780 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS The functions w 1 , , w n–1 are thus assumed to be expressible in terms of w n .From (15.14.3.24), in view of (15.14.3.25), we find the following system of n – 1 ordinary differential equations: dw k dw n = F k (w 1 , , w n ) F n (w 1 , , w n ) , k = 1, , n – 1.(15.14.3.26) Further, assuming that a solution of system (15.14.3.26) has been obtained and the functions (15.14.3.25) are known, we substitute them into (15.14.3.22)–(15.14.3.23) to arrive at the system of two equations ∂ ∂t u + g(w n ) + ∂u ∂x = 0, ∂w n ∂t = f n (w n )u, (15.14.3.27) where g(w n )=G w 1 (w n ), , w n–1 (w n ), w n , f n (w n )=F n w 1 (w n ), , w n–1 (w n ), w n . Introducing the new dependent variable w = g(w n ) ≡ G w 1 (w n ), , w n–1 (w n ), w n in (15.14.3.27), we obtain system (15.14.3.11) in which the function f = f (w)isdefined parametrically, f = g (w n )f n (w n ), w = g(w n ), with w n being the parameter. Remark 1. Solutions of the above special form arise, for example, in problems with initial and boundary conditions, w 1 = ···= w n = u = 0 at t = 0, u = 1 at x = 0. (The initial conditions correspond to the absence of particle in the reservoir at the initial time and the boundary condition corresponds to a given suspended concentration in the injected fluid.) In this case, the system ordinary differential equations (15.14.3.26) is solved under the following initial conditions: w 1 = ···= w n–1 = 0 at w n = 0. Remark 2. In problems on flows of n-component fluids through porous media, the mass balance for retained and suspended particles is governed by equation (15.14.3.22) in which the function G is the sum of individual components: G(w 1 , , w n )= n k=1 w k ; equations (15.14.3.23) define the capture kinetics. 15.14.4. First-Order Hyperbolic Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type ∗ 15.14.4-1. Systems of two equations. Systems in the form of conservation laws. 1 ◦ . Consider the system of two quasilinear equations of the form f 1 (u, w) ∂u ∂t + g 1 (u, w) ∂w ∂t + h 1 (u, w) ∂u ∂x + k 1 (u, w) ∂w ∂x = 0, f 2 (u, w) ∂u ∂t + g 2 (u, w) ∂w ∂t + h 2 (u, w) ∂u ∂x + k 2 (u, w) ∂w ∂x = 0, (15.14.4.1) * Subsection 15.14.4 was written by A. P. Pires and P. G. Bedrikovetsky. 15.14. NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 781 where x and t are independent variables, u = u(x, t)andw = w(x, t) are two unknown functions, and f k (u, w)andg k (u, w) are prescribed functions (k = 1, 2, 3, 4). For any f k (u, w)andg k (u, w), system (15.14.4.1) admits the following simplest solu- tions: u = C 1 , w = C 2 ,(15.14.4.2) where C 1 and C 2 are arbitrary constants. 2 ◦ . The main mathematical models in continuum mechanics and theoretical physics have the form of systems of conservation laws. Usually mass, momentum, and energy for phases and/or components are conserved. A quasilinear system of two conservation laws in two independent variables has the form ∂F 1 (u, w) ∂t + ∂G 1 (u, w) ∂x = 0, ∂F 2 (u, w) ∂t + ∂G 2 (u, w) ∂x = 0. (15.14.4.3) It is a special case of system (15.14.4.1). It admits simple solutions of the form (15.14.4.2). 15.14.4-2. Self-similar continuous solutions. Hyperbolic systems. System (15.14.4.1) is invariant under the transformation (x, t) → (ax, at), where a is any number (a ≠ 0). Therefore, it admits a self-similar solution of the form u = u(ξ), w = w(ξ), ξ = x t .(15.14.4.4) The substitution of (15.14.4.4) into (15.14.4.1) results in the following system of ordi- nary differential equations: (h 1 – ξf 1 )u ξ +(k 1 – ξg 1 )w ξ = 0, (h 2 – ξf 2 )u ξ +(k 2 – ξg 2 )w ξ = 0, (15.14.4.5) where the arguments of the functions f m , g m , h m ,andk m (m = 1, 2) are omitted for brevity. System (15.14.4.5) may be treated as an algebraic system for the derivatives u ξ and w ξ ; for it to have a nonzero solution, its determinant must be equal to zero: (f 1 g 2 – f 2 g 1 )ξ 2 –(f 1 k 2 + g 2 h 1 – f 2 k 1 – g 1 h 2 )ξ + h 1 k 2 – h 2 k 1 = 0.(15.14.4.6) System (15.14.4.1) is called strictly hyperbolic (or, for short, hyperbolic) if the discrim- inant of the quadratic equation (15.14.4.6), with respect to ξ, is positive: (f 1 k 2 + g 2 h 1 – f 2 k 1 – g 1 h 2 ) 2 – 4(f 1 g 2 – f 2 g 1 )(h 1 k 2 – h 2 k 1 )>0.(15.14.4.7) In this case, equation (15.14.4.6) has two different roots: ξ 1,2 = ξ 1,2 (u, w). (15.14.4.8) Substituting either root (15.14.4.8) into (15.14.4.5) and taking u to be a parameter along the integral curve, we obtain (k 1 – ξ 1,2 g 1 ) dw du + h 1 – ξ 1,2 f 1 = 0.(15.14.4.9) 782 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS The ordinary differential equation (15.14.4.9) provides two sets of continuous solutions for system (15.14.4.1). The procedure for constructing these solutions consists of two steps: (i) the dependence w = w(u), different for each set, is found from (15.14.4.9); and (ii) this dependence is then substituted into formula (15.14.4.8), in the left-hand side of which ξ 1,2 is replaced by x/t. This results in a solution defined in implicit form: ξ 1,2 (u, w)=x/t, w = w(u). Example 1. The system of equations describing one-dimensional longitudinal oscillations of an elastic bar consists of the equations of balance of mass and momentum: ∂u ∂t – ∂w ∂x = 0, ∂w ∂t – ∂σ(u) ∂x = 0. (15.14.4.10) Here, u is the deformation gradient (strain), v is the strain rate, and σ(u) is the stress. Self-similar solutions are sought in the form (15.14.4.4). This results in the system of ordinary differential equations ξu ξ + w ξ = 0, σ u (u)u ξ + ξw ξ = 0. (15.14.4.11) Equating the determinant of this system to zero, we obtain the quadratic equation ξ 2 = σ u (u), whose roots are ξ 1,2 = σ u (u). (15.14.4.12) The system is assumed to be hyperbolic, which implies that σ u (u)>0. Substituting (15.14.4.12) into (15.14.4.11) yields a separable first-order ordinary differential equation: w u σ u (u)=0. Integrating gives its general solution w = C σ u (u) du,(15.14.4.13) where C is an arbitrary constant (two different constants, corresponding to the plus and minus sign). Formula (15.14.4.13) together with σ u (u)=x/t (15.14.4.14) defines two one-parameter self-similar solutions of system (15.14.4.10) of the form (15.14.4.4) in implicit form. 15.14.4-3. Simple Riemann waves. In the case of self-similar solutions of the form (15.14.4.4), one of the unknowns can be expressed in terms of the other by eliminating the independent variable ξ to obtain, for example, w = w(u). (15.14.4.15) Assuming initially that the unknowns are functionally related via (15.14.4.15), one can obtain a wider class of exact solutions to system (15.14.4.1) than the class of self-similar solutions. Indeed, substituting (15.14.4.15) into (15.14.4.1) gives two equations for one function u = u(x, t): (f 1 + g 1 w u ) ∂u ∂t +(h 1 + k 1 w u ) ∂u ∂x = 0, (f 2 + g 2 w u ) ∂u ∂t +(h 2 + k 2 w u ) ∂u ∂x = 0. (15.14.4.16) 15.14. NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 783 The dependence (15.14.4.15) is chosen so that the two equations of (15.14.4.16) coincide. This condition results in the following first-order ordinary differential equation for w = w(u) quadratically nonlinear in the derivative: (g 1 k 2 – g 2 k 1 )(w u ) 2 +(f 1 k 2 + g 1 h 2 – f 2 k 1 – g 2 h 1 )w u + f 1 h 2 – f 2 h 1 = 0.(15.14.4.17) Treating (15.14.4.17) as a quadratic equation in w u , we assume its discriminant to be positive: (f 1 k 2 + g 1 h 2 – f 2 k 1 – g 2 h 1 ) 2 – 4(f 1 h 2 – f 2 h 1 )(g 1 k 2 – g 2 k 1 )>0. This condition is equivalent to condition (15.14.4.7) and implies hat the system in question, (15.14.4.1), is strictly hyperbolic. In this case, equation (15.14.4.17) has two different real roots and is reducible to first-order ordinary differential equations of the standard form w u = Λ m (u, w), m = 1, 2.(15.14.4.18) Having found a solution w = w(u) of this equation for any m, we substitute it into the first (or the second) equation (15.14.4.16). As a result, we obtain a quasilinear first-order partial differential equation for u(x, t): (f 1 + g 1 Λ m ) ∂u ∂t +(h 1 + k 1 Λ m ) ∂u ∂x = 0, w = w(u). (15.14.4.19) The general solution of this equation can be obtained by the method of characteristics (see Subsection 13.1.1). This solution depends on one arbitrary function and is called a simple Riemann wave. To each of the two equations (15.14.4.18) there corresponds an equation (15.14.4.19) and a Riemann wave. Suppose equations (15.14.4.18) have the integrals R m (u, w)=C m , m = 1, 2, where C 1 and C 2 are arbitrary constants. The functions R m (u, w) are called Riemann invariants. Another definition of Riemann functions is given in Paragraph 15.14.4-6. Example 2. We look for exact solutions of system (15.14.4.10) of the special form (15.14.4.15), implying that one of the unknown functions can be expressed via the other. Substituting (15.14.4.15) into (15.14.4.10) gives ∂u ∂t – w u ∂u ∂x = 0, w u ∂u ∂t – σ u (u) ∂u ∂x = 0.(15.14.4.20) Requiring that the two equations coincide, we get the condition (w u ) 2 = σ u (u). It follows that there are two possible relations between the unknowns: w σ u (u) du = C m , m = 1, 2.(15.14.4.21) The left-hand sides of these relations represent Riemann functions. On eliminating w from (15.14.4.20), using (15.14.4.21), we arrive at the quasilinear first-order partial differential equation ∂u ∂t σ u (u) ∂u ∂x = 0.(15.14.4.22) A detailed analysis of this equation is given in Subsection 13.1.3; the general solution is found by the method of characteristics and can be represented in implicit form: x t σ u (u)=Φ(u), (15.14.4.23) where Φ(u) is an arbitrary function. 784 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Formulas (15.14.4.21) and (15.14.4.23) define two exact solutions of system (15.14.4.10), each containing one arbitrary function and one arbitrary constant. In the special case Φ(u) ≡ 0, this solution turns into the self-similar solution considered above. In the case of a linear system, with σ(u)=a 2 u, formulas (15.14.4.23) give two traveling-wave solutions u = Ψ 1,2 (x at), which represent waves propagating in opposite directions and having an arbitrary undistorted profile. The evolution of a Riemann wave with the initial profile u = ϕ(x)att = 0 (–∞ < x < ∞)(15.14.4.24) for the lower sign in equation (15.14.4.22) is described parametrically by the formulas x = ξ + F(ξ)t, u = ϕ(ξ), (15.14.4.25) where F(ξ)= √ σ u (u) u=ϕ(ξ) . The second unknown, w, is expressed in terms of u by formula (15.14.4.21). Example 3. A one-dimensional ideal adiabatic (isentropic) gas flow is governed by the system of two equations ∂ρ ∂t + ∂(ρv) ∂x = 0, ∂(ρv) ∂t + ∂[ρv 2 + p(ρ)] ∂x = 0. (15.14.4.26) Here, ρ = ρ(x, t) is the density, v = v(x, t) is the velocity, and p is the pressure. The first equation (15.14.4.26) represents the law of conservation of mass in fluid mechanics and is referred to as a continuity equation. The second equation (15.14.4.26) represents the law of conservation of momentum. The equation of state is given in the form p = p(ρ). For an ideal polytropic gas, p = Aρ γ , where the constant γ is the adiabatic exponent. Exact solutions of system (15.14.4.26) of the form v = v(ρ) are given by the formulas v = p ρ (ρ) dρ ρ + C m , x t p ρ (ρ) dρ ρ + p ρ (ρ)+C m = Φ(ρ), where Φ(ρ) is an arbitrary function and the C m are arbitrary constants (different solutions have the different constants, m = 1, 2). The Riemann functions are obtained from the first relation by expressing C m via ρ and v. For an ideal polytropic gas, which corresponds to p = Aρ γ , the exact solution of system (15.14.4.26) with γ ≠ 1 becomes v = 2 γ – 1 Aγ ρ γ–1 2 + C m , x t γ + 1 γ – 1 Aγ ρ γ–1 2 + C m = Φ(ρ). Remark. System (15.14.4.26) with ρ = h and p(ρ)= 1 2 gh 2 ,wherev is the horizontal velocity averaged over the height h of the water level and g is the acceleration due to gravity, governs the dynamics of shallow water. 15.14.4-4. Linearization of gas dynamic systems by the hodograph transformation. The hodograph transformation is used in gas dynamics and the theory of jets for the linearization of equations and finding solutions of certain boundary value problems. For a quasilinear system of two equations (15.14.4.1), the hodograph transformation has the form x = x(u, w), t = t(u, w); (15.14.4.27) this means that u, w are now treated as the independent variables and x, t as the dependent variables. 15.14. NONLINEAR SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS 785 Differentiating each relation in (15.14.4.27) with respect to x and t (as composite functions) and eliminating the partial derivatives u t , w t , u x , w x from the resulting relations, we obtain ∂u ∂t =–J ∂x ∂w , ∂w ∂t = J ∂x ∂u , ∂u ∂x = J ∂t ∂w , ∂w ∂x =–J ∂t ∂u ,(15.14.4.28) where J = ∂u ∂x ∂w ∂t – ∂u ∂t ∂w ∂x is the Jacobian of the functions u = u(x, t)andw = w(x, t). Replacing the derivatives in (15.14.4.1) with the help of relations (15.14.4.28) and dividing by J, we arrive at the linear system g 1 (u, w) ∂x ∂u – k 1 (u, w) ∂t ∂u – f 1 (u, w) ∂x ∂w + h 1 (u, w) ∂t ∂w = 0, g 2 (u, w) ∂x ∂u – k 2 (u, w) ∂t ∂u – f 2 (u, w) ∂x ∂w + h 2 (u, w) ∂t ∂w = 0. (15.14.4.29) Remark. The hodograph transformation (15.14.4.27) is unusable if J ≡ 0. In this degenerate case, the quantities u and w are functionally related, and hence they cannot be taken as independent variables. In this case, relation (15.14.4.15) holds; it determines simple Riemann waves. This is why in using the hodograph transformation (15.14.4.27), one loses solutions corresponding to simple Riemann waves. 15.14.4-5. Cauchy and Riemann problems. Qualitative features of solutions. Cauchy problem (t ≥ 0,–∞ < x < ∞). Find functions u = u(x, t), w = w(x, t) that solve system (15.14.4.1) for t > 0 and satisfy the initial conditions u(x, 0)=ϕ 1 (x), w(x, 0)=ϕ 2 (x), (15.14.4.30) where ϕ 1 (x)andϕ 2 (x) are prescribed functions. The Cauchy problem is also often referred to as an initial value problem. Riemann problem (t ≥ 0,–∞ < x < ∞). Find functions u = u(x, t), w = w(x, t)that solve system (15.14.4.1) for t > 0 and satisfy piecewise-smooth initial conditions of the special form u(x, 0)= u L if x < 0, u R if x > 0, w(x, 0)= w L if x < 0, w R if x > 0. (15.14.4.31) Here, u L , u R , w L ,andw R are prescribed constant quantities. Since equations (15.14.4.1) and the boundary conditions (15.14.4.31) do not change under the transformation (x, t) → (ax, at), where a is any positive number, the solution of the Riemann problem (15.14.4.1), (15.14.4.31) is self-similar. The unknown functions are sought in the form (15.14.4.4) and satisfy the ordinary differential equations (15.14.4.5). When passing to the self-similar variable ξ = x/t, the initial conditions (15.14.4.31) are transformed into boundary conditions of the form u → u L , w → w L as ξ → –∞; u → u R , w → w R as ξ →∞. Solutions of the Riemann problem (15.14.4.1), (15.14.4.31) can be either continuous (such solutions are called rarefaction waves) or discontinuous (in this case, they describe shock waves). In both cases, the solution for each of the unknowns (e.g., for u) consists of two types of segments: those where the solution is constant, u = const, and those where it smoothly changes according to the law u = u(x/t). The endpoints of these segments 786 NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS lie on straight lines x = a n t in the plane (x, t). At these points, individual segments of solutions either join together (for continuous solutions) or agree with one another based on conservation laws (for discontinuous solutions); for details see Paragraphs 15.14.4-8 and 15.14.4-10. Let us explain why shock waves, corresponding to discontinuous solutions, arise by an example. Consider the system of equations for one-dimensional longitudinal oscillations of an elastic bar (15.14.4.10). It was shown above that this system admits simple Rie- mann waves, which are described by the quasilinear first-order partial differential equation (15.14.4.22). This equation coincides, up to notation, with the model equation of gas dy- namics (13.1.3.1), which was studied in detail in Subsection 13.1.3. If the function F(ξ) in solution (15.14.4.25) is nonmonotonic (this function is determined by the initial profile of the wave), then the characteristic lines defined by the first formula in (15.14.4.25) will intersect in the (x, t) plane for various ξ. This results in nonuniqueness in determining the function u = u(x, t), which contradicts physical laws. Therefore, in order to avoid nonuniqueness of physical quantities, discontinuous solutions have to be considered. 15.14.4-6. Reduction of systems to the canonical form. Riemann invariants. 1 ◦ . Eliminating the derivative w t and then the derivative u t from (15.14.4.1), we arrive after simple rearrangements to the canonical form of a gas dynamics system: ∂u ∂t + p 1 (u, w) ∂u ∂x + q 1 (u, w) ∂w ∂x = 0, ∂w ∂t + p 2 (u, w) ∂u ∂x + q 2 (u, w) ∂w ∂x = 0, (15.14.4.32) where p 1 = h 1 g 2 – g 1 h 2 f 1 g 2 – g 1 f 2 , q 1 = k 1 g 2 – g 1 k 2 f 1 g 2 – g 1 f 2 , p 2 = f 1 h 2 – h 1 f 2 f 1 g 2 – g 1 f 2 , q 2 = f 1 k 2 – k 1 f 2 f 1 g 2 – g 1 f 2 . The functions f m , g m , h m , k m , p m ,andq m (m = 1, 2) depend on u and w; it is assumed that f 1 g 2 – g 1 f 2 0. 2 ◦ . For further simplifications of system (15.14.4.32), we proceed as follows. Multiply the first equation by b 1 = b 1 (u, w) and the second by b 2 = b 2 (u, w) and add up to obtain b 1 ∂u ∂t + b 2 ∂w ∂t +(b 1 p 1 + b 2 p 2 ) ∂u ∂x +(b 1 q 1 + b 2 q 2 ) ∂w ∂x = 0.(15.14.4.33) Impose the following constraints of the functions b 1 and b 2 : b 1 p 1 + b 2 p 2 = λb 1 , b 1 q 1 + b 2 q 2 = λb 2 . (15.14.4.34) These constraints represent a linear homogeneous algebraic system of equations for b m . For this system to have nontrivial solutions, its determinant must be zero. This results in a quadratic equation for the eigenvalues λ = λ(u, w): λ 2 –(p 1 + q 2 )λ + p 1 q 2 – p 2 q 1 = 0.(15.14.4.35) . Linearization of gas dynamic systems by the hodograph transformation. The hodograph transformation is used in gas dynamics and the theory of jets for the linearization of equations and finding solutions of. Systems of Quasilinear Equations. Systems of Conservation Laws of Gas Dynamic Type ∗ 15.14.4-1. Systems of two equations. Systems in the form of conservation laws. 1 ◦ . Consider the system of two. C 1 and C 2 are arbitrary constants. 2 ◦ . The main mathematical models in continuum mechanics and theoretical physics have the form of systems of conservation laws. Usually mass, momentum, and