Steady compressible Oseen flow with slip boundary conditions
Mathematical Institute, Polish Academy of Sciences
ul.Sniadeckich 8, 00-956 Warszawa
We prove the existence of solution in a class
to steady compressible Oseen system
with slip boundary conditions in a two dimensional, convex domain with the boundary
of class .
The method is to regularize a weak solution
obtained via the Galerkin method. The problem of regularization
is reduced to a problem of solvability of a certain transport equation by application
of the Helmholtz decomposition.
The method works under additional assumption on the geometry of the boundary.
MSC: 35Q10, 76N10
Keywords: Compressible Navier-Stokes flow, Slip boundary conditions
In this paper we consider a system of Stokes-type equations describing steady flow of a barotropic, compressible fluid in a two dimensional, convex domain with - boundary, supplied with inhomogeneous slip boundary conditions with nonnegative friction coefficient. The system can be considered as a linearization of a Navier-Stokes system for compressible fluid around a constant flow , thus we will call it compressible Oseen system. The slip boundary conditions involving friction enable to describe the interactions between the fluid and the boundary of the domain. It also turns out that they allow to extract some information on the vorticity of the velocity, that can be used to show that the velocity has higher regularity. Such approach has been applied in  and  to incompressible flows. In this paper we follow these ideas, modifying them in a way that they can be applied to the compressible system. A significant feature of this system is its elliptic-hyperbolic character: the momentum equation is elliptic in the velocity, while the continuity equation is hyperbolic in the density. Therefore we can prescribe the values of the density only on the part of the boundary where the flow enters the domain and a singularity appears in the points where the inflow and outflow parts of the boundary meets.
We show existence of a solution . A method we apply is to regularize a weak solution obtained via the Galerkin method. Analysing the vorticity of the velocity we can show that the density is in fact solution to a certain transport equation, obtained via elimination of the velocity from the continuity equation. The problem of regularization is thus reduced to a problem of solvability of a transport equation. The values of the density are prescribed on the part of the boundary where the flow enters the domain, and the density can be found as a solution to the transport equation via method of characteristics, thus singularities appear in points where the inflow and outflow parts of the boundary coincides. We show that the solvability of this transport equation is relied with the geometry of the boundary near the singularity points, thus we can define classes of domains where our method of regularization can or cannot be applied.
Since similar difficulties resulting from the mixed character of the problem appear in the analysis of steady compressible Navier-Stokes system, it is likely that the results of this paper will turn out useful in future analysis of the nonlinear problem. Now let us formulate the problem more precisely.
The steady compressible Oseen system reads:
where is a bounded, convex domain in with a boundary of class . is the velocity of the fluid and is the density. denotes outward unit normal to . We assume that , and are given functions. and are viscosity constants satisfying and is a friction coefficient (note that if then the conditions (1.1) reduce to a homogeneous Dirichlet condition). The system (1.6) can be considered as a linearization of a steady compressible Navier-Stokes system around a constant flow . More precisely, the perturbed flow satisfies inhomogeneous boundary conditions and , but if we assume that and are regular enough we can reduce the problem to homogeneous boundary conditions (1.1). Thus we distiguish the inflow and outflow parts of the boundary as the parts where the perturbed flow enters and leaves the domain:
Let us also denote . We assume that consist of two points: and (see Fig. 1). Due to the convexity of we can define functions and for in the following way:
is given as a -function of . We will denote these functions by and
respectively (Fig. 2)
For convenience we will denote
The main result of this paper is
Assume that and is large enough. Assume further that the boundary near the singularity points satisfies the following condition
Then the system (1.6) has a unique solution and
The geometric condition (1.2) may look strange since it is formulated in a general form, but it has a clear meaning. Namely, the boundary near the singularity points can not be too flat, more precisely, our method works if the boundary is less flat than a graph of a function around zero for some . We also show (lemma 13 (b)) that the method does not work if the boundary behaves like or is more flat. The limit case if the boundary is more flat that the graph of for all , but less flat than . An example of such a function is . In lemma 14 we show that our method doesn’t work in such case. The proof of theorem 1 is divided into several steps. In section 2 we show existence of a weak solution in a class using the Galerkin method (theorem 3). To obtain a weak solution it is enough to assume that , and no further constraint on the geometry of is required. The constraint (1.2) arises when we want to show that the weak solution belongs to class , and we also need . The issue of regularity of the weak solution is treated in section 3. First we prove that the vorticity of the velocity belongs to (lemma 7). Such approach has been applied to incompressible Navier-Stokes equations in  and . In the incompressible case we can next solve a div-rot system to show higher regularity of the velocity, but in the compressible case we have to extract some information on the density. The idea is to use the Helmholtz decomposition in , that means express the velocity as a sum of a divergence-free vector function and a gradient. The standard theory of elliptic equations enables us to show that the divergence-free part belongs to , and in order to show higher regularity of the gradient part it is enough to show that . In lemma 10 we show that , thus we have to show that . The method is to show that the density is a solution to the transport equation
Thus the problem of regularization of the weak solution is reduced a problem of solvability of the transport equation (1.4). The boundary condition (1.6) prescribes the values of the density on the inflow part of the boundary and (1.4) can be solved via method of characteristics, thus a singularity appears in the points and , which we will call the singularity points. It turns out that we can solve the equation (1.4) provided that the singularity is not too strong, what is reflected in the constraint (1.2). We will finish the introductory part removing inhomogeneity on the boundary. Let us construct a function satisfying
such that . Then a pair , where , satisfies
Obviously we have
thus from now on we can work with the system (1.6) denoting , , and .
2 Weak solution
In order to define a weak solution to the system (1.6) consider a space
and equipped with the norm . Consider also a space
with the norm .
Now we want to introduce a weak formulation of (1.6). First, observe that for regular enough we have
where for .
Thus taking in (1.6) and multiplying it by a function we get
Multiplying (1.6) by a regular function we get
The above considerations leads to a natural definition of a weak solution to the system (1.6).
We want to show existence of a weak solution using the Galerkin method. In order to show existence of solutions to approximate problems in section 2.1 we apply well known result (lemma 1). This result automatically gives uniform boundedness of the sequence of approximate solutions, what enables us to show convergence of approximate solutions to the weak solution in section 2.2.
2.1 Approximate solutions
In order to construct a Galerkin approximation of a weak solution let us introduce an orthonormal basis of : and finite dimensional spaces: . We will search for a sequence of approximations to the velocity in the form
Let us denote . Taking , and where
in (2) we get
For we obtain a system of equations on coefficients . If a function of a form (2.4) satisfies the equations (2.1) for , it means that a pair satisfies (2)-(2.3) for each . We will call such a pair an approximate solution to (2) - (2.3).
Let be a finitely dimensional Hilbert space and let be a continuous operator satisfying
In order to apply lemma 1 we will need some auxiliary results in spaces and .
(Poincare inequality in )
Assume that (2.12) doesn’t hold. Then such that . Without loss of generality we can assume , thus
Clearly is a bounded sequence in and thus thanks to boundedness of the compact embedding theorem implies that it contains a subsequence that is a Cauchy sequence in . But (2.8) implies that is also a Cauchy sequence in . Thus is a Cauchy sequence in , hence for some . Obviously and , thus is constant almost everywhere. But also , and since is a bounded set with regular boundary, the unit normal takes all the values from the unit sphere on . Therefore
what contradicts ∎
Now we will use the Poincare inequality to show that in a following modification of the Korn inequality holds:
Assume that is large enough. Then for :
The proof is based on a proof of a different version of the Korn inequality in . We have
The second term of the r.h.s is equal to and the third term vanishes since , thus from (2.1) we get
but we have and thus using the Poincare inequality (2.7) we get
and the last term will be positive provided that is large enough. ∎
The last inequality we need is the Poincare inequality in .
(Poincare inequality in )
The proof is straightforward using density of smooth functions in and the Jensen inequality. ∎
The following theorem gives a solution to the system (2.1)
For and there exists a solution to the system (2.1), . The function satisfies
In order to apply Lemma 1 we have to define an appropriate operator
. For convenience let us define :
Now (2.1) can be rewritten as and thus it is natural to define
We have to verify the assumptions of Lemma 1. Obviously and it is a continuous operator. For we have
Using the definition of we can rewrite (2.1) as
Using the Korn inequality (2.9) we get for large enough. Now let us denote
Then and we have
Combining these bounds we get
thus is a solution to (2.1) ∎
2.2 Existence of weak solution
Now we show that the sequence constructed in previous section converges to the weak solution of our problem.
Assume that and is large enough. Then there exists a weak solution to (1.6) satisfying the estimate
Since the sequence is bounded in , there exists a subsequence and a function such that . Now let us denote for simplicity . It is bounded in , thus there exists a subsequence for some function . Now we need to show that , but this is quite obvious. We have
It is a bit more complicated to show the existence of . The estimate (2.17) gives boundedness in of the sequences , , and boundedness in of . Thus up to a subsequence
for some and some . On the other hand, since the sequence is bounded in , the compactness theorem yields up to a subsequence for some . We want to show that in fact and that satisfies (2) - (2.3). But we have :
thus . Similarily we can verify that
Thus , and the pair satisfies (2) - (2.3) . The density of in implies that it also satisfies (2) - (2.3) . Thus indeed is a weak solution. The estimate (2.16) is obtained in a standard way taking and in (2) - (2.3) and then applying the Korn inequality (2.9) and the Poincare inequality in (2.12). ∎
In this section we will show that the weak solution belongs to a class . The idea of the proof has been outlined in the introduction. We start with showing that if is a weak solution then .
Note that on we have .
Let us denote .
Since for , thus for (2) takes the form
For we have
where and denotes the curvature of .
For we have
where is the curvature of .
Since we have , and using the definition of we can write
Integration by parts yields