On singular limit equations for incompressible fluids in moving thin domains
We consider the incompressible Euler and Navier-Stokes equations in a three-dimensional moving thin domain. Under the assumption that the moving thin domain degenerates into a two-dimensional moving closed surface as the width of the thin domain goes to zero, we give a heuristic derivation of singular limit equations on the degenerate moving surface of the Euler and Navier-Stokes equations in the moving thin domain and investigate relations between their energy structures. We also compare the limit equations with the Euler and Navier-Stokes equations on a stationary manifold, which are described in terms of the Levi-Civita connection.
Key words and phrases:Moving thin domain, moving surface, singular limit, incompressible fluid, surface fluid
2010 Mathematics Subject Classification:Primary: 35Q35, 35R01, 76M45; Secondary: 76A20
Fluid flows in a thin domain appear in many problems of natural sciences, e.g. ocean dynamics, geophysical fluid dynamics, and fluid flows in cell membranes. In the study of the incompressible Navier-Stokes equations in a three-dimensional thin domain mathematical researchers are mainly interested in global existence of a strong solution for large data since a three-dimensional thin domain with sufficiently small width can be considered “almost two-dimensional.” It is also important to investigate the behavior of a solution as the width of a thin domain goes to zero. We may naturally ask whether we can derive limit equations as a thin domain degenerates into a two-dimensional set and compare properties of solutions to the original three-dimensional equations and the corresponding two-dimensional limit equations. There are several works studying such problems with a three-dimensional flat thin domain [15, 16, 29, 33] of the form
for small , where is a two-dimensional domain and and are functions on , and a three-dimensional thin spherical domain  which is a region between two concentric spheres of near radii. (We also refer to  for the strategy of analysis of the Euler equations in a flat and spherical thin domain and its limit equations.) However, mathematical studies of an incompressible fluid in a thin domain have not been done in the case where a thin domain and its degenerate set have more complicated geometric structures. (See  for the mathematical analysis of a reaction-diffusion equation in a thin domain degenerating into a lower dimensional manifold.)
In this paper we are concerned with the incompressible Euler and Navier-Stokes equations in a three-dimensional thin domain that moves in time. The purpose of this paper is to give a heuristic derivation of singular limits of these equations as a moving thin domain degenerates into a two-dimensional moving closed surface. We also investigate relations between the energy structures of the incompressible fluid systems in a moving thin domain and the corresponding limit systems on a moving closed surface.
Here let us explain our results on limit equations and strategy to derive them. Let be an evolving closed surface in and and its (scalar) outward normal velocity and unit outward normal vector field, respectively. We assume that does not change its topology. Also, let be a tubular neighborhood of of radius in with sufficiently small . We consider the Euler equations
and the Navier-Stokes equations with (perfect slip) Navier boundary condition
Here and denote the unit outward normal vector field and the (scaler) outward normal velocity of . Also, is the viscosity coefficient and is the strain rate tensor with the transpose of the gradient matrix . We suppose that admits the normal coordinate system for , where is the closest point mapping onto and is the signed distance from increasing in the direction of . Based on the normal coordinates, we expand the velocity field on in powers of the signed distance as
and the pressure similarly. We substitute them for the equations in and determine equations on that the zeroth order term in (1.8) satisfies. Then we obtain limit equations of the Euler equations (1.1)–(1.3):
Here is the material derivative along the velocity field and and denote the tangential gradient and the surface divergence on , respectively (see Section 2 for their definitions). Similarly, we get limit equations of the Navier-Stokes equations (1.4)–(1.7):
Here and is the orthogonal projection onto the tangent plane of . Note that if we take the average of (1.8) in the normal direction of then
Therefore, formally speaking, our limit equations are equations satisfied by the limit of the average in the thin direction of a solution to the original Euler or Navier-Stokes equations in as goes to zero. (The above method is also applied in  to derive a limit equation of a nonlinear diffusion equation in a moving thin domain.)
In the equations (1.9) and (1.12) the scalar function , which comes from the normal derivative of the bulk pressure (see the expansion (3.5) of and (3.17) in the proof of Theorem 3.1), is determined by the normal component of (1.9) and (1.12). Therefore, the limit Euler system (1.9)–(1.11) is intrinsically equivalent to
We note that these tangential surface fluid systems were also derived in [17, 18] recently. The derivation of the Navier-Stokes equations on a moving surface in  is based on local conservation laws of mass and linear momentum for a surface fluid. On the other hand, the authors of  applied a global energetic variational approach to derive several kinds of equations for an incompressible fluid on an evolving surface.
The viscous term in the momentum equation (1.12) of the limit Navier-Stokes system appears in the Boussinesq-Scriven surface fluid model which was first described by Boussinesq  and generalized by Scriven  to an arbitrary curved moving surface (see also [1, Chapter 10] for derivation of the Boussinesq-Scriven surface fluid model). In  the Boussinesq-Scriven surface fluid model was considered to formulate a continuum model for fluid membranes in a bulk fluid, which contains equations for a viscous fluid on a curved moving surface, and study the effect of membrane viscosity in the dynamics of fluid membranes. It was also studied in the context of two-phase flows [5, 6, 25] in which equations for a surface fluid are considered as the boundary condition on a fluid interface.
Since we consider an incompressible fluid on a moving surface or in its tubular neighborhood, some constraints on the motion of the surface are necessary. For the existence of a surface incompressible fluid it is required that the area of the moving surface is preserved in time. To consider a bulk incompressible fluid in the -tubular neighborhood of the moving surface for all sufficiently small, we need another constraint on the moving surface besides the area preserving condition. However, it is automatically satisfied by the Gauss-Bonnet theorem and the assumption that the moving surface does not change its topology. See Remark 3.3 for details.
When the surface does not move in time, our tangential limit system (1.15) of the Euler equations is the same as the Euler system on a fixed manifold derived by Arnold [2, 3], who applied the Lie group of diffeomorphisms of a manifold (see also Ebin and Marsden ). Also, for a stationary surface our tangential limit system (1.16) of the Navier-Stokes equations is the same as the Navier-Stokes system on a manifold derived by Taylor , although the authors of  claim that (1.16) is different from Taylor’s system (see Remark 4.3). For detailed comparison of our limit systems and the systems derived in previous works see Remarks 3.2 and 4.2. We further note that the function in the limit momentum equations (1.9) and (1.12), which is determined by the normal component of these equations, does not vanish even if the surface is stationary. See Remarks 3.2 and 4.2 for details.
Finally we note that our results are based on formal calculations and thus mathematical justification is required. There are a few works that present rigorous derivation of limit equations in the case where a degenerate set is a hypersurface or a manifold. Temam and Ziane  derived limit equations for the Navier-Stokes equations in a thin spherical domain by characterizing the thin width limit of a solution to the original equations as a solution to the limit equations. In , Prizzi, Rinaldi, and Rybakowski compared the dynamics of a reaction-diffusion equation in a thin domain and that of a limit equation when a thin domain degenerates into a lower dimensional manifold. Recently, the present author derived a limit equation of the heat equation in a moving thin domain shrinking to a moving closed hypersurface by characterization of the thin width limit of a solution . Although there are several tools and methods introduced in the above papers, it seems that mathematical justification of our results is difficult because of the nonlinearity of the equations and the evolution of the shape of the degenerate surface, and that we need some new techniques.
This paper is organized as follows. In Section 2 we give notations and formulas on quantities related to a moving surface and a moving thin domain. In Sections 3 and 4 we derive the limit equations of the Euler and Navier-Stokes equations in a moving thin domain, respectively. In Section 5 we derive the energy identities of the Euler and Navier-Stokes equations and the corresponding limit equations and investigate relations between them. In Appendices A and B we give proofs of lemmas in Section 2 involving the differential geometry of a surface embedded in the Euclidean space.
We fix notations on various quantities of a moving surface and give formulas on them. All functions appearing in this section are assumed to be sufficiently smooth.
Lemmas in this section are proved by straightforward calculations. To avoid making this section too long we give proofs of them in Appendix A, except for the proofs of Lemmas 2.4 and 2.5. Also, a proof of the formula (2.15) in Lemma 2.4 is given in Appendix B. Although we are concerned with a two-dimensional surface in this paper, all notations and formulas in this section apply to hypersurfaces of any dimension with easy modifications.
2.1. Moving surfaces and moving thin domains
Let , be a two-dimensional closed (i.e. compact and without boundary), connected, and oriented moving surface in . The unit outward normal vector and the (scalar) outward normal velocity of are denoted by and , respectively. Also, let be a space-time hypersurface associated with . We assume that is smooth at each and moves smoothly in time. In particular, does not change its topology. By the smoothness assumption on , the (outward) principal curvatures and of are bounded uniformly with respect to . Hence there is a tubular neighborhood
of radius independent of that admits the normal coordinate system
where is the closest point mapping onto and is the signed distance function from (see e.g. [11, Lemma 2.8]). Moreover, the mapping and the signed distance are smooth in the closure (in ) of a space-time noncylindrical domain . We assume that increases in the direction of . Therefore,
Moreover, differentiating both sides of
For a sufficiently small we define a moving thin domain in as
and a space-time noncylindrical domain and its lateral boundary as
Since is a tubular neighborhood of , the unit outward normal vector and the outward normal velocity of its boundary are given by
2.2. Notations and formulas for quantities on fixed surfaces
In this subsection we fix and suppress the time . Hence denotes a two-dimensional closed, connected, oriented and smooth surface in . Let us give notations and formulas for several quantities on the fixed surface . (In the sequel we use the same notations given in this subsection for the moving surface .) Let be the orthogonal projection onto the tangent plane of at each point on given by
where is the identity matrix of three dimension and for denotes the tensor product of and given by
For a function on we define its tangential gradient as
Here is an extension of to satisfying . Note that the tangential gradient of is independent of the choice of its extension (see e.g. [11, Lemma 2.4]). Also, it is easy to see that and hold on . The tangential derivative operators are given by
so that , which are again independent of the choice of an extension of . For example, we may take the constant extension in the normal direction of given by for .
For vector fields on and on , we define the gradient matrix and the divergence of as
and the tangential gradient matrix and the surface divergence of as
These notations are consistent with the formula on , where is an arbitrary extension of to with . For a function on we denote by the tangential Hessian matrix of whose -entry is given by (). Let be a matrix-valued function defined on or on of the form
We define the divergence on or the surface divergence on as a vector field whose -th component is given by
Finally we set
and call them the Weingarten map of , the Laplace-Beltrami operator on , (twice) the mean curvature of , and the Gaussian curvature of , respectively. The usual Laplacian and the Laplace-Beltrami operator acting on vector fields are understood to be componentwise operators.
For all we have
Also, is symmetric (i.e. ) and holds on by (2.9).
The tangential derivatives () are noncommutative in general. An exchange formula for them includes the unit outward normal of the surface.
Let be a function on . For each we have
Here denotes the -th component of the vector field .
The next formula is a consequence of (2.11), which we use in Section 4 to express a viscous term of limit equations of the Navier-Stokes equations in terms of the Laplace-Beltrami operator. For a vector field on we set
The matrices and are called a tangential strain rate and a projected strain rate in , respectively.
Let be a (not necessarily tangential) vector field on . Then
holds on (note that on the right-hand side is tangential).
To compare our limit systems with the incompressible fluid systems on a fixed manifold derived by Arnold [2, 3] and Taylor  we need formulas on the Levi-Civita connection. Let be the Levi-Civita connection on with respect to the metric on induced by the Euclidean metric of (see e.g. [9, Section 2.3] and [24, Sections 3.3.1 and 4.1.2] for the definition of the Levi-Civita connection). Hence for tangential vector fields and on the covariant derivative of along is denoted by , which is again a tangential vector field on . The Levi-Civita connection is considered as a mapping
where and are the tangent and cotangent bundle of , respectively, and for a vector bundle over we denote by the set of all smooth sections of . (Hence denotes the set of all smooth tangential vector fields on . We refer to [20, Chapter 10] for the definitions of a vector bundle and a section.) Also, for a tangential vector field on the notation stands for a mapping from into itself. Then we write for the formal adjoint operator of (see [24, Section 10.1.3]) and set . The operator is called the Bochner Laplacian (note that there is another definition of the Bochner Laplacian where the sign is taken opposite).
Let and are tangential vector fields on . Then
hold on . Here is an extension of to with and denotes the directional derivative of along in , i.e.
Also, the left-hand side of (2.14) is independent of the choice of the extension .
The formula (2.14) is well-known as the Gauss formula (see e.g. [9, Section 4.2] and [19, Section VII.3]) and we omit its proof. Note that on since is tangential. Hence the Gauss formula (2.14) is also expressed as
for tangential vector fields and on . We also call (2.16) the Gauss formula.
A proof of the formula (2.15) is given in Appendix B. Note that (2.15) is useful by itself since it gives a global expression under the fixed Cartesian coordinate system of the Bochner Laplacian acting on tangential vector fields on , which is originally defined intrinsically and represented under only local coordinate systems.
Combining Lemmas 2.3 and 2.4 we get the following formula on the surface divergence of the projected strain rate, which is crucial for comparison of our limit Navier-Stokes system and the incompressible viscous fluid system on a manifold derived by Taylor  (see Remark 4.2).
For a tangential vector field on satisfying we have
Let be a tangential vector field on satisfying . Then
by and . Applying this and
to the formula (2.13), and observing that is tangential, we have
Moreover, since is symmetric and has the eigenvalues , , and , where the eigenvector corresponding to the eigenvalue is (see Lemma 2.1), for each we can take an orthonormal basis of the tangent plane of at such that , . (The vectors and are called the principal directions at . See e.g. [19, Section VII.5] for details.) Expressing the tangential vector as a linear combination of and and using and we easily obtain . Applying this and (2.15) to (2.18) we obtain (2.17). ∎
Besides derivation of limit equations, we are also interested in thin width limits of energy identities for the Euler and Navier-Stokes equations. To derive limit energy identities we give change of variables formulas for integrals over level-set surfaces and tubular neighborhoods of . For and we set
Here the second equality follows from the definition of the Gaussian curvature and (2.10). The function is the Jacobian appearing in the following change of variables formulas (see [13, Section 14.6] or Appendix A).
For a function on we have
Here denotes the two-dimensional Hausdorff measure.
2.3. Material derivatives and differentiation of composite functions with the closest point mapping
Now let us return to the moving surface . We first give a material time derivative of a function on . Let be a vector field on with . Suppose that there exists the flow map of , i.e. is a diffeomorphism onto its range for each and
Note that is a diffeomorphism from onto for each since the normal component of is equal to the outward normal velocity of the moving surface , which completely determines the change of the shape of . We define the material derivative of a function on along the velocity field as
By the chain rule of differentiation it is also represented as
where is an arbitrary extension of to satisfying . We write for with and call it the normal time derivative. Note that the normal time derivative of a function on is equal to the time derivative of its constant extension in the normal direction, i.e.
Also, for a tangential vector field on the material derivative of along the velocity field of the form is expressed as
In the following sections we frequently differentiate the composition of a function on and the closest point mapping . To avoid repetition of the same calculations we give several formulas on derivatives of composite functions with .
Let be a function on . Based on the normal coordinate system for , we expand in powers of the signed distance :
Here , , and the coefficients of higher order terms in are considered as functions on . Also, for we write for the terms of order higher than with respect to small , i.e.
In the sequel, we also use Landau’s symbol (as ) for a nonnegative integer , i.e. is a quantity satisfying for small with a constant independent of . Note that, contrary to , we may differentiate with respect to and since it just stands for the higher order terms in the expansion (2.24) with respect to small , and the -th order derivative of is for . Also, for and by on . We use the same notations on the expansion (2.24) for functions on with each fixed .
Let be a scalar- or vector-valued function on . The derivatives of the composite function with respect to and are of the form
for . Here we abbreviate to .
We also give an expansion formula for the divergence of a matrix-valued function which we need to derive limit equations of the Navier-Stokes equations.
Let and be matrix-valued functions on with each fixed . For we set
Then we have
for . Here denotes the transpose of the matrix .
3. Limit equations of the Euler equations
We consider the incompressible Euler equations in :
Here is the velocity of a bulk fluid and is the pressure. The goal of this section is to derive limit equations of the Euler equations as goes to zero. According to the normal coordinate system (2.1), we expand and with respect to the signed distance as
Here we used the notation (2.24). The limit equations are given as the principal term in the expansion with respect to of the Euler equations in .
Let and satisfy the Euler equations (3.1)–(3.3) in the moving thin domain . Then the normal component of the zeroth order term in the expansion (3.4) is equal to the outward normal velocity of the moving surface , i.e. . Moreover, and the zeroth order term and the first order term in the expansion (3.5) satisfy
Let us explain how the limit equations (3.6) and (3.7) determine , , and . As stated in Theorem 3.1, the normal component of is equal to the outward normal velocity of the moving surface. The tangential component of and the scalar function are determined by the equations
Finally the scalar function is given just by the inner product of (3.6) and :
Note that comes from the normal derivative of the pressure of the bulk fluid in the moving thin domain (see (3.17) below).
The system (3.8) is the same as the incompressible Euler system (II) in  with the constant density. When the surface is stationary, the limit velocity is tangential () and holds on by the Gauss formula (2.14), where is the covariant derivative. From this and the fact that is independent of the time it follows that
Hence the tangential limit system (3.8) becomes