Abstract

The existence and uniqueness results for the Tricomi problem of Chaplygin’s hodograph equation are shown, in the case that the domain considered is close to the parabolic degenerate line, by adopting the energy integral methods and choosing judiciously suitable multipliers.

1. Introduction

In this paper we consider the Tricomi problem of the following second-order linear partial differential equation. Considerwhereand is a constant. Equation (1) is of elliptic type for and , hyperbolic type for and , and parabolic degenerate on the line . We are interested in this equation because it is actually an equivalent form of Chaplygin’s hodograph equation (with as the unknown). (In this paper we will use the subscripts like and to denote the partial derivatives and )where the function (called sonic speed in gas dynamics) is given by the Bernoulli law [1, page 23] with being a positive constant, , and the adiabatic exponent for polytropic gas. One can easily show that, by taking and , (3) is transformed to (1), with (cf. [2, page 72]).

The significance of Chaplygin’s hodograph equation (3) lies in the fact that it is the hodograph transform of the following compressible Euler equations of isentropic irrotational flows:where is the density of mass of the flow and is the velocity of the flow along the coordinates of the Euclidean plane. Since in this case the sonic speed , then is a function of given by the Bernoulli law. Some fundamental problems in gas dynamics, such as detached shocks in supersonic flow past blunt bodies and subsonic jets (cf. [1, 3]), could be considered more favorably by using hodograph equation (3) (or (1)) rather than Euler equations (5), because the latter are generally a quasi-linear mixed elliptic-hyperbolic system, which is still far beyond the ability of present-day analytical tools to study.

In a previous work [2], the authors have studied a mixed boundary value problem of (3) in the sonic circle , with an artificial Dirichlet boundary condition on part of the sonic line , to understand the regularity and behavior of solutions of (3) in the elliptic region and near the degenerate line. Now, we continue our project in this paper to investigate the Tricomi problem of (1), that is, to find a function satisfying (in certain sense to be specified later) (1) in a planar domain which is simply connected, containing a segment of the -axis, and bounded by the characteristic curves (by definition, a characteristic curve of (1) satisfies equation for ) and lying in the lower half plane and a Jordan curve lying in the upper half plane , with Dirichlet conditions on and (see Figure 1). Here emanates from the origin and intersects the horizontal line at a point , where is sufficiently small. The characteristic curve emanates from the point and intersects the -axis at a point . The arc has two endpoints and . The Dirichlet conditions on and are, respectively,where is the arc-length parameter of the boundary curve so that the point moves counterclockwise on as increases. Then the outward unit normal along is given by Note that one can require to be piecewise smooth except at the point , where at best the curve is . (We thank a referee for pointing out this fact. Here as usual we use to denote the Hölder space of -times continuously differential real-valued functions on whose th order derivatives are all Hölder continuous with the exponent .) Let be a given function in the standard Sobolev space . The functions and are the traces of on and , respectively. Then it is obvious that their union belongs to

Figure 1 

The domain and its boundary in the formulations of the Tricomi problem.

It is well known that the Tricomi problem was firstly proposed and studied by Tricomi in [4] for the now so-called Tricomi equation , by using singular integral equations and the matching technique. Tricomi’s study of this problem was mainly motivated to understand second-order mixed elliptic-hyperbolic type equations from a purely mathematical point of view. Later it was discovered that the Tricomi equation may be considered in certain sense as a simple approximation near the sonic line of Chaplygin’s hodograph equation in transonic aerodynamics (cf. [5]) and the Tricomi problem is physically relevant to determining some flow field in transonic flows, such as the detached bow shock and the mixed subsonic-supersonic flow ahead of a blunt body [6]. More general linear mixed elliptic-hyperbolic equations and more general formulations of boundary conditions (such as generalized Tricomi problem, Frankl problem, and generalized Frankl problem) were also considered. For example, Morawetz [7] proved the uniqueness for smooth solutions using Noether’s theorem on conservation laws for the equation Rassias [8] studied weak solutions for the equation Osher [9] showed the existence for Lavrentiev-Bitsadze equation . Aziz and Schneider [10] investigated the existence of weak solutions of the Gellerstedt problem and the Gellerstedt-Neumann problems for the equation Lupo et al. [11] proved existence of weak solutions for Tricomi problem with closed Dirichlet boundary conditions. Lupo et al. [12] considered the existence, uniqueness, and qualitative properties of weak solutions to the degenerate hyperbolic Goursat problem on characteristic triangles for linear and semilinear equations of Tricomi type. See also, for example, [13–16] for works on the nonlinear Tricomi problems. We recommend the introduction in the monograph [17] for a review of the status of mixed-type equations around the 1970s. Morawetz [18] also reviewed the existence and uniqueness theorems for mixed-type equations and their applications to transonic flows, and Chen [19] introduced more recent progress.

However, to the best of our knowledge, there is not any result on the Tricomi problem of Chaplygin’s hodograph equation (1), which is relevant to many physical problems in transonic aerodynamics. So we will devote this work to establishing some basic properties of such problems. The main result is the following theorem.

Theorem 1. There is a positive constant determined only by such that if the domain is contained in the strip , then the Tricomi problem (1) and (8) has a quasi-regular distributional solution. Furthermore, the solution is unique in .

For the definition of quasi-regular distributional solution, see Definition 2. The constant is given in (77).

Our proof depends on the classical energy methods, or the method of multipliers (see [8, 20, 21]). Besides the method of singular integral equations, these seem to be the only general way to study well-posedness of mixed-type equations. (However, see also [22] for regularity of solutions of Tricomi equation by using the methods from harmonic analysis.) Although the idea of energy method is rather simple, it is usually very technical to choose appropriate multipliers to a physically interesting equation, like (1), as shown in this paper.

We remark that there is another type of mixed elliptic-hyperbolic equations, firstly studied by Maria Cinquini-Cibrario, now called Keldysh type, whose canonical form is (see [23, page 11]). An up-to-date review of studies of Keldysh-type mixed equations was presented in [24]. It is possible now to study directly many boundary value problems of quasi-linear Keldysh-type equations; for example, see [5, 25, 26] for studies of steady continuous subsonic-supersonic flows in (approximate) de Laval nozzles.

The rest of the paper is organized as follows. In Section 2 we will define a quasi-regular distributional solution to our Tricomi problem and show that it satisfies the equation and boundary conditions in the ordinary sense if it is a classical solution. We will establish the uniqueness of classical solutions in Section 3. Finally, in Section 4, the existence of a quasi-regular distributional solution is proved by the dual method in functional analysis.

2. Definition of Solutions

Denote the linear operator by with It is a formal dual operator of .

Definition 2. Let and and for some . A function is a quasi-regular distributional solution of the equationsubjected to boundary conditions (8), iffor all .

Now we show that a quasi-regular distributional solution satisfies (1) and boundary conditions (8) in the classical pointwise sense if it belongs to . In fact, using integration by parts and (15), we get, for all , thatChoosing particularly that , all the three boundary integrals vanish, and (16) is reduced to Since is dense in , we getand hence almost everywhere in

Next, by employing (16) and (18), we have, for all , Since , it follows that Therefore, for all . Taking that vanishes in a neighborhood of and , we infer that , and furthermore, for all , there holdsRecall that , and we have Hence we get on , orwhere is a normalizing factor. Thus, we have by using (24). Since on and is a characteristic curve, then and . Thus, we haveby using (22). Since is an arbitrary function, we see .

3. Uniqueness of Classical Solutions

Assume that are two solutions of Tricomi problem (1) and (8), and takeThen solvesWe will show that in .

Set where , , and are sufficiently smooth functions to be determined (cf. Remark 4). Since we obtain that The goal is to show that all integrals , , , and are nonnegative by choosing suitable functions , , and .

One observes that the integral ifand the integral if the following conditions hold in :

By using (28), we haveSince on , it follows that Then Recall that and , we haveWe also note that and henceTherefore, we getby using (37) and (39). It follows thatby using (34). Since and on , we have Thus So by using (43) provided that