On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel
The problem of integrating the Laplace equation in a changing 3-dimensional region, with the initial and boundary conditions, is investigated. The paper is mainly devoted to the problem arising in dynamics of an inviscid incompressible fluid which partially fills a moving vessel and is in irrotat...
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irk-123456789-1747602021-01-28T01:27:24Z On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel Zolotenko, G.F. The problem of integrating the Laplace equation in a changing 3-dimensional region, with the initial and boundary conditions, is investigated. The paper is mainly devoted to the problem arising in dynamics of an inviscid incompressible fluid which partially fills a moving vessel and is in irrotational absolute motion. In this case the considered space region is bounded by the rigid vessel’s walls and the unknown free surface of fluid. The boundary conditions consist of the Neyman conditions on the rigid walls and the nonlinear kinematic and dynamic conditions on the free surface. Besides, the condition of a constancy of the region’s volume is imposed. The concept of a solution of this problem is analyzed. One distinguishes a certain class of solutions and proves their existence. An example of such a solution is given. 2001 Article On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel / G.F. Zolotenko // Нелінійні коливання. — 2001. — Т. 4, № 4. — С. 560-573. — Бібліогр.: 11 назв. — англ. 1562-3076 AMS Subject Classification: 76B03, 76B07 http://dspace.nbuv.gov.ua/handle/123456789/174760 en Нелінійні коливання Інститут математики НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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The problem of integrating the Laplace equation in a changing 3-dimensional region, with the
initial and boundary conditions, is investigated. The paper is mainly devoted to the problem
arising in dynamics of an inviscid incompressible fluid which partially fills a moving vessel and
is in irrotational absolute motion. In this case the considered space region is bounded by the
rigid vessel’s walls and the unknown free surface of fluid. The boundary conditions consist of
the Neyman conditions on the rigid walls and the nonlinear kinematic and dynamic conditions
on the free surface. Besides, the condition of a constancy of the region’s volume is imposed.
The concept of a solution of this problem is analyzed. One distinguishes a certain class of
solutions and proves their existence. An example of such a solution is given. |
| format |
Article |
| author |
Zolotenko, G.F. |
| spellingShingle |
Zolotenko, G.F. On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel Нелінійні коливання |
| author_facet |
Zolotenko, G.F. |
| author_sort |
Zolotenko, G.F. |
| title |
On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel |
| title_short |
On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel |
| title_full |
On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel |
| title_fullStr |
On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel |
| title_full_unstemmed |
On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel |
| title_sort |
on solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel |
| publisher |
Інститут математики НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/174760 |
| citation_txt |
On solutions of general nonlinear initial boundary-value problem of inviscid fluid's dynamics in moving vessel / G.F. Zolotenko // Нелінійні коливання. — 2001. — Т. 4, № 4. — С. 560-573. — Бібліогр.: 11 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT zolotenkogf onsolutionsofgeneralnonlinearinitialboundaryvalueproblemofinviscidfluidsdynamicsinmovingvessel |
| first_indexed |
2025-07-15T11:55:26Z |
| last_indexed |
2025-07-15T11:55:26Z |
| _version_ |
1837713879831937024 |
| fulltext |
Nonlinear Oscillations, Vol. 4, No. 4, 2001
ON SOLUTIONS OF GENERAL NONLINEAR INITIAL
BOUNDARY-VALUE PROBLEM OF INVISCID FLUID’S
DYNAMICS IN MOVING VESSEL
G. F. Zolotenko
Institute of Mathematics, NAS of Ukraine
Tereshchenkivs’ka St., 3, Kyiv, 01601, Ukraine
e-mail: zolot@imath.kiev.ua
The problem of integrating the Laplace equation in a changing 3-dimensional region, with the
initial and boundary conditions, is investigated. The paper is mainly devoted to the problem
arising in dynamics of an inviscid incompressible fluid which partially fills a moving vessel and
is in irrotational absolute motion. In this case the considered space region is bounded by the
rigid vessel’s walls and the unknown free surface of fluid. The boundary conditions consist of
the Neyman conditions on the rigid walls and the nonlinear kinematic and dynamic conditions
on the free surface. Besides, the condition of a constancy of the region’s volume is imposed.
The concept of a solution of this problem is analyzed. One distinguishes a certain class of
solutions and proves their existence. An example of such a solution is given.
AMS Subject Classification: 76B03, 76B07
1. The Initial Boundary-Value Problem
One considers the following nonlinear initial boundary-value problem for the Laplace
equation in the changing region Ω(t) (t denotes time) of the Euclidean space R3:
4u(x, y, z, t) = 0, (x, y, z) ∈ Ω(t), t ∈ [t0, t1], (1)
∂u
∂n
= h1(x, y, z, t, n), (x, y, z) ∈ S(t), (2)
∂u
∂n
= h2(x, y, z, t, n), (x, y, z) ∈ Σ(t), (3)
Φ(x, y, z, t,∇u, ut) = 0, (x, y, z) ∈ Σ(t), (4)∫
Ω(t)
dΩ(t) = const ∀t ∈ [t0, t1], (5)
u(x, y, z, t0) = u0(x, y, z), ζ(x, y, t0) = ζ0(x, y) (4u0 = 0, (x, y, z) ∈ Ω(t0)). (6)
Here the operators 4, ∇ act with respect to the variables x, y, z which are the coordinates
of any point of the space R3 in a certain coordinate system Oxyz; the normal derivatives are
taken in the direction of the unit vector n = n(x, y, z, t) of the outward normal to the boundary
Γ(t) of the region Ω(t); the letter indices denote the partial derivatives with respect to the
corresponding variables. The segment [t0, t1] is finite.
560 c© G. F. Zolotenko, 2001
ON SOLUTIONS OF GENERAL NONLINEAR INITIAL BOUNDARY-VALUE PROBLEM .. . 561
As we shall see below, the boundary functions h1, h2 depend on the normal vector n explic-
itly. In the general case of the unsmooth boundary surface Γ(t), this circumstance makes the
boundary conditions (2), (3) irregular; thus it is essential in theoretical sense. To underline this,
here and further the vector n is pointed out among the arguments of the functions h1, h2.
The domain Ω(t) is simply connected. It is inserted in the domain Π which is bounded by
the fixed closed surface ∂Π and is simply connected too. Thus the following inclusion is valid:
Ω(t) ⊆ Π ∀t ∈ [t0, t1].
The surface ∂Π is known. S(t) is a surface of the intersection of the unknown boundary Γ(t)
and the known surface ∂Π. In the designation S(t), the symbol t reflects the motion of S(t)
along the surface ∂Π. Those cases are eliminated when the set S(t) becomes empty.
The boundary Γ(t) is presented as
Γ(t) = S(t) ∪ Σ(t), S(t) ∩ Σ(t) = l(t),
where Σ(t) is the free surface to be found; l(t) is the intersection curve of the considered pieces
of the boundary. The unknown surface Σ(t) can change its configuration and location relative
to the vessel. We suppose that the surface Σ(t) is described by the equation
z = ζ(x, y, t).
In the boundary condition (2) the function h1 is known and defined by the formula
h1(x, y, z, t, n) = [v(t) + ω(t)× r] · n(x, y, z, t), (7)
where v(t), ω(t): [t0, t1] → R3 are the known vector-valued functions; r = (x, y, z). The
symbols ”·” and ”×” denote the scalar and vector product, respectively. (In function (7) the
normal n(x, y, z, t) is taken in the points of the surface S(t).)
The functions h2 and Φ from the boundary conditions (3), (4) are of the following form:
h2(x, y, z, t, n) = h1(x, y, z, t, n) + (1 + ζ2
x + ζ2
y )−
1
2 ζt(x, y, t), (8)
Φ(x, y, z, t,∇u, ut) = ut +
1
2
(∇u)2 −∇u · (v(t) + ω(t)× r)− g(t) · r +
p0
ρ
− F (t). (9)
Here g(t): [t0, t1] → R3 is the known vector-valued function; p0, ρ > 0 are certain real constant
parameters; F (t) is an arbitrary function of time. (In contrast to (7), the normal n(x, y, z, t) of
the function h1 in (8) is taken in the points of the free surface Σ(t).)
562 G.F. ZOLOTENKO
The sought for quantity of the problem (1) – (6) is the pair of functions (u, ζ). The function
F (t) is unknown too, but as we shall see below it is determined from additional considerations.
The formulated problem is known from dynamics of the bounded volume of an inviscid
incompressible fluid under the vessel’s arbitrary motion in gravity field [1]. In equations (1) –
(8) u(r, t) is a potential of the fluid’s absolute velocity (∇u gives the projections of the fluid’s
absolute velocity on the axes of the moving coordinate system Oxyz); equation (2) is the im-
permeability condition of the vessel’s rigid walls for fluid; the nonlinear relations (3), (4) are
the kinematic and dynamic conditions on the fluid free surface, respectively; equation (5) is the
condition of constancy of the fluid volume; equations (6) are the initial conditions. The function
F (t) is an arbitrary function from the Lagrange – Cauchy integral. The domain Π is nothing else
but a cavity of the vessel and Ω(t) is the fluid volume.
In a partial case, when Ω(t) coincides with Π, the function ζ(x, y, t) ≡ ζ(x, y), which is of
practical interest too (this problem constitutes the well-known Zhukovsky problem).
In general, the problem (1) – (9) is very difficult. An existence of its solutions has not been
proved hitherto. The obstacles are connected with nonlinearity of the boundary conditions,
variability of the solutions’ domain and indeterminacy of the domain’s boundaries. There exists
another less evident difficulty, namely, the functions h1 on S(t) and h2 on Σ(t) are irregular as
a consequence of their dependence on the normal vector n which, in the general case of the
unsmooth surfaces Γ(t), must be considered as a distribution (generalized function). Under
this circumstances, as far as we know, even the solution’s notion has not been yet formulated
accurately.
On the other hand, there exist many papers which are devoted to the development of ap-
proximate methods of solving the considered problem in different particular cases (of the ves-
sel’s shapes and the laws of the vessel’s motion). However, each of such methods must not
contradict to a priori mathematical properties of an exact solution about which there exist little
information in the case under consideration. As a pay for not meeting this requirement, there
may be either an instability of the numerical method for solving the problem or an absence of
the convergence of the approximate solution to the exact solution.
In connection with the indicated circumstances, the purpose of this paper is formulated as
follows: to clarify the concept of the solution of the problem (1) – (9), to prove its existence
under certain conditions and to show the example of such a solution.
One uses the S. L. Sobolev’s [2] and J. Nečas’ [3] results which are most suitable regarding
the equation, the boundary functions, and unsmoothness of the boundary surfaces (see, for
example, [4 – 8]).
2. The Region Ω(t)
The properties of space region in which the initial boundary-value problem is considered
play an important role in the solvability questions of this problem. In linear hydrodynamics the
case of the Lipschitzian region is the most general [9, p. 31]. Let us extent this approach to the
nonlinear case.
We shall assume that the boundary ∂Π of the region Π consists of a finite number M of the
surface pieces Sk, k = 1, . . . ,M , each of which is described by the following relations
y
(k)
3 = f (k)(y
(k)
1 , y
(k)
2 ), (y
(k)
1 , y
(k)
2 ) ∈ D(k), k = 1, . . . ,M, (10)
ON SOLUTIONS OF GENERAL NONLINEAR INITIAL BOUNDARY-VALUE PROBLEM .. . 563
where (y
(k)
1 , y
(k)
2 , y
(k)
3 ) is the local Cartesian coordinate system; f (k)(y
(k)
1 , y
(k)
2 ) is a continuous
function in the square (α > 0 is some constant)
D(k) = {(y(k)
1 , y
(k)
2 ) : | y(k)
i |< α, i = 1, 2}
which satisfies the Lipschitz condition
| f (k)(ξ)− f (k)(η) |≤ Ld(ξ, η) ∀ξ, η ∈ D(k) ⊂ R2 (L > 0). (11)
Here d(·, ·) denotes the Euclidean distance between the vectors.
In addition, we assume that the surface ∂Π has not any singular points, i.e., there exists
β > 0 such that the points (y
(k)
1 , y
(k)
2 , y
(k)
3 ) for which | y(k)
i |< α, i = 1, 2, and the coordinate
y
(k)
3 satisfies one of the conditions
f (k)(y
(k)
1 , y
(k)
2 )− β < y
(k)
3 < f (k)(y
(k)
1 , y
(k)
2 ),
f (k)(y
(k)
1 , y
(k)
2 ) < y
(k)
3 < f (k)(y
(k)
1 , y
(k)
2 ) + β (12)
lie inside the open region Π or outside the closed region
−
Π= Π ∪ ∂Π, respectively.
The totality of relations (10) – (12) defines the Lipschitzian surface in the sense of definition
of paper [3, p. 14, 15].
Next, let us fix the origin O of the coordinate system Oxyz at any point inside Π. Then
considering Σ(t) as a family of surfaces which depend on t and are situated inside Π, we shall
define each of these surfaces as
z = ζ(x, y, t), (x, y) ∈ D(ζ), t ∈ [t0, t1], (13)
where D(ζ) is the orthogonal projection of the surface Σ(t) to the plane Oxy; the function
ζ(x, y, t) is continuous with respect to x, y in the region D(ζ) and satisfies the Lipschitz condi-
tion
| ζ(ξ, t)− ζ(η, t) |≤ Ld(ξ, η) ∀ξ, η ∈ D(ζ), t ∈ [t0, t1]. (14)
We note that for all t the relation D(ζ) ⊆ D0 holds, where D0 is the orthogonal projection of
the region Π to the plane Oxy.
From the two subsets into which the surface Σ(t) divides the region Π, the one we select
for Ω(t) is that which is situated, entirely or partially, under the surface Σ(t) (along the axis
Oz). For example, if the fluid filling a rolling rectangular vessel touches the vessel’s lid, Ω(t) is
situated partially under Σ(t) only. If it does not, it is situated under Σ(t) entirely.
Remark 1. In connection with the suggested parametrization of the boundary surfaces in
the considered hydrodynamic problem it is appropriate to note the following. Since the points
of the surface pieces Sk are determined by the local coordinates (y
(k)
1 , y
(k)
2 , y
(k)
3 ), the projections
x, y, z of the corresponding vector r to the axis x, y, z will be functions of these local coordinates
if formula (7) is considered together with the boundary conditions (2). This also refers to the
vector r in the expressions (8), (9) if they are considered together with the boundary conditions
564 G.F. ZOLOTENKO
(3), (4), but here the local coordinates for the free surface Σ(t) coincide with x, y, z. We should
also add that in all cases the normal n to the considering surfaces will be taken in projections
to the axes of the ”support” coordinate system Oxyz, because the input vectors v0(t), ω(t) are
usually given in this coordinate system.
Besides the condition (14), one also supposes that a condition on the construction of the β-
neighbourhood of the surface z = ζ(x, y, t), which is analogous to (12), is fulfilled, namely: the
points (x, y, z) for which | x |< α, | y |< α and the coordinate z satisfies one of the conditions
ζ(x, y, t)− β < z < ζ(x, y, t), ζ(x, y, t) < z < ζ(x, y, t) + β (15)
are situated inside the open region Ω(t) or outside the closed region
−
Ω (t) = Ω(t) ∪ Γ(t),
respectively.
To ensure the Lipschitz condition for the entire surface Γ(t), we also introduce the following
condition (in short, the l-condition): at the points of the line l(t) in which the tangent planes to
both the surface S(t) and the surface Σ(t) exist, the angles between this tangent planes are not
equal to zero.
In general, the l-condition may be not fulfilled. Indeed, the points of the line l(t) are situated
on the boundaries of three media (the fluid, the gas, and the rigid body). As is known, the
angles between the contact surfaces of this media at the mentioned points are functions of the
corresponding coefficients of the surface tension [10, p. 32]. Theoretically, these coefficients
can be such that the mentioned angles will be equal to zero. In our case, however, the surface
tension is neglected and therefore one can say nothing about the values of these angles. But as
we shall see below, there exists an example for which the l-condition holds.
The class of the surfaces Σ(t) which are defined by the explicit equations (13) is narrower
than the class of the surfaces which are defined both in the implicit form f(x, y, z, t) = 0 and
in the parametric form x = x(τ1, τ2), y = y(τ1, τ2), z = z(τ1, τ2) (τ1, τ2 are the parameters).
(The surfaces (13) make up a subset both of the set of the implicitly defined surfaces when
f(x, y, z, t) = z − ζ(x, y, t)) and of the set of the surfaces which are defined parametrically
when τ1 = x, τ2 = y, x(τ1, τ2) = x, y(τ1, τ2) = y, z(τ1, τ2) = ζ(x, y, t).) At the same time this
class of surfaces contains a sufficiently large number of practical cases. In connection with this,
it is advisable to extract the corresponding regions Ω(t) as a separate class.
Definition. A changing region Ω(t) is called a ζ-region if for all t the fixed boundary surface
S(t) satisfies the conditions (10) – (12), the free boundary surface Σ(t) is set by the explicit equa-
tion (13) and satisfies the conditions (14), (15), and at the line of intersection of the surfaces S(t)
and Σ(t) the l-condition is fulfilled.
By the classification of paper [6], the ζ-regions belong to the Hölder class Cl,λ, where l = 0,
λ = 1 (since one requires only a continuity of the functions which define the pieces of the
boundary (the derivatives’ continuity is not obligatory) and the Lipschitz conditions coincide
with the corresponding Hölder conditions for which the exponent λ is one).
Let us note some peculiarity of the ζ-regions which is important in what follows and refers
to the connection between structure and properties of its boundary.
Proposition 1. The ζ-region is a locally star-shaped region at any time t.
ON SOLUTIONS OF GENERAL NONLINEAR INITIAL BOUNDARY-VALUE PROBLEM .. . 565
Proof. Indeed, this property for the Lipschitz regions is proved in [11, p. 308] but the
Lipschitz property of the ζ-region at any t follows from Definition.
The locally star-shaped structure of the ζ-regions will be used below when applying Sobolev’s
results to the problem (1) – (9).
Remark 2. The assumptions (13), (14) and the l-condition concern the unknown math-
ematical objects (the sought for surface Σ(t) and the line l(t) of its intersection with S(t)).
Therefore, there exists a certain risk to make a mistake if we endow these objects with the
noted properties beforehand. An exception is the condition (13) which is fulfilled uncondition-
ally, because it is present in the relation (3) implicitly. As for the rest of the conditions, the
example constructed below shows that the initial problem (1) – (9) can have a solution (u, ζ)
component ζ of which satisfies these conditions and, consequently, the corresponding Ω(t) is a
ζ-region.
3. The Class of Functions for Possible Solutions
In this section one clarifies an important question regarding the classes of functions in
which the solutions of the considered initial boundary-value problem must be sought.
A priory information about the solutions of the problem (1) – (9) can be obtained if one
notes that any its solution (u0, ζ0) (if it exists) necessarily is a solution of the shortened problem
(1) – (3), (5). After the substitution of ζ0(x, y, t) for ζ(x, y, t) this shortened problem turns into
the Neyman problem in a changing region. (We note that the shortened problem (1) – (3), (5)
is of an independent practical interest too. In hydrodynamics it corresponds to the case when
at any time t the free surface is known and it is necessary to find the potential and the pressure
in the fluid.)
When t is fixed the problem (1) – (3) is a classical Neyman problem for the Laplace equa-
tion. For the class of regions with the so-called ”simple” boundary (or else, with the piecewise
smooth boundary) its solution is given by S.L. Sobolev [2, p. 124]. In this case, the conditions
on the boundary functions are formulated in the form of their belonging to the space L2(Γ).
O.A. Ladyzhenskaya and N.N. Uraltzeva [6] considered the regions with a piecewise smooth
boundary too, but they supposed that the boundary functions are defined at once in the en-
tire domain Ω + ∂Γ (p. 25), which cannot be said about the functions h1 and h2. The cases
of irregular boundary functions from the space W−1/2
2 (Γ) ⊃ L2(Γ) are investigated by J.-L.
Lions and E. Magenes [7] but they restricted themselves to boundaries Γ from the class C∞ of
infinitely-differentiated surfaces [p. 48]. The Neyman problem for the Laplace equation (1) –
(3) with ”hydrodynamic” specific character may be investigated if one combines Sobolev’s [2]
and Nečas’ [3] results; this is carried out below.
Before solving the question regarding the class of functions for the solutions, we consider
properties of the vector-valued function n(x, y, z, t) which is included in the boundary condi-
tions (2), (3) and makes them irregular.
Let n = (n1, n2, n3) be the vector-valued function n in projections to the axes of the local
coordinate systems, where
ni =
ni(y
(k)
1 , y
(k)
2 , y
(k)
3 , t), if (x(k), y(k), z(k)) ∈ Sk,
ni(x, y, z, t), if (x, y, z) ∈ Σ(t),
566 G.F. ZOLOTENKO
i = 1, 2, 3, k = 1, . . . ,M.
In the first row of this formula, the coordinates x(k), y(k), z(k) of the points of the surface Sk
in the coordinate system Oxyz are connected with the local coordinates y(k)
1 , y
(k)
2 , y
(k)
3 of these
points by the following relations:
x(k)
y(k)
z(k)
=
x
(k)
0
y
(k)
0
z
(k)
0
+A(k)
y
(k)
1
y
(k)
2
y
(k)
3
,
where x(k)
0 , y
(k)
0 , z
(k)
0 are coordinates of the origin of the k-th local coordinate system in the
axes x, y, z; A(k) is the matrix consisting of cosines of the angles between the axes x, y, z and
y
(k)
1 , y
(k)
2 , y
(k)
3 .
For every piece Sk of the surface Γ(t), the matrix A(k) has its own form and is constant,
i.e., independent of the space variables and time. For the piece Σ(t) of the surface Γ(t), an
analogous relation holds, where one must assume x(k)
0 = y
(k)
0 = z
(k)
0 = 0 andA(k) is the identity
matrix.
In addition, let n = (n
(k)
x , n
(k)
y , n
(k)
z ) be the unit normal vector to the surface Sk in projec-
tions to the axes x, y, z (see Remark 1). It is evident that n
(k)
x
n
(k)
y
n
(k)
z
= A(k)
n1(y
(k)
1 , y
(k)
2 , y
(k)
3 )
n2(y
(k)
1 , y
(k)
2 , y
(k)
3 )
n3(y
(k)
1 , y
(k)
2 , y
(k)
3 )
. (16)
Proposition 2. Let the region Ω(t) be a ζ-region for all t. Then at any time t there exists a
vector n(x, y, z, t) normal to its boundary Γ(t) almost everywhere on Γ(t) and its components
nx, ny, nz , in the coordinate system Oxyz, are measurable and bounded functions of x, y, z de-
fined almost everywhere on Γ(t).
Proof. For the Lipschtzian region which does not depend on time, existence almost every-
where, measurability, and boundedness of the projections n1, n2, n3 follow from Lemma 4.2 of
Paper [3, p. 88]. (Here and later, the measurability is regarded in the sense of Lebesgue.) For
the boundary Γ(t), this remains valid because Γ(t) is Lipschitzian for all t in virtue of Definition.
Since nx, ny, nz are linear combinations of n1, n2, n3 with piecewise continuous coefficients on
Γ(t) (in virtue of (16)), the projections of the normal n(x, y, z, t) to the axes of the coordinate
system Oxyz also exist almost everywhere and are measurable, bounded functions on Γ(t),
which is what was to be proved.
The measurability and boundedness of the function n(x, y, z, t) will be used to determine
the class of functions for the functions h1, h2 from the boundary conditions (2), (3).
To prove Theorem of the present paper, it will also be necessary to use the following known
result of Sobolev concerning solution of the Neyman problem [2, p. 124] and formulated in
terms of belonging the boundary functions to the space L2(Γ).
Proposition 3 [8, p. 323]. The Neyman problem for the Laplace equation in some finite
region Ω of the Euclidean space Rn has a unique weak solution from the space W 1
2 (Ω) if Ω is
ON SOLUTIONS OF GENERAL NONLINEAR INITIAL BOUNDARY-VALUE PROBLEM .. . 567
locally star-shaped and the boundary function h(x) (x ∈ Rn) satisfies the following conditions
h ∈ Lq∗(Γ),
1
q
+
1
q∗
= 1, q <
2(n− 1)
n− 2
, (17)
∫
Γ
h(x)dΓ = 0, (18)
where Γ is the boundary of the region Ω.
Now, taking into account the concrete form of the functions h1, h2, we can prove the fol-
lowing statement.
Theorem. Let Ω(t) be a ζ-region, the functions h1(x, y, z, t), h2(x, y, z, t) be defined by the
formulae (7), (8), and the derivative ζ0
t (x, y, t) satisfy the condition
∀t ∈ [t0, t1] the function (x, y) 7→ ζ0
t (x, y, t) is measurable and bounded on D(ζ). (19)
Then there exists a unique weak solution u0(x, y, z, t) of the problem (1) – (3), (5) which, for any
t, belongs to the Sobolev space W 1
2 (Ω(t)).
Proof. Let us fix t, suppose n = 3 in the relation (17) and replace Γ with Γ(t) in (18). We
also note that in the considered case,
h = h1(x, y, z, t) for (x, y, z) ∈ S(t), h = h2(x, y, z, t) for (x, y, z) ∈ Σ(t).
Then, first of all, according to Proposition 1 the region Ω(t) is locally star-shaped.
Let us apply Proposition 3 and prove that the conditions (17), (18) are fulfilled for the
considered shortened hydrodynamic problem. Indeed, let q = 2. Then we should prove that
h ∈ L2(Γ(t)).
However, the function h1 is a scalar product of the vector-valued functions [v(t) +ω(t)× r] and
n with the first of them being continuous with respect to the space variables x, y, z in virtue of
Remark 1 and definition of the Lipschitzian surface. At the same time, it follows from Propo-
sition 2 that the components of the vector n are measurable and bounded functions of x, y, z.
Hence, h1 is measurable and bounded, too, and h1 ∈ L2(S(t)).
In turn, h2 is the sum of h1 and
ζ0
t√
1 + (ζ0
x)2 + (ζ0
y )2
.
However, the function h1 on Σ(t) is an element of the space L2(Σ(t)) in virtue of the above
argument. The derivative ζ0
t is measurable and bounded in virtue of the condition (19) of
Theorem. The denominator
√
1 + (ζ0
x)2 + (ζ0
y )2 is measurable, bounded, and nonzero because,
568 G.F. ZOLOTENKO
in the coordinate system Oxyz (which is local for the surface Σ(t)), the normal n to the free
surface Σ(t) is of the form
n(x, y, z, t) =
(−ζ0
x,−ζ0
y , 1)√
1 + (ζ0
x)2 + (ζ0
y )2
,
and is measurable and bounded in virtue of Proposition 2. Hence, h2 ∈ L2(Σ(t)).
At last, since S(t) and Σ(t) intersect themselves at most along the line l(t), one can write∫
Γ(t)
h2dΓ(t) =
∫
S(t)
h2
1dS(t) +
∫
Σ(t)
h2
2dΣ(t).
Remark 3. The integrals over the surface Γ(t) everywhere are understood in the Lebesgue
sense and defined with the help of partition of unity. The latter is associated with the description
of Sk (in the general case) in different local coordinate systems.
It follows from this that h ∈ L2(Γ(t)), and hence the conditions (17) are satisfied.
Let us consider the condition (18). We have the relation∫
Γ0(t)
hdΓ0(t) =
∫
S0(t)
[v(t) + ω(t)× r] · ndS0(t)
+
∫
Σ0(t)
{
[v(t) + ω(t)× r] · n+ [1 + (ζ0
x)2 + (ζ0
y )2]−
1
2 ζ0
t
}
dΣ0(t). (20)
Here Γ0(t), S0(t), and Σ0(t) correspond to the surface z = ζ0(x, y, t). By the divergence
theorem (which is true in the case of the Lipschitzian regions too), we can find that∫
Γ0(t)
[v(t) + ω(t)× r] · ndΓ0(t) = 0. (21)
Then we shall use the additional integral condition (5). By differentiating (5) with respect
to t (such differentiation is possible for the regions which are bounded by piecewise smooth
surfaces), we obtain the following relations:
d
dt
∫
Ω(t)
dΩ(t) =
∫
Γ(t)
VndΓ(t) =
∫
S(t)
VndS(t) +
∫
Σ(t)
VndΣ(t) ≡ 0, (22)
where Vn = Vn(x, y, z, t) is the normal component of the velocity of the point Γ(t) as it moves
relative to the coordinate system Oxyz. However, it follows from equations (10) and (13) for
the parts of the surface Γ(t) that
Vn ≡ 0 on S0(t), Vn = [1 + (ζ0
x)2 + (ζ0
y )2]−
1
2 ζ0
t on Σ0(t). (23)
ON SOLUTIONS OF GENERAL NONLINEAR INITIAL BOUNDARY-VALUE PROBLEM .. . 569
From (22) using (23) we find that∫
Σ0(t)
[1 + (ζ0
x)2 + (ζ0
y )2]−
1
2 ζ0
t dΣ0(t) ≡ 0. (24)
Substituting (21) and (24) into (20) we obtain the equality (18).
Up to now we supposed that twas fixed. Since it was arbitrary, the statement of the theorem
is true for any t ∈ [t0, t1]. Theorem is proved.
Thus, for the input problem (1) – (9) there exists a reason to look for the solutions (u, ζ)
among the functions that satisfy the following conditions (the arguments that are not speci-
fied in the following formulae vary on the sets indicated in the notations of the corresponding
function space):
Ω(t) is a ζ-region,
u(r, t) : Ω(t)× [t0, t1] → R, ζ(x, y, t) : D(ζ)× [t0, t1] → R;
the properties with respect to the space variables are
∀ t ∈ [t0, t1] :
u(·, t), ut(·, t) ∈W 1
2 (Ω(t)),
ux(·, t), uy(·, t), uz(·, t) ∈ L2(Ω(t)),
ζ(·, ·, t), ζt(·, ·, t) ∈ C(D(ζ)) and is Lipschitzian,
ζx(·, ·, t), ζy(·, ·, t) exist almost everywhere and are measurable and bounded in D(ζ);
the properties with respect to time t are
∀ r ∈ Ω(t) :
u(r, ·), ux(r, ·), uy(r, ·), uz(r, ·) ∈ C([t0, t1]),
ut(r, ·) is piecewise continuous in [t0, t1] and has a finite number of discontinuities of the
first kind,
∀ (x, y) ∈ D(ζ) :
ζ(x, y, ·), ζx(x, y, ·), ζy(x, y, ·) ∈ C1([t0, t1]),
ζt(x, y, ·) ∈ C([t0, t1]).
One can verify that the properties of u, ζ and their derivatives as functions of t agree with
the properties of the given functions v(t), ω(t), g(t) imposed by physical reasons and consist in
the following:
v(t), ω(t), g(t) : [t0, t1] ⊂ R → R3,
v(t), ω(t), g(t) ∈ C([t0, t1]),
v̇(t), ω̇(t) are piecewise continuous and have a finite number of discontinuities of the first
kind.
570 G.F. ZOLOTENKO
This assumptions correspond to the translational and angular motions of the vessel with
jumps (discontinuities) of acceleration. Such motions occur, for example, during an acceler-
ation, retardation, instantaneous stopping, pushes of the vessel, and so on. The ”ordinary”
motions with continuously changing acceleration are also included here.
In the formulated requirements on smoothness of the sought for functions, the conditions
on ζx(·, t), ζy(·, t) and u(·, t), ux(·, t), uy(·, t), uz(·, t), ut(·, t) follow from Proposition 2 and The-
orem, respectively. The condition on derivative ζt(·, t), evidently, agrees with the requirement
(19) of Theorem on ζt. All the rest of the properties either follow from physical reasons or are
their corollaries.
4. Example
Let us show that the set of the pairs of functions (u0, ζ0) mentioned in Theorem is not
empty. To this end, we shall construct one exact solution of the input problem (1) – (9).
Let the region Π be a cylinder the axis of which coincides with the coordinate axis Ox, i.e.,
Π =
{
(x, y, z) : | x |< a, y2 + z2 < R2
}
,
∂Π =
{
(x, y, z) : x = ±a, y2 + z2 < R2
}
∪
{
(x, y, z) : | x |< a, y2 + z2 = R2
}
,
where R, a is the radius and a half of length of the cylinder, respectively. The origin O of the
coordinate system Oxyz is placed in the cylinder center, the axis Oz is directed in an opposite
way to the vector g (the location of the Oy axis is determined, in this case, uniquely).
Let us consider the following family of surfaces Σ(t) (13) depending on parameter t:
ζ(x, y, t) = h(t) + k(t)y, (x, y) ∈ D(ζ) ⊂ D0, t ∈ [t0, t1].
Here h(t), k(t) are certain functions of time and the sets D(ζ), D0 are rectangles in the plane
Oxy which are determined by the relations
D(ζ) = {(x, y) : | x |< a, y1(t) < y < y2(t)}, D0 = {(x, y) : | x |< a, | y |< R}.
In the penultimate formula, the functions y1(t), y2(t) are solutions for y of the following system
of equations:
y2 + z2 = R2, z − h(t)− k(t)y = 0,
and are of the form
y1,2(t) =
{−hk ± [(1 + k2)R2 − h2]
1
2 }
1 + k2
.
Here the minus and plus sings correspond to the functions y1(t) and y2(t), respectively. Further
we suppose h(t), k(t), R to be such that
y1(t) < 0, y2(t) > 0.
ON SOLUTIONS OF GENERAL NONLINEAR INITIAL BOUNDARY-VALUE PROBLEM .. . 571
Thus, the function ζ(x, y, t) is independent of the space variable x and, as a function of
y, is determined in the changing region D(ζ) which is bounded by abscissas y1(t), y2(t) of the
intersection points of the circle and the straight line.
The region Ω(t), in this particular case, can be presented in the following form:
Ω(t) = Ω1(t) ∪ Ω2(t),
where
Ω1(t) = {(x, y, z) : | x |< a, −R < z < z1(t), | y |<
√
R2 − z2},
Ω2(t) =
{
(x, y, z) : | x |< a, z1(t) < z < z2(t),
z − h(t)
k(t)
< y <
√
R2 − z2
}
.
Here the functions z1(t), z2(t) are the ordinates of the above-mentioned intersection points of
the circle and straight line, correspond to the abscissas y1(t), y2(t), and are determined by the
formulae
z1,2(t) = {h± k[(1 + k2)R2 − h2]
1
2 }/(1 + k2).
The minus and the plus sings correspond to the functions z1(t), z2(t), respectively. If k(t) > 0,
then z1(t) < z2(t), which is supposed further.
The surface S(t) consists of two plane sets and the cylindrical surface so that we can write
S(t) =
3⋃
j=1
Sj(t),
where
S1,2(t) = {(x, y, z) : x = ±a, y2 + z2 < R2, −R < z < h(t) + k(t)y},
S3(t) = {(x, y, z) : | x |< a, y2 + z2 = R2, −R < z < h(t) + k(t)y}.
The surface S(t) is evidently piecewise smooth for any t.
The line l(t) of intersection of the surfaces S(t) and Σ(t) is a set of the following form:
l(t) =
4⋃
j=1
lj(t),
where
l1,2(t) = {(x, y, z) : | x |< a, y = y1,2(t), z = z1,2(t)},
l3,4(t) = {(x, y, z) : x = ±a, y1(t) < y < y2(t), z = h(t) + k(t)y}.
572 G.F. ZOLOTENKO
The l-condition on the line l(t) can be violated only if h(t) ≡ ±R when k(t) ≡ 0 (in this case
the l-line degenerates into segments coinciding with the generator of the cylindrical surface).
Further we shall suppose that | h(t) |< R. With this inequality, the l-condition is always
fulfilled.
Thus, in the considered example the region Ω(t) is a ζ-region.
In respect to the functions v(t), ω(t) in (7), we suppose that in projections to the axes x, y, z,
they are of the following form:
v(t) = (0, vy(t), vz(t)),
vy(t) = (C1 + C2t) cosα(t) + (C3 + C4t) sinα(t),
vz(t) = −(C1 + C2t) sinα(t) + (C3 + C4t) cosα(t),
ω(t) =
(
dα
dt
, 0, 0
)
,
whereα(t) ∈ C1([t0, t1]) is an arbitrary function and also−π/2 < α(t) < π/2;Cj , j = 1, 2, 3, 4,
is a certain real constant.
Under this conditions, we can give a solution of the shortened problem (1) – (3), (5) which
simultaneously will be a solution of the initial problem (1) – (9). Indeed, one can directly verify
that for the function
ζ0(x, y, t) = h0
√
1 + k2(t) + k(t)y,
where
k(t) =
C2 cosα(t) + (g + C4) sinα(t)
C2 sinα(t)− (g + C4) cosα(t)
,
and h0 is a certain real constant, the function
u0(x, y, z, t) = [(C1 + C2t) cosα(t) + C3 + C4t) sinα(t)]y
+ [(C3 + C4t) cosα(t)− (C1 + C2t) sinα(t)]z,
is a solution of the shortened problem (1) – (3), (5). The constant h0 is determined from condi-
tion (5).
Supposing (we use arbitrariness of F (t) from the Lagrange – Cauchy integral)
F (t) =
p0
ρ
+
1
2
[
(C1 + C2t)
2 + C3 + C4t)
2
]
+ [C2 sinα(t)− (g + C4) cosα(t)]h0
√
1 + k2(t)
and substituting u0 and ζ0 in (4), one can verify that a pair of the constructed functions (u0, ζ0)
is in fact a solution of the initial nonlinear problem (1) – (9) in this special case.
ON SOLUTIONS OF GENERAL NONLINEAR INITIAL BOUNDARY-VALUE PROBLEM .. . 573
It is obvious that the condition (15) is satisfied. Besides, it is clear that the constructed
functions u0, ζ0 satisfy all the formulated smoothness conditions. Thus the set of a pairs of the
functions (u0, ζ0) which figure in Theorem is indeed nonempty.
We note that the constructed solution is a solution in the usual sense although Theorem
guarantees existence of a more general solution only (more precisely, a weak solution for the
function u0). It should also be added that the important question of the parameters (v0(t), ω(t),
g(t), and so on) for which the conditions of Theorem are realized remains to be solved.
5. Conclusion
The general nonlinear initial boundary-value problem of the potential absolute motion of
an inviscid incompressible fluid partially filling a Lipschitzian cavity of a certain moving vessel
has a solution (u, ζ) for which the potential u(r, t) is a function of both the Sobolev space with
respect to r and the space of continuous functions with respect to t and the surface ζ(x, y, t)
is a function of both the Lipschitzian class with respect to x, y and the class of continuously
differentiable functions with respect to t.
The constructed example of an exact solution may be used for testing and graduating the
corresponding computing methods.
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Received 28.09.2000
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