Fluid dynamics solutions obtained from the Riemann invariant approach.
The generalized method of characteristics is used to obtain rank-2 solutions of the classical equations of hydrodynamics in (3+1) dimensions describing the motion of a fluid medium in the presence of gravitational and Coriolis forces. We determine the necessary and sufficient conditions which guarantee the existence of solutions expressed in terms of Riemann invariants for an inhomogeneous quasilinear system of partial differential equations. The paper contains a detailed exposition of the theory of simple wave solutions and a presentation of the main tool used to study the Cauchy problem. A systematic use is made of the generalized method of characteristics in order to generate several classes of wave solutions written in terms of Riemann invariants.
This work is dedicated to the memory of professor Marek Burnat.
keywords: generalized method of characteristics, Riemann invariants, multiwave solutions, fluid dynamics equations.
Mathematics Subject Classification (2000): 35B06, 35F50, 35F20
I Introduction. Inhomogeneous fluid dynamics system
The compressible flow of an ideal fluid in the presence of gravitational and Coriolis forces is governed by the Euler equations in (3+1) dimensions
which constitute a hyperbolic quasilinear system in four independent variables (namely time and the three space variables and five dependent variables: is the density of the fluid, is the pressure of the fluid, is the vector field of the fluid velocity and is the adiabatic exponent. The ideal fluid flow is subjected to the gravitational force and the Coriolis force in a non-inertial coordinate system.
This system admits two distinct families of characteristics associated with entropic and acoustic waves. These waves play an essential role from the point of view of a physical and mathematical analysis of the initial system (1). It is convenient to choose characteristic coordinates as new independent variables instead of the Euler or Lagrange coordinates (see e.g. [11, 12, 14, 19, 20, 21]). The existence of Riemann invariants which remain constant along these characteristics considerably simplifies the problem of constructing and investigating wave solutions admitted by the system (1). An advantage of the presence of Riemann invariants (1) is that, at least in certain favorable cases, they lead to expressions for which the general integrals are given in closed form. The objective of this paper is to look for certain classes of solutions describing the propagation of waves that satisfy the Euler equations (1). Such classes of solutions are particularly interesting from the physical point of view because they cover a wide range of nonlinear wave phenomena arising in the presence of the external forces that are observed in fluid dynamics. The methodological approach assumed in this work is based on the generalized method of characteristics initiated by M. Burnat [1, 2] and next developed by Z. Peradzynski [15, 16] for homogeneous nonelliptic quasilinear systems. A specific feature of that approach is an algebraization of the partial differential equations (PDEs) under consideration by representing the general integral elements as linear combinations of some special rank-1 elements associated with certain vector fields which generate characteristic curves in the spaces of independent and dependent variables, respectively. The introduction of those rank-1 elements (also called simple elements, see definition 2.1) proved to be a useful tool for constructing solutions in the case of the inhomogeneous Euler equations. These simple integral elements are in one-to-one correspondence with Riemann wave solutions (also called simple waves). By means of the Cartan theory for involutive systems, it was shown in [5, 15] that these elements serve as building blocks for constructing certain classes of solutions expressed in terms of Riemann invariants which can be interpreted as multiple wave superpositions of two or more single Riemann waves [1, 2, 9, 16].
Riemann  demonstrated that simple wave solutions for hydrodynamics-type systems, even with arbitrary smooth initial data, usually cannot be extended indefinitely in time but blows up after a certain finite period. The first derivative of a simple wave solution of (2) becomes unbounded after some finite time, , and for times smooth solutions of the initial Cauchy problem do not exist. Therefore, we deal here with the phenomenon known as the gradient catastrophe. Riemann also investigated the problem of extending the simple wave solutions in some generalized sense beyond the time of the blow up. On the basis of the laws of conservation of mass and momentum, Riemann introduced solutions based on discontinuous functions which can be interpreted as shock waves. He demonstrated a link between the wave front velocity and parameters of the fluid state before and behind that discontinuity. The problem of the propagation and superposition of simple waves has been extensively developed by many authors (e.g. see [9, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] and references therein).
In this paper we concentrate on the simplest case, namely on the propagation of a single simple wave admitted by the inhomogeneous system (1) and show that this leads to several new classes of interesting solutions. We also solve the problem of determining the necessary and sufficient conditions on the initial data for the corresponding Cauchy problem for the inhomogeneous quasilinear system in order that the solution evolve as a Riemann wave. Further, we obtain certain formulas for this type of solution in terms of the initial data. These solutions may in turn be useful in the study of more complex solutions, i.e. nonlinear superposition of simple waves, and in the investigation of the global existence and uniqueness of those solutions as well as the nature of the gradient catastrophe. This topic is much better understood in the case where the Riemann wave problem reduces to the examination of a certain exterior differential system expressed in terms of Riemann invariants, known to be in involution in the sense of Cartan . In this paper, these theoretical considerations are systematically used to generate all nonlinear wave propagations admitted by the fluid dynamical system (1) in (3+1) dimensions. A broad review of recent developments in this subject can be found in books such as A. Jeffrey , A. Madja , Z.Peradsynski , B. Rozdestvenski and Y. Janenko and reference therein.
In what follows we assume that all mappings, tensor fields and manifolds are smooth enough to make our considerations relevant and we use the summation convention unless otherwise stated.
This paper is organized as follows. Section II contains a detailed account of rank-2 solutions expressed in terms of Riemann invariants. These results are used in Section III to formulate and solve the Riemann wave Cauchy problem for inhomogeneous systems. Section IV contains a detailed account of the algebraic properties of the inhomogeneous Euler system (1). In section V, we describe in detail a procedure for constructing solutions of the Euler equations (1) and illustrate this procedure through examples.
Ii Rank-2 solutions
In this section we consider the possibility of adapting the Riemann invariant approach for quasilinear systems of PDEs to the construction of a propagation of a simple wave allowed by an inhomogeneous system. This topic has already been discussed by the authors in [5, 6, 7, 8, 9]. However our present approach goes deeper into the algebraic and geometric aspects which will later enable us to obtain several new classes of exact solutions of the inhomogeneous system of fluid dynamics (1). We will show that these solutions contain arbitrary functions of one variable.
Recall that in order to construct simple waves of the inhomogeneous quasilinear system of partial differential equations (PDE)s in independent variables and unknown functions
we must find two sets of functions , and , satisfying the system of algebraic equations
where we denote by and the spaces of dependent variables and independent variables , respectively. To obtain these functions , and , one proceeds by requiring that the matrix have rank less than and that the matrix satisfy
at some generic point . These conditions obviously provide some algebraic restrictions on the functions and . Suppose that we have obtained the two sets of functions and satisfying the above rank conditions for which we assume a linearly independent set of vectors at each point of some open subset . For each and so obtained, we can solve the system (3) for and , respectively.
A rank-2 solution of the inhomogeneous quasilinear system (2) is said to be a propagation of a simple wave on a simple state if the matrix can be decomposed as
for each point , where denotes the tensor product and is an arbitrary function of .
To show that the matrix has the correct decomposition for all indices and , we multiply on the left by the matrix to obtain
Hence the system (2) holds because of the way that the so-called simple elements and have been constructed. The physical meaning of the linear combination of these simple elements and are quite different [9, 15]. While the homogeneous element is usually attributed to certain waves, which can propagate in the medium, the inhomogeneous element leads to some special solution which will be called a simple state and which in general may not be attributed to a wave. In the literature [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21] solutions of are actually called simples waves. We use the word "wave" for solutions which are interpreted as physical waves. For instance, elliptic inhomogeneous systems also have solutions of rank one, but we cannot call them "waves". Instead, they are called "modes" . This is why we have chosen to call solutions of the system (2) such that "simple states". In this paper, we look for solutions of the form (5), where the matrix of the tangent mapping is the sum of a homogeneous and an inhomogeneous element. Note that a correct choice of the element of the form (5) leaves much freedom and requires us to study the structure of its components as well as the corresponding solution. A physical interpretation of this type of solution may be thought of as an interaction of waves with a medium in some state. The necessary and sufficient conditions for the existence of solutions of (5) were derived in . These conditions guarantee the propagation of a single wave solution described by the inhomogeneous quasilinear system (2). At this point let us summarize our approach for constructing rank-2 solutions of (2) subjected to (5). We make the assumption that we choose a holonomic system for the vector fields by requiring a proper length for each vector such that
This requirement means that there exists a parametrization of a surface immersed in the space of dependent variables
which is obtained by solving the system of PDEs
where and are parameters along the respective integral trajectories of the vector fields and on . Assuming that we have the parametric representation of the surface in the space , we consider the functions , that is, the functions pulled back to the surface . The then become functions of the parameters and on . In order to simplify the notation we denote by . Thus, taking the differential of the expression (8), we get
where and . This system of 1-forms is an involutive system in the sense of Cartan if the conditions 
are satisfied. It was shown in  that the conditions (12) are necessary and sufficient for the existence of a rank-2 solution of (2) describing a propagation of a simple wave on a simple state depending on two arbitrary analytic functions of one variable.
In order to illustrate our method let us consider the case with two independent variables and . After the elimination of the variable in the system (11), we obtain
Now we show that a solution of the system (13) describes the propagation of a single simple wave and we justify the notion "simple wave on a simple state" for this situation. It was proved [4, 14] that if the initial data is sufficiently small, then there exists a time interval in which the gradient catastrophe for the solution of the system (13) does not occur since the function is constant along the characteristic of the system (13). If we choose in the space of independent variables the initial condition for the function in such a way that the derivative has compact support
then for arbitrary time , is contained in the strip between characteristics of the family passing through the ends of the interval In this case the strip containing divides the remaining part of the space into two disjoint regions. In the region the solution of the system (13) is described by the simple state. In this region holds and the solution satisfies equation (13.ii) with . From the compatibility of the equations (13.ii), we obtain
which means that the direction of does not depend on the variable , so it is constant on . Choosing the parametrization of the curve , in such a way that the covector does not depend on the parameter , we can express the solution on region in the form of a simple state, i.e.
where is a constant vector.
For the general case (an arbitrary number of independent variables), it is convenient from the computational point of view to express the integral conditions (12) in explicit form because there exist functions and such that
hold. In order to construct rank-2 simple wave solutions of the inhomogeneous system (2) we consider two separate cases, namely, when the coefficient of and in (17) does not vanish anywhere and when this coefficient is identically equal to zero.
A The case when
Let the function be the solution of the equation . Then we obtain
where we have denoted and . So we can study the system (11) in the form
for which the integrability conditions (17) are automatically satisfied. According to the definition presented in , the result of a propagation of a simple wave on a simple state is strictly nonlinear if for all such that the conditions hold. This situation takes place when we have in the expression (12) and consequently the wave vectors and take the forms (18).
We now show that solutions of the system (19) can be expressed in the implicit form
To simplify the formulae, we use the following notation
Differentiating equations (20), we obtain
or equivalently, written in a matrix form,
So we have
So we have
From these equations we find
This is a consequence of equation (22.a). Finally, we obtain the system of differential equations for the unknown functions and
Note that equation (23.c) is a closure condition for the one-form , that is the integrability condition of the system (17.a) and (17.b) for the function . The general solution of equation (17.c) has the form
where the constant of integration with respect to , depending on , is, for convenience, denoted by . Given the function we can solve the system of equations (23.a) and (23.b). Integrating the closed form , we get
where is a broken line with vertices 0, , . Thus the general solution of the system (23) has the form
So we have the following proposition:
All solutions of the system (19) can be obtained by solving the implicit system of equations
with respect to the variables , . Here is an arbitrary differentiable function of .
B The case when
where is the space of linear forms and is a differentiable function of one variable. By virtue of equation (17.c) for arbitrary functions and we have
So for the quantities , , we take the solutions of the system
(which always exist locally), and obtain
where is a differentiable function of .
In the case when the wave vectors and are given by the expressions (26) and (27). Hence, the superposition is nonlinear since for some such that the wave vectors and satisfy the conditions . Thus, in this case, system (11) has the form
So we have
where is a differentiable function of . We have the following proposition:
The general integral of the inhomogeneous system (28) has the implicit form
where is an arbitrary differentiable function of , and
The above solutions can be generalized to the case of many simple waves in the inhomogeneous system
where , are linearly independent 1-forms. The general integral of the system (30) has the form
where are arbitrary functions of their arguments.
Iii Formulation of the Cauchy problem for the case of the propagation of a simple wave on a simple state
Let us now study an example of the formulation of the Cauchy problem for the Pfaffian system of the form
According to the preceding section, the integrability conditions (17) are automatically satisfied. We are looking for a solution in the form
Then we have
from which we get
So, we have
and we obtain the system of equations
It follows that so . Then , where is an arbitrary differentiable function of . Finally we have
Suppose that, on some curve , the values of the functions are given by . Let us assume that the functions and are invertible. Then, as a parameter of this curve, we can choose the value , i.e.
or the value , i.e.
Then the functions and are the inverses of each other. Moreover, by virtue of the identity we have
Taking equation (35) into account, it follows that
Thus we get the following proposition:
If on some curve the Cauchy conditions and are given for the equation (31) in such a way that:
the functions and are strongly monotonic,
for a vector tangent to the curve does not belong to the annihilator of the forms and ,
then there exists a tubular neighborhood of the curve for which the Cauchy problem of (31) has (locally) exactly one rank-2 solution. This solution represents a simple wave on a simple state.