ ■ \
+ dxm4r + — [9' x k]%m4i)
'dg 4 + ( —m4r + g
0
dxr 4dm4r
dxdt.
_ dxr dxr
According to criterion (2.3) it must be
$1^NVUh,g^— $1^NUg=0 for all u e D(N), h,g e D(N'U).
Due to an arbitrary choice of the functions h%, i = 1, 4, by (2.7) we obtain ( dg%, s dg1 dg4 dgl
î=i \
dt
dxr
dx%
dx%
\ f d g4 dg4 \
+ glQ\,ir - y x k]% mri j + ( -Q^m4r + dx^m44 + g4Qi,4r I = 0,
r = 1, 3,
3 r 3 /
EE
r=1 i=1 ^
d d d 4 ~^—m4r + -—mir + -—m44 + -—m4r
dt dxr dxr dxr
+ 9ldm - y X kfm4 ) +
M
d 4 4 m4
d x
4
;m# + g
dxr
0.
Hence, thanks to the arbitrary choice of the functions g%, i = 1, 4, we get
' mir + mri = 0,
^¿4 = 0,
< mri = 0,
m4 = 0, m44 = 0,
where i = 1,3, r = 1,3. Finally, mij(x, t) = 0, = 1,4, and therefore, M contradicts to our initial assumption. The proof is complete.
3. Construction of a semibounded functional
0. This □
We have already proved that operator (1.1) is not potential with respect to nonlocal bilinear form (2.4) and there is no matrix variational multiplier of the given type. We need the following theorem later on.
We consider an arbitrary equation
N(u) = 0v, u G D(N) ÇU ÇV, (3.1)
where the operator N in the general case is nonpotential with respect to a fixed nonlocal bilinear form $1(-, ■) = (■, ■) : V X V ^ R.
Theorem 3.1. [7} Let
1) N : D(N) Ç U ^ V be a twice Gâteaux differentiable operator on the convex set D(N);
2) (■, ■) : V X V ^ R be a given nonlocal bilinear form;
3) C : D(C) D R(N) ^ V be an arbitrary invertible linear symmetric operator, such that for all fixed elements u G D(N) and g,h G D(NU) the function p(e) = (N(u + eh), CN'u+ehg) is in C 1[0,1}. Then the operator N is potential on D(N) with respect to the following local bilinear form
$(u; u, g) = (v,CNUg). (3.2)
The corresponding functional is given by
Fn [u} = 1 (N (u),CN (u))
(3.3)
We observe that
8Fn[u, h] = $(u; N(u),h) = (N(u),CN'h). Denoting the adjoint operator for N'u by N'' and assuming that R(C) C D(N'*), by the above identity we obtain:
8Fn[u, h] = ( N'*CN(u),h)
for all u G D(N), h G D(N'U).
Assuming that D(N'U) = U and (■, ■) : V xV ^ R is a nonsingular continuous in each variable nonlocal bilinear form, we get 5Fn [u, h] = 0 u E D(N) for all h E D(NU) if and only if
Ni(u) = NU*CN(u) = 0y, u ed(n).
(3.4)
Thus, the operator N1 is potential on D(N) with respect to the nonlocal bilinear form $1 and the operator N is potential on D(N) with respect to bilinear form (3.2).
If NU* is an invertible operator, then problems (3.1) and (3.4) are equivalent in the following sense: if U is a solution to one of them, then U is a solution to the other, that is,
N(U) = 0y if and only if Ni(U) = 0y.
In this case functional (3.3) provides an indirect variational statement of problem (3.1).
If the operator C is positive definite with respect to a nonlocal bilinear form (■, ■) : V xV ^ R, i.e.,
(v,Cv) ^k |M| for all v E D(C),
where k > 0, then
Fn [u] ^ 0 for all u ED(N)
and Fn[U] = 0 U is a solution of (3.1). Thus, in this case formula (3.3) specifies a semi-bounded functional whose mininum is attained on the solutions to problem (3.1).
Note that functional (3.3) was obtained in another way in [8] while resolving one of the statements of the inverse problem in the calculus of variations.
We return back to problem (1.1), (1.2). We find the adjoint operator for NU:
N
( d dt 1 0
-1 d dt 0
0 0 d dt
d d d
d x1 dx2 - d x3
__d_\
d§1
dg2 dx3 0 /
D(N'*) = {(u1, v2, v3, v4) :v1 E C 1(QT), i = 1, 3), v4 E C 1(Q);
vllt=0 = 0, vllt=T = 0(t = M), w4|dn = 0}. We define an operator C on R(N) by the formula
( Cv)j(x, t) = J K(x,t,y, t)4P(x, t)4P(y, r)vj(y, t) dydr
Qt
=
1, 4,
(3.5)
where
K(x, t,y, t) = K = exp V^ x%y% + tr
(3.6)
where (\>%, i= 1,4, are arbitrary functions in the class C 1(QT) such that
f(x, i) = 0 as (x, t) E Qt, (|t=o = 0, =t = 0, i = 1,3;
(4|
on
0.
With the above choice of the functions (1, (2, (3, (4 we have Cv E D(NU*). It is also easy to see that operator (3.5) is symmetric on R(N).
We are going to show that it is positive definite. In order to do this, we find the expansion of function (3.6) into the Maclaurin series
K
Z
H=o
1
( x1)
1\«i
(x3)«3 t«4 (y 1)«1 ■■■ (y 3)«3T'
3\a,3 a, 4
Here a = (a\, ■■■ ,a4), ai, i = 1, 4, are nonnegative integers, |a| = ~Yhai, a! = aj ...a4!.
i=1
Using this series, we find
$i(i> ,Cv ) = vj(x, t) K (x,t,y, r)(j (x, t)(j (y, r)vj(y, r)dydrdxdt
Qt
Qt 3=1 4 ro
= E E (a!)2 J i=1 M=o v ; qt
1
(x1)"1 ■ ■ ■ (x3)"3ta4( (x, t)v\x, t) dxdt
x I (y 1)a1 ■■■ (y3T3Ta4C (y, ry(y, r)dydr
Qt
4 ro
E E ( Mai"^)2 ^0.
3=1 |"|=0
We note that all Mai^"a4:> vanish simultaneously if and only if vJ = 0, j = 1,4 in Qt [9]. Therefore, if v = 0v then $1(w, Cv) > 0. Thus, the operator C of form (3.5) is positive definite and invertible.
Denoting K = K(x,t,y, r), from (1.3) and (3.5) we get
( CN(n)Y(x, t) = J K((x, t)('(y, r) Qt
( CN(u))4(x, t) = j K(4(x, t)(4(y, T) Qt
Using the chain rule, we obtain
duKy,T) — Hy, r) x kf+ ^ — F
dr
dy1
dydr, i = 1, 3,
^ dui(y, t) — F4
=1
d d .
( CN(u))(x, t)
Qt
— (l(x, t)ui(y, r)DT [K(i(y, r)] — K(\x, t)((y, r)[u(y, t) x k] — p(y)(l(x, t)Dyi [K(i(y, r)]
— Kf(x, t)((y, r)Fi
dy dr, i = 1, 3,
(3.7)
( CN(u))4(x, t)
Qt
3
^(4(x, t)H(y, r)Dyj [K(4(y, r)] =1
— K(4(x, t)(4(y, t)F4
d d .
Using formulae (1.3), (3.3), (3.7), we find the needed functional as follows
Fn [u] = 2//IE ( — (l(x, f)ul(v, t)Dt [K(i(y, r)]
Qt Qt
— K(i(x, t)ji(y, r)[u(y, t) x kf — p(yW(x, t)Dyi [K((y, r)} —
— Kp(x, tW(y, T)F*)(— [u(x, t) xkr + d4xxl — F%)
d
dx%
^T (4(x, t)U (y, r)Dyj [K(4(y, r)] + K(4(x, t)(4(y, r)F4]
3=1 '
^ ^(M) _ F4^ | dydrdxdt.
Using the chain rule, we get
Fn[u] = 2 I f \ ^ (ul(x, i)AM + ([u(x, t) x k] + F) A2, +p(x)A^)
Qt Qt
=1
+ I ^ Ui(x, t)Blti + F452,i I > dydrdxdt,
=1
-)}
(3.8)
where
and
AM =u%, r)Dt [(?(x, t)DT [K(i(y, r)]] + [u(y, r) x k]l(l(y, r)Dt [K(\x, t)}
+ p(y)Dt [(?(x, t)Dyi [K(\y, r)]] + , r)Dt [K(i(x, t)F*] , A2,i =u\y, r)(\x, t)DT [Ktf (y, r)] + [u(y, t) x kiKf(x, t)((y, r)
+ p(y)(i(xit )Dyi [K(\y, r)] +K(\x, t)(\y )F\ A3, =u\y, r)Dx* [(\x, ¿)dt [K(4(y, r)]] +p(y)Dx, |/(x, t)Dyi [K(\y, r)]]
+ [u(y, t) x k]((\y, r)Dxi [K(l(x, t)} + , r)Dx* [K('(x, t)F"] ,
3
BM = (y, r)Dxi K4(x, t)Dy] (K(4(y, r))] +(4(y, r)Dx* [K(4(x, t)F4} ,
=1
3
B
2,i
(y, r)(4(x, t)Dy> (.K(4(y, r))+K(4(x, t)(4(y, r)F4. =1
Thus, we have proved the following theorem.
Theorem 3.2. The functional of form (3.8) is semibounded on the solutions of problem (1.1), (1.2).
Remark 3.1. Functional (3.8) possesses the following properties:
1) it is bounded below on set (1.2);
2) it takes a minimum value only on the solutions of problem (1.1), (1.2);
3) it does not involve the derivatives of unknown functions;
4) the set of its stationary points contains the solution set of problem (1.1), (1.2).
For specific details on solvability conditions for the type of problems under consideration, uniqueness theorems, Lp-estimates for solutions the reader can see, for instance, [2].
4. Conclusions and future directions The results of this paper can be summarized as follows.
(i) We studied the potentiality of the operator of the boundary value problem for the Sobolev system of partial differential equations. We showed that it is not potential with respect to the classical bilinear form. It means that the considered system cannot be obtained from Hamilton variational principle.
(ii) The problem of the existence of a matrix variational multipler for (1.1) was studied. We showed that there is no matrix variational multiplier with elements depending on x and .
(iii) We posed the question that whether there exists a functional semibounded on solutions of the given boundary value problem. We proposed an algorithm for the constructive determination of such functional.
The main advantage of constructed functional (3.8) is in applications of direct variational methods and its numerical performance.
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Vladimir Mikhailovieh Saevhin,
S.M. Nikol'skii Institute of Mathematics at RUDN University, Miklukho-Maklava str,, 6, 117198, Moscow, Russia E-mail: [email protected]
Phuoc Toan Trinh,
S.M. Nikol'skii Institute of Mathematics at RUDN University,
Miklukho-Maklava str., 6,
117198, Moscow, Russia
E-mail: tr. phuoctoanSgmail. com
