# Qualitative Behaviour of Solutions

for the
Two-Phase Navier-Stokes Equations

with Surface Tension

###### Abstract.

The two-phase free boundary value problem for the isothermal Navier-Stokes system is studied for general bounded geometries in absence of phase transitions, external forces and boundary contacts. It is shown that the problem is well-posed in an -setting, and that it generates a local semiflow on the induced phase manifold. If the phases are connected, the set of equilibria of the system forms a -dimensional manifold, each equilibrium is stable, and it is shown that global solutions which do not develop singularities converge to an equilibrium as time goes to infinity. The latter is proved by means of the energy functional combined with the generalized principle of linearized stability.

Mathematics Subject Classification (2000):

Primary: 35R35, Secondary: 35Q30, 76D45, 76T10.

Key words: Two-phase Navier-Stokes equations, surface tension,
well-posedness, stability, compactness, generalized principle of linearized stability, convergence.

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## 1. Introduction

In this paper we consider a free boundary problem that describes the motion of two isothermal, viscous, incompressible Newtonian fluids in . The fluids are separated by an interface that is unknown and has to be determined as part of the problem.

More precisely, we consider two fluids that fill a region . Let be a given surface which bounds the region occupied by a viscous incompressible fluid, , the dispersed phase and let be the complement of the closure of in , corresponding to the region occupied by a second incompressible viscous fluid, , the continuous phase. Note that the dispersed phase is assumed not to be in contact with the boundary of . We assume that the two fluids are immiscible, and that no phase transitions occur. The velocity of the fluids is denoted by , and the pressure field by .

Let denote the position of at time . Thus, is a sharp interface which separates the fluids occupying the regions and , respectively. We denote the normal field on , pointing from into , by . Moreover, and mean the normal velocity and the curvature of with respect to , respectively. Here the curvature is negative when is convex in a neighborhood of , in particular the curvature of a sphere is . The motion of the fluids is governed by the following system of equations for

(1.1) | ||||||

Here, is the stress tensor defined by

and

denotes the jump of the quantity , defined on the respective
domains , across the interface .

Given are the initial position
of the interface and the initial velocity
.
The unknowns are the velocity field
,
the pressure field ,
and the free boundary .

The constants and denote the densities
and the viscosities of the respective fluids,
and the constant stands for the surface tension. In the sequel
we drop the index since there is no danger of confusion; however, we keep in mind
that and have jumps across the interface, in general.

System (1.1) comprises the two-phase Navier-Stokes equations with surface tension. The corresponding one-phase problem is obtained by setting and discarding . Here we concentrate the discussion on the two-phase problem.

There are several papers in the literature dealing with problem (1.1); cf. [5, 6, 7, 8, 9, 25, 26, 27]. All of them employ Lagrangian coordinates to obtain local well-posedness. This way it seems difficult to establish smoothing of the unknown interface, and this method is hardly useful in case phase transitions have to be taken into account. Here we employ a different approach, namely the Direct Mapping Method via a Hanzawa transform, which has been quite efficient in the study of Stefan problems, i.e. phase transitions involving temperature, only.

In a recent paper [19] we have shown that problem (1.1) is locally well-posed in an -setting provided and the initial interface is sufficiently close to a flat configuration. In addition, the interface as well as the solution are proved to become instantaneously real analytic. This result is based on a careful analysis of the underlying linear problem. Building on the latter results we show in this paper local well-posedness for arbitrary bounded geometries as described above. This induces a local semiflow on a well-defined nonlinear phase manifold.

It is known that the set of equilibria of the system are zero velocities, constant pressures in the components of the phases and the dispersed phase is a union of disjoint balls. Concentrating on the case of connected phases, we prove that equilibria are stable and any solution starting in a neighbourhood of such a steady state exists globally and converges to another equilibrium.

The energy of the system serves as a strict Ljapunov functional, hence the limit sets of the solutions are contained in the set of equilibria . Combining these results we show that any solution which does not develop singularities converges to an equilibrium in the topology of the phase manifold.

## 2. Transformation to a Fixed Domain

Let be a bounded domain with boundary of class , and suppose is a hypersurface of class , i.e. a -manifold which is the boundary of a bounded domain ; we then set . Note that is connected, but maybe disconnected, however, it consists of finitely many components only, since by assumption is a manifold, at least of class . Recall that the second order bundle of is given by

Here denotes the surface gradient on . Recall also the Haussdorff distance between the two closed subsets , defined by

Then we may approximate by a real analytic hypersurface , in the sense that the Haussdorff distance of the second order bundles of and is as small as we want. More precisely, for each there is a real analytic closed hypersurface such that . If is small enough, then bounds a domain with , and we set .

It is well known that such a hypersurface admits a tubular neighbourhood, which means that there is such that the map

is a diffeomorphism from onto . The inverse

of this map is conveniently decomposed as

Here means the orthogonal projection of to and the signed distance from to ; so and iff . In particular we have .

Note that on the one hand, is determined by the curvatures of , i.e. we must have

where mean the principal curvatures of at . But on the other hand, is also connected to the topology of , which can be expressed as follows. Since is a compact manifold of dimension it satisfies the ball condition, which means that there is a radius such that for each point there are , , such that , and . Choosing maximal, we then must also have .

Setting , we may use the map to parametrize the unknown free boundary over by means of a height function via

for small , at least. Extend this diffeomorphism to all of by means of

Here denotes a suitable cut-off function; more precisely, , , for , and for . This way is transformed to the fixed domain . Note that for , and

in particular,

Now we define the transformed quantities

the pull backs of and . This gives the following problem for .

(2.1) |

Here , and denote the transformed Laplacian, gradient and curvature, respectively. More precisely, we have

and

and

with

Note that

hence

With the curvature tensor and the surface gradient we have

and

Employing this notation, we have

and

where denotes the projection onto the tangent space of . Thus, is boundedly invertible, if and are sufficiently small. The curvature becomes

a differential expression involving second order derivatives of only linearly. Its linearization is given by

Here denotes the Laplace-Beltrami operator on .

It is convenient to decompose the stress boundary condition into tangential and normal parts. Multiplying the stress interface condition with we obtain

for the normal part of the stress boundary condition, and

for the tangential part. Note that the latter neither contains the pressure jump nor the curvature!

We rewrite this problem in quasilinear form, dropping the bars and collecting its principal linear part on the left hand side.

(2.2) | |||

The right-hand sides in this problem are either lower order terms or are of the same order appearing on the left, but carrying a factor or , which are small by construction. In fact, since is approximated by in the second order bundle we have smallness of , , and even of , uniformly on . All terms on the right-hand side are at least quadratic. More precisely, besides the which have been introduced before, the nonlinearities have the following form:

The idea of our approach can be described as follows. We consider the transformed problem (2). Based on maximal -regularity of the linear problem given by the left hand side of (2), we employ the contraction mapping principle to obtain local well-posedness of the nonlinear problem. The solutions of the transformed problem will belong to the following class:

This program will be carried out in the next sections.

## 3. The Linearized Problem

We consider now the inhomogeneous linear problem

(3.1) |

on a finite time-interval . We choose the same regularity classes for and as before, i.e.

and

Then

and

Therefore the equation for the height function lives in the trace space for the components of , i.e.

hence the natural space for is given by

Here the last space comes from the curvature term in the stress boundary condition, which induces an additional order in spatial regularity. Assuming that belongs to the trace space of , i.e.

we have the additional regularity for the pressure jump across the interface . The function is given; we will choose appropriately in Section 4.

There is another hidden regularity which comes from the divergence equation. To identify it, let . An integration by parts yields

Set and define the functional by means of

Then we have

Since this implies . Observe that this condition contains the compatibility condition

which appears choosing .

In the particular case we have if and only if and .

The main theorem of this section states that problem (3.1) admits maximal regularity, in particular, it defines an isomorphism between the solution space and the space of data.

###### Theorem 3.1.

Let , a bounded domain with , a closed hypersurface of class and , , be positive constants, ; set , and suppose

Then the two-phase Stokes problem (3.1) admits a unique solution with regularity

if and only if the data satisfy the
following regularity and compatibility conditions:

(a) , ,
and ;

(b) , and ;

(c) ,

(d) ;

(e) ;

(f) ;

(g) , and
;

(h) , and
.

The solution map is continuous between the corresponding spaces.

The proof will be carried out in the following subsections.

In general the pressure has no more regularity as stated in Theorem 3.1. However, there are situations where enjoys extra time-regularity, as stated in the following