556 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 13.1.2. Cauchy Problem. Existence and Uniqueness Theorem 13.1.2-1. Cauchy problem. Consider two formulations of the Cauchy problem. 1 ◦ . Generalized Cauchy problem. Find a solution w = w(x, y) of equation (13.1.1.1) satisfying the initial conditions x = h 1 (ξ), y = h 2 (ξ), w = h 3 (ξ), (13.1.2.1) where ξ is a parameter (α ≤ ξ ≤ β)andtheh k (ξ) are given functions. Geometric interpretation: find an integral surface of equation (13.1.1.1) passing through the line defined parametrically by equation (13.1.2.1). 2 ◦ . Classical Cauchy problem. Find a solution w = w(x, y) of equation (13.1.1.1) satisfying the initial condition w = ϕ(y)atx = 0,(13.1.2.2) where ϕ(y) is a given function. It is convenient to represent the classical Cauchy problem as a generalized Cauchy problem by rewriting condition (13.1.2.2) in the parametric form x = 0, y = ξ, w = ϕ(ξ). (13.1.2.3) 13.1.2-2. Procedure of solving the Cauchy problem. The procedure of solving the Cauchy problem (13.1.1.1), (13.1.2.1) involves several steps. First, two independent integrals (13.1.1.2) of the characteristic system (13.1.1.3) are deter- mined. Then, to fi nd the constants of integration C 1 and C 2 , the initial data (13.1.2.1) must be substituted into the integrals (13.1.1.2) to obtain u 1 h 1 (ξ), h 2 (ξ), h 3 (ξ) = C 1 , u 2 h 1 (ξ), h 2 (ξ), h 3 (ξ) = C 2 .(13.1.2.4) Eliminating C 1 and C 2 from (13.1.1.2) and (13.1.2.4) yields u 1 (x, y, w)=u 1 h 1 (ξ), h 2 (ξ), h 3 (ξ) , u 2 (x, y, w)=u 2 h 1 (ξ), h 2 (ξ), h 3 (ξ) . (13.1.2.5) Formulas (13.1.2.5) are a parametric form of the solution of the Cauchy problem (13.1.1.1), (13.1.2.1). In some cases, one may succeed in eliminating the parameter ξ from relations (13.1.2.5), thus obtaining the solution in an explicit form. Example 1. Consider the Cauchy problem for linear equation ∂w ∂x + a ∂w ∂y = bw (13.1.2.6) subjected to the initial condition (13.1.2.2). The corresponding characteristic system for equation (13.1.2.6), dx 1 = dy a = dw bw , has two independent integrals y – ax = C 1 , we –bx = C 2 .(13.1.2.7) Represent the initial condition (13.1.2.2) in parametric form (13.1.2.3) andthensubstitutethe data (13.1.2.3) into the integrals (13.1.2.7). As a result, for the constants of integration we obtain C 1 = ξ and C 2 = ϕ(ξ). Sub- stituting these expressions into (13.1.2.7), we arrive at the solution of the Cauchy problem (13.1.2.6), (13.1.2.2) in parametric form: y – ax = ξ, we –bx = ϕ(ξ). By eliminating the parameter ξ from these relations, we obtain the solution of the Cauchy problem (13.1.2.6), (13.1.2.2) in explicit form: w = e bx ϕ(y – ax). 13.1. LINEAR AND QUASILINEAR EQUATIONS 557 13.1.2-3. Existence and uniqueness theorem. Let G 0 be a domain in the xy-plane and let G be a cylindrical domain of the xyw-space obtained from G 0 by adding the coordinate w, with the condition |w| < A 1 being satisfied. Let the coefficients f, g,andh of equation (13.1.1.1) be continuously differentiable functions of x, y,andw in G and let x = h 1 (ξ), y = h 2 (ξ), and w = h 3 (ξ) be continuously differentiable functions of ξ for |ξ| < A 2 defining a curve C in G with a simple projection C 0 onto G 0 . Suppose that (h 1 ) 2 +(h 2 ) 2 ≠ 0 (the prime stands for the derivative with respect to ξ)and fh 2 – gh 1 ≠ 0 on C. Then there exists a subdomain G 0 ⊂ G 0 containing C 0 where there exists a continuously differentiable function w = w(x, y) satisfying the differential equation (13.1.1.1) in G 0 and the initial condition (13.1.2.1) on C 0 . This function is unique. It is important to note that this theorem has a local character, i.e., the existence of a solution is guaranteed in some “sufficiently narrow,” unknown neighborhood of the line C (see the remark at the end of Example 2). Example 2. Consider the Cauchy problem for Hopf’s equation ∂w ∂x + w ∂w ∂y = 0 (13.1.2.8) subject to the initial condition (13.1.2.2). First, we rewrite the initial condition (13.1.2.2) in the parametric form (13.1.2.3). Solving the characteristic system dx 1 = dy w = dw 0 ,(13.1.2.9) we find two independent integrals, w = C 1 , y – wx = C 2 . (13.1.2.10) Using the initial conditions (13.1.2.3), we find that C 1 = ϕ(ξ)andC 2 = ξ. Substituting these expressions into (13.1.2.10) yields the solution of the Cauchy problem (13.1.2.8), (13.1.2.2) in the parametric form w = ϕ(ξ), (13.1.2.11) y = ξ + ϕ(ξ)x. (13.1.2.12) The characteristics (13.1.2.12) are straight lines in the xy-plane with slope ϕ(ξ) that intersect the y-axis at the points ξ. On each characteristic, the function w has the same value equal to ϕ(ξ) (generally, w takes different values on different characteristics). For ϕ (ξ)>0, different characteristics do not intersect and, hence, formulas (13.1.2.11) and (13.1.2.12) define a unique solution. As an example, we consider the initial profile ϕ(ξ)= ⎧ ⎨ ⎩ w 1 for ξ ≤ 0, w 2 ξ 2 + εw 1 ξ 2 + ε for ξ > 0, (13.1.2.13) where w 1 < w 2 and ε > 0. Formulas (13.1.2.11)–(13.1.2.13) give a unique smooth solution in the entire half-plane x > 0. In the domain filled by the characteristics y = ξ + w 1 x (for ξ ≤ 0), the solution is constant, i.e., w = w 1 for y/x ≤ w 1 . (13.1.2.14) For ξ > 0, the solution is determined by relations (13.1.2.11)–(13.1.2.13). Let us look how this solution is transformed in the limit case ε → 0, which corresponds to the piecewise- continuous initial profile ϕ(ξ)= w 1 for ξ ≤ 0, w 2 for ξ > 0, where w 1 < w 2 . (13.1.2.15) We further assume that ξ > 0 [for ξ ≤ 0, formula (13.1.2.14) is valid]. If ξ = const ≠ 0 and ε → 0, it follows from (13.1.2.13) that ϕ(ξ)=w 2 . Hence, in the domain filled by the characteristics y = ξ + w 2 x (for ξ > 0), the solution is constant, i.e., we have w = w 2 for y/x ≥ w 2 (as ε → 0). (13.1.2.16) For ξ → 0, the function ϕ can assume any value between w 1 and w 2 depending on the ratio of the small parameters ε and ξ;thefirst term on the right-hand side of equation (13.1.2.12) can be neglected. As a result, 558 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS we find from equations (13.1.2.11) and (13.1.2.12) that the solution has the following asymptotic behavior in explicit form: w = y/x for w 1 ≤ y/x ≤ w 2 (as ε → 0). (13.1.2.17) By combining relations (13.1.2.14), (13.1.2.16), and (13.1.2.17) together, we obtain the solution of the Cauchy problem for equation (13.1.2.8) subject to the initial conditions (13.1.2.15) in the form w(x, y)= w 1 for y ≤ w 1 x, y/x for w 1 x ≤ y ≤ w 2 x, w 2 for y ≥ w 2 x. (13.1.2.18) Figure 13.1 shows characteristics of equation (13.1.2.8) that satisfy condition (13.1.2.15) with w 1 = 1 2 and w 2 = 2.Thisfigure also depicts the dependence of w on y (for x = x 0 = 1). In applications, such a solution is referred to as a centered rarefaction wave (see also Subsection 13.1.3). x y w y w 2 w 1 y 2 y 1 x 0 0 0 ywx 220 = ywx 110 = Figure 13.1. Characteristics of the Cauchy problem (13.1.2.8), (13.1.2.2) with the initial profile (13.1.2.15) and the dependence of the unknown w on the coordinate y for w 1 = 1 2 , w 2 = 2,andx 0 = 1. Remark. If there is an interval where ϕ (ξ)<0, then the characteristics intersect in some domain. According to equation (13.1.2.11), at the point of intersection of two characteristics defined by two distinct values ξ 1 and ξ 2 of the parameter, the function w takes two distinct values equal to ϕ(ξ 1 )andϕ(ξ 2 ), respectively. Therefore, the solution is not unique in the domain of intersecting characteristics. This example illustrates the local character of the existence and uniqueness theorem. These issues are discussed in Subsections 13.1.3 and13.1.4inmoredetail. 13.1.3. Qualitative Features and Discontinuous Solutions of Quasilinear Equations 13.1.3-1. Model equation of gas dynamics. Consider a quasilinear equation of the special form* ∂w ∂x + f(w) ∂w ∂y = 0,(13.1.3.1) * Equations of the general form are discussed in Subsection 13.1.4. 13.1. LINEAR AND QUASILINEAR EQUATIONS 559 which represents a conservation law of mass (or another quantity) and is often encountered in continuum mechanics, gas dynamics, hydrodynamics, wave theory, acoustics, multiphase flows, and chemical engineering. This equation is a model for numerous processes of mass transfer: sorption and chromatography, two-phase fl ows in porous media, flow of water in river, street traffic development, flow of liquid films along inclined surfaces, etc. The independent variables x and y in equation (13.1.3.1) usually play the role of time and spatial coordinate, respectively, w is the density of the quantity being transferred, and f(w)isthe rate of w. 13.1.3-2. Solution of the Cauchy problem. Rarefaction wave. Wave “overturn.” 1 ◦ . The solution w = w(x, y) of the Cauchy problem for equation (13.1.3.1) subject to the initial condition w = ϕ(y)atx = 0 (–∞ < y < ∞)(13.1.3.2) can be represented in the parametric form y = ξ + F(ξ)x, w = ϕ(ξ), (13.1.3.3) where F(ξ)=f ϕ(ξ) . Consider the characteristics y = ξ + F(ξ)x in the yx-plane for various values of the parameter ξ. These are straight lines with slope F(ξ). Along each of these lines, the unknown function is constant, w = ϕ(ξ). In the special case f = a = const, the equation in question is linear; solution (13.1.3.3) can be written explicitly as w = ϕ(y – ax), thus representing a traveling wave with a fixed profile. The dependence of f on w leads to a typical nonlinear effect: distortion of the profile of the traveling wave. We further consider the domain x ≥ 0 and assume* that f > 0 for w > 0 and f w > 0. In this case, the greater values of w propagate faster than the smaller values. If the initial profile satisfies the condition ϕ (y)>0 for all y, then the characteristics in the yx-plane that come from the y-axis inside the domain x > 0 are divergent lines, and hence there exists a unique solution for all x > 0. In physics, such solutions are referred to as rarefaction waves. Example 1. Figures 13.2 and 13.3 illustrate characteristics and the evolution of a rarefaction wave for Hopf’s equation [for f(w)=w in (13.1.3.1)] with the initial profile ϕ(y)= 4 π arctan(y – 2)+2.(13.1.3.4) It is apparent that the solution is smooth for all x > 0. 4 4 80 y x Figure 13.2. Characteristics for the Hopf’s equation (13.1.2.8) with the initial profile (13.1.3.4). * By the change x =–x the consideration of the domain x ≤ 0 can be reduced to that of the domain x ≥ 0. The case f < 0 can be reduced to the case f > 0 by the change y =–y. 560 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 0 4 8 4 x = 0 1 2 y w Figure 13.3. The evolution of a rarefaction wave for the Hopf’s equation (13.1.2.8) with the initial profile (13.1.3.4). 2 ◦ . Let us now look at what happens if ϕ (y)<0 on some interval of the y-axis. Let y 1 and y 2 be points of this interval such that y 1 < y 2 .Thenf (y 1 )>f(y 2 ). It follows from the first relation in (13.1.3.3) that the characteristics issuing from the points y 1 and y 2 intersect at the “time instant” x ∗ = y 2 – y 1 f(w 1 )–f(w 2 ) ,wherew 1 = ϕ(y 1 ), w 2 = ϕ(y 2 ). Since w has different values on these characteristics, the solution cannot be continuously extended to x > x ∗ .Ifϕ (y)<0 on a bounded interval, then there exists x min =min y 1 ,y 2 x ∗ such that the characteristics intersect in the domain x > x min (see Fig. 13.4). Therefore, the front part of the wave where its profile is a decreasing function of y will “overturn” with time. The time x min when the overturning begins is defined by x min =– 1 F (ξ 0 ) , where ξ 0 is determined by the condition |F (ξ 0 )| =max|F (ξ)| for F (ξ)<0,andthewave is also said to break. A formal extension of the solution to the domain x > x min makes this solution nonunique. The boundary of the uniqueness domain in the yx-plane is the envelope of the characteristics. This boundary can be represented in parametric form as y = ξ + F(ξ)x, 0 = 1 + F (ξ)x. 0 2 4 24 68 y x x = 5 0.5 1.0 1.5 x = 0 Figure 13.4. Characteristics for the Hopf’s equation (13.1.2.8) with the initial profile (13.1.3.5). 13.1. LINEAR AND QUASILINEAR EQUATIONS 561 24 6 2.0 1.5 w y x = 0 0.5 1.0 1.5 2.0 2.5 Figure 13.5. The evolution of a solitarywave for the Hopf’s equation (13.1.2.8) with theinitialprofile (13.1.3.5). Example 2. Figure 13.5 illustrates the evolution of a solitary wave with the initial profile ϕ(y)=cosh –2 (y – 2)+1 (13.1.3.5) for equation (13.1.3.1) with f(w)=w. It is apparent that for x > x min ,wherex min = 3 4 √ 3 ≈ 1.3,thewave “overturns” (the wave profile becomes triple-valued). 13.1.3-3. Shock waves. Jump conditions. In most applications where the equation under consideration is encountered, the unknown function w(x, y) is the density of a medium and must be unique for its nature. In these cases, one has to deal with a generalized (nonsmooth) solution describing a step-shaped shock wave rather than a continuous smooth solution. The many-valued part of the wave profile is replaced by an appropriate discontinuity, as shown in Fig. 13.6. It should be emphasized that a discontinuity can occur for arbitrarily smooth functions f(w)andϕ(y) entering equation (13.1.3.1) and the initial condition (13.1.3.2). w y sx() Figure 13.6. Replacement of the many-valued part of the wave profile by a discontinuity that cuts off domains with equal areas (shaded) from the profile of a breaking wave. In what follows, we assume that w(x, y) experiences a jump discontinuity at the line y =s(x)intheyx-plane. On bothsides ofthe discontinuity the function w(x, y)issmoothand single-valued; as before, it is described by equations (13.1.3.3). The speed of propagation of the discontinuity, V , is expressed as V = s (the prime stands for the derivative) and must satisfy the condition V = F (w 2 )–F (w 1 ) w 2 – w 1 , F (w)= f(w) dw,(13.1.3.6) where the subscript 1 refers to values before the discontinuity and the subscript 2 to those after the discontinuity. In applications, relation (13.1.3.6), expressing a conservation law at discontinuity, is conventionally referred to as the Rankine–Hugoniot jump condition (this condition is derived below in Paragraph 13.1.3-4). 562 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS The continuous wave “overturns” (breaks), thus resulting in a discontinuity if and only if the propagation velocity f(w) decreases as y increases, i.e., the inequalities f(w 2 )<V < f (w 1 )(13.1.3.7) are satisfied. Conditions (13.1.3.7) have the geometric meaning that the characteristics issuing from the x-axis (these characteristics “carry” information about the initial data) must intersect the line of discontinuity (see Fig. 13.7). In this case, the discontinuous solution is stable with respect to small perturbations of the initial profile (i.e., the corresponding solution varies only slightly). y x line of discontinuity characteristics characteristics Figure 13.7. Mutual arrangement of characteristics and lines of discontinuity in the case of a stable shock wave. The position of the point of discontinuity in the yw-plane may be determined geometri- cally by following Whitham’s rule: the discontinuity must cut off domains with equal areas from the overturning wave profile (these domains are shaded in Fig. 13.6). Mathematically, the position of the point of discontinuity can be determined from the equations s(x)=ξ 1 + F 1 x, s(x)=ξ 2 + F 2 x, w 2 F 2 – w 1 F 1 = F (w 2 )–F (w 1 )+ F 2 – F 1 ξ 2 – ξ 1 ξ 2 ξ 1 wdξ. (13.1.3.8) Here, w and F are defined as functions of ξ by w = ϕ(ξ)andF = f (w), the function F (w) is introduced in equation (13.1.3.6), and the subscripts 1 and 2 refer to the values of the corresponding quantities at ξ = ξ 1 and ξ 2 . Equations (13.1.3.8) permit one to determine the dependences s = s(x), ξ 1 = ξ 1 (x), and ξ 2 = ξ 2 (x). It is possible to show that the jump condition (13.1.3.6) follows from the last equation in (13.1.3.8). Example 3. For Hopf’s equation, which corresponds to f(w)=w in equation (13.1.3.1), the jump condition (13.1.3.6) can be represented as V = w 1 + w 2 2 . Here, we take into account the relation F (w)= 1 2 w 2 . System (13.1.3.8), which determines the position of the point of discontinuity, becomes s(x)=ξ 1 + ϕ(ξ 1 )x, s(x)=ξ 2 + ϕ(ξ 2 )x, ϕ(ξ 1 )+ϕ(ξ 2 ) 2 = 1 ξ 2 – ξ 1 ξ 2 ξ 1 ϕ(ξ) dξ, where the function ϕ(ξ) specifies the initial wave profile. . f, g,andh of equation (13.1.1.1) be continuously differentiable functions of x, y,andw in G and let x = h 1 (ξ), y = h 2 (ξ), and w = h 3 (ξ) be continuously differentiable functions of ξ for. slightly). y x line of discontinuity characteristics characteristics Figure 13.7. Mutual arrangement of characteristics and lines of discontinuity in the case of a stable shock wave. The position of the point of. point of intersection of two characteristics defined by two distinct values ξ 1 and ξ 2 of the parameter, the function w takes two distinct values equal to ϕ(ξ 1 )and (ξ 2 ), respectively. Therefore,