Table Of ContentLecture Notes in Mathematical Fluid Mechanics
I. V. Denisova
V. A. Solonnikov
Motion of a
Drop in an
Incompressible
Fluid
Advances in Mathematical Fluid Mechanics
Lecture Notes in Mathematical Fluid Mechanics
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I. V. Denisova • V. A. Solonnikov
Motion of a Drop in an
Incompressible Fluid
I.V.Denisova V.A.Solonnikov
InstituteofProblems St.PetersburgDepartment
ofMechanicalEngineering ofSteklovMath.Institute
RussianAcademyofSciences RussianAcademyofSciences
St.Petersburg,Russia St.Petersburg,Russia
Translation from Russian language edition: Dvizhenie kapli v neszhimaemoy zhidkosti:
monografiyabyI.V.DenisovaandV.A.Solonnikov,©Izdatel’stvolan’2020,Izdatel’stvo
lan’.AllRightsReserved.
ISSN2297-0320 ISSN2297-0339 (electronic)
AdvancesinMathematicalFluidMechanics
ISSN2510-1374 ISSN2510-1382 (electronic)
LectureNotesinMathematicalFluidMechanics
ISBN978-3-030-70052-2 ISBN978-3-030-70053-9 (eBook)
https://doi.org/10.1007/978-3-030-70053-9
MathematicsSubjectClassification:35Q30,76D05,35R35
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Contents
1 Introduction.......................................................... 1
1.1 Statement of the Problem, Definition of the
H¨older Spaces................................................... 4
2 A Model Problem with Plane Interface and with
Positive Surface Tension Coefficient............................ 11
2.1 Auxiliary Propositions ......................................... 11
2.2 An Explicit Solution of a Homogeneous Model Problem .... 13
2.3 Theorems on the Fourier Multipliers in the Ho¨lder Spaces.. 17
2.4 An Estimate of the Solution of Problem (2.2.1).............. 25
2.5 The Problem for the Inhomogeneous Stokes System......... 33
3 The Model Problem Without Surface Tension Forces ..... 47
3.1 Statement of the Problem and Formulation of
Existence Theorem ............................................. 47
3.2 Preliminary Considerations.................................... 49
3.3 The Homogeneous Problem: An Explicit Solution........... 56
3.4 The Proof of Theorem 3.3.1 ................................... 60
3.4.1 The Analysis of d...................................... 61
3.4.2 The Estimate of π..................................... 64
3.4.3 The Estimate of the Vector f =− 1 ∇p in D3 ..... 68
ρ± T
3.5 The Proof of Theorem 3.1.1 ................................... 71
4 A Linear Problem with Closed Interface Under
Nonnegative Surface Tension ................................... 79
4.1 Auxiliary Propositions: The Statement of Results........... 79
4.2 A Priori Estimates of the Solution of Problem (1.1.7)....... 82
4.3 The Solvability of Problem (1.1.7): Constructing a
Regularizer...................................................... 93
v
vi Contents
5 Local Solvability of the Problem in Weighted
H¨older Spaces........................................................ 103
5.1 Weighted H¨older Spaces: Formulation of the Local
Existence Theorem for the Nonlinear Problem............... 103
5.2 Weighted Estimates for Linear Problem (1.1.7) ............. 108
5.3 Solvability of a Linearized Problem on a Finite
Time Interval ................................................... 117
5.4 The Proof of the Solvability of Nonlinear Problem (5.1.1) .. 131
6 Global Solvability in the H¨older Spaces for the
Nonlinear Problem Without Surface Tension................ 139
6.1 Statement of the Main Result ................................. 139
6.2 A Linear Problem with Closed Interface
Between the Fluids ............................................. 141
6.3 A Linearized Problem.......................................... 143
6.4 Global Solvability of Problem (1.1.1) with σ =0 ............ 145
7 Global Solvability of the Problem Including
Capillary Forces: Case of the Ho¨lder Spaces ................. 155
7.1 Setting of the Problem, Statement of the Main Result...... 155
7.2 Energy Estimate of the Solution .............................. 160
7.3 A Linearized Problem.......................................... 170
7.4 Global Classical Solvability of Problem (7.1.3), (1.0.3)...... 172
8 Thermocapillary Convection Problem ........................ 183
8.1 Setting of the Problem and Statement of Results............ 183
8.2 Linearized Problems............................................ 186
8.3 The Solvability of Problem (8.1.2) ........................... 191
8.4 The Problem in R3 with a Constant Temperature
Value at Infinity ................................................ 201
9 Motion of Two Fluids in the Oberbeck-Boussinesq
Approximation ...................................................... 205
9.1 Setting of the Problem and the Statement of Its
Local Solvability................................................ 205
9.2 A Linearized Problem and Estimates of
the Initial Pressure ............................................. 208
9.3 Local Solvability of Problem (9.1.3), (9.1.4) ................. 214
9.4 Global Solvability of the Oberbek-Boussinesq Problem ..... 222
9.4.1 Energy Estimate of the Solution ..................... 225
9.4.2 Global Solvability of
Problem (9.1.1), (9.4.1), (1.0.3) ...................... 227
10 Local L2-Solvability of the Problem with
Nonnegative Coefficient of Surface Tension .................. 233
10.1 The Definition of the Sobolev-Slobodetskiˇı Spaces
and the Introduction of Equivalent Norms ................... 233
Contents vii
10.2 L2-estimates of a Solution of the Model Problem
with Plane Interface Between the Fluids ..................... 237
10.2.1 The Homogeneous Problem........................... 237
10.2.2 The Nonhomogeneous Problem with
Plane Interface......................................... 240
10.3 A Priori Estimates of a Solution of the Problem
with a Closed Interface......................................... 246
10.4 L2-Solvability of the Linearized Problem with
Closed Interface in R3.......................................... 252
10.4.1 The Solvability of Problem (1.1.7) in a
Simplified Case ........................................ 253
10.4.2 The Case of Nonzero r, b(cid:2) and v0 =0 ............... 264
10.4.3 Problem (1.1.7) in the General Case................. 265
10.5 Local Solvability of the Nonlinear Problem
in the L2-Setting................................................ 268
11 Global L2-Solvability of the Problem Without
Surface Tension...................................................... 277
11.1 The Statement of Global Existence Theorem ................ 277
11.2 Auxiliary Propositions ......................................... 279
11.3 The Proof of the Existence of a Global Solution............. 283
12 L2-Theory for Two-Phase Capillary Fluid.................... 285
12.1 The Statement of the Theorem on Global Solvability ....... 286
12.2 Global Solvability of a Linear Problem ....................... 291
12.3 The Nonlinear Problem ........................................ 298
Conclusions ................................................................ 309
References.................................................................. 311
Chapter 1
Introduction
Abstract A literature review, problem statement, and definition of spaces
are given. Free boundary problems for the Navier–Stokes system are set.
Boundary conditions on an unknown interface are discussed. The transfor-
mationfromtheEulercoordinatestotheLagrangianonesiscarriedoutwhich
leads to a problem in a known domain but with nonlinear coefficients. This
system is one of the main problems studied in the monograph the content of
which is described at the end of the chapter.
The problem on the unsteady motion of two viscous incompressible immis-
cible liquids with an unknown closed interface belongs to the class of free
boundary problems. The theory of these problems for the Navier–Stokes
equations has in their development only about four decades, although their
statement goes back to the classical works of the nineteenth century. Most
authors working in this direction consider stationary problems. This also
appliestotheproblemofthemotionofafinitevolumeofonefluidinanother.
This problem was first studied in the works of J. Hadamard [38] and
W. Rybczynski [70]. They obtained an analytical solution to the Stokes
system corresponding to the axially symmetric fall of a spherical drop at a
constantspeed.Stationarymotionoftwoliquidswithafreeclosedsurfaceof
their interface was investigated by V. Y. Rivkind [65–67] and J. Bemelmans
[8]. In work [67] based on a priori estimates, the solution of the Navier–
Stokes system was justified by approximate methods by which the motion
andshapeofadropletinaliquidmedium[65,66]werecalculated.Assuming
fluid density is small, J. Bemelmans established the existence of a unique
solution to the nonlinear problem of a drop falling with a constant speed.
Unsteady problems on the motion of fluids with free boundaries are more
difficult to research. They are less studied. The general formulation of these
problems is as follows (see [63]): find the region Ω ⊂ Rn(n = 2,3) occupied
t
fluid with viscosity ν > 0 and density ρ > 0 at the moment time t > 0; find
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I.V.Denisova,V.A.Solonnikov,Motion of a Drop in an Incompressible
Fluid,AdvancesinMathematicalFluidMechanics,
https://doi.org/10.1007/978-3-030-70053-9 1
2 1 Introduction
the vector field v = (v1,·,vn) and pressure function p satisfying the initial-
boundary value problem for the Navier–Stokes equations:
1
D v−νΔv+(v·∇)v+ ∇p=f, ∇·v =0 inΩ , t>0, (1.0.1)
t ρ t
v|t=0 =v0(x), x∈Ω0, v|x∈Σ =0, Tn|Γt =σHn. (1.0.2)
Here Dt = ∂/∂t, ∇ = (∂/∂x1,∂/∂x2,∂/∂x3), Σ is a solid boundary, Γt =
∂Ωt\Σ is the free surface, f is the field of external forces, v0 is the vector
field of initial velocity in the given region Ω0, T is the stress tensor with the
elements
T =−δkp+μ(∂v /∂x +∂v /∂x ), i,k (cid:2)n;
ik i i k k i
n is the outward normal vector to Γ at the point x, σ (cid:3)0 is surface tension
t
coefficient, and H(x,t)/(n−1) is the mean curvature of Γ (H < 0 at the
t
points of convexity of Γ to the outside of the fluid); the dot denotes the
t
scalar product in Rn.
In addition, to exclude mass loss across the free surface Γ , it is assumed
t
tobealwayscomposedbythesameliquidparticles.Itisanalytically written
like this: for t > 0, Γ consists of such points x(ξ,t) whose corresponding
t
radius vector x(ξ,t) is a solution to the Cauchy problem
Dtx=v(x(t),t), x|t=0 =ξ, ξ ∈Γ0, t>0, (1.0.3)
where Γ0 = ∂Ω0\Σ is the surface given at the initial moment. On a solid
surface, the non-slip condition is usually set.
UnsteadymotionofaheavyviscousfluidoverasolidbottomΣwasstudied
intheworksofJ.Beale[6,7]andG.Allain[4].Inthethree-dimensionalcase,
local-in-time unique solvability was proved for the problem (1.0.1)–(1.0.3)
with σ = 0 in Sobolev spaces [6], and, moreover, for σ > 0 and small initial
data, global-in-time solvability was done also in these spaces [7]. Without
limiting the size of the initial data, G. Allain established a local existence
theorem for this problem with σ >0, n=2 [4].
The most advanced problem is that on the motion of an isolated liquid
mass. In this case, the surface Σ is absent, and the free boundary Γ is
closed. Thus, V. O. Bytev [11] and O. M. Lavrentieva [50] considered
problem(1.0.1)–(1.0.3)inaringwithtwofreeboundaries,whilethevectorv0
was considered axially symmetric. In [11], it was shown that Ω can expand
t
to infinity for σ =0 in some cases. For σ >0, it was established in [50] that
Ωt can turn into a circle for t=t0 <∞ and save this form for all t>t0.
Problem (1.0.1)–(1.0.3) for a simply connected domain Ω with a closed
t
boundary Γ was studied in the works of one of the authors. In the Sobolev–
t
Slobodetskiˇı space, the existence of a global solution was proved for σ > 0
