Dynamics of propagation andinteraction of δ-shock waves in conservation law systems

Transcription

1 J. Differential Equations Dynamics of propagation andinteraction of δ-shock waves in conservation law systems V.G. Danilov a, V.M. Shelkovich b,,1 a Department of Mathematics, Moscow Technical University of Communication and Informatics, Aviamotornaya, 8a, , Moscow, Russia b Department of Mathematics, St.-Petersburg State Architecture and Civil Engineering University, 2 Krasnoarmeiskaya 4, , St. Petersburg, Russia Received22 December 2003; revised20 December 2004 Abstract We introduce a new definition of a δ-shock wave type solution for a class of systems of conservation laws in the one-dimensional case. The weak asymptotics method developed by the authors is usedto construct formulas describing the propagation andinteraction of δ-shock waves. The dynamics of merging two δ-shocks is described by explicit formulas continuously in time Elsevier Inc. All rights reserved. MSC: primary 35L65; secondary 35L67, 76L05 Keywords: Hyperbolic systems of conservation laws; Interaction of δ-shocks; The weak asymptotics method 1. Introduction In [3 9,29,30] see also [2] the weak asymptotics method for studying the dynamics of propagation andinteraction of different singularities infinitely narrow δ-solitons, shocks, δ-shocks of nonlinear equations andhyperbolic systems of conservation laws Corresponding author. addresses: danilov@miem.edu.ru V.G. Danilov, shelkv@vs1567.spb.edu V.M. Shelkovich. 1 Supportedin part by DFG Project 436 RUS 113/593/3 andgrant of Russian Foundation for Basic Research /$ - see front matter 2005 Elsevier Inc. All rights reserved. doi: /j.jde

2 334 V.G. Danilov, V.M. Shelkovich / J. Differential Equations was developed see [3, Introduction]. One of the main ideas of this methodis basedon the ideas of Maslov s approach that permits deriving the Rankine Hugoniot conditions directly from the differential equations considered in the weak sense [2,21,24] see also Whitham [35, 2.7.,5.6.]. In fact, Maslov s algebras of singularities [2,22,23] underlie our method. In this paper we introduce a new definition of a δ-shock wave type solution for a class of hyperbolic systems of conservation laws. Using this definition, in the framework of the weak asymptotics method, we describe the propagation and interaction of δ-shock waves. The subject of this paper was presentedat the IXth International Conference on Hyperbolic Problems [6]. Here we give the full version of this work. Consider the system of conservation laws L 1 [u, v] =u t + F u, v x = 0, L 2[u, v] =v t + Gu, v = 0, 1.1 x where F u, v and Gu, v are smooth functions, linear with respect to v; u = ux, t, v = vx, t R; x R. As is well known, even in the case of smooth and, certainly, in the case of discontinuous initial data u 0 x, v 0 x, this system may have discontinuous solutions. In this case, it is said that a pair ux, t, vx, t L R 0, ; R 2 is a generalizedsolution of the Cauchy problem 1.1 with the initial data u 0 x, v 0 x if the integral identities uφ t + F u, vφ x dx dt + u 0 xφx, 0dx = 0, 0 vφ t + Gu, vφ x dx dt + v 0 xφx, 0dx = holdfor all compactly supportedtest functions φx, t DR [0,, where dx denotes an improper integral dx. It is well known [6 9,34,10,12,14,16,17,29,30,32] that there are nonclassical situations when the Cauchy problem for system 1.1 does not possess a weak L -solution except for some particular initial data. In contrast to the standard results of existence of weak solutions to strictly hyperbolic systems, here the linear component of the solution v may contain Dirac measures andmust be sought in the space of measures, while the first component u has bounded variation. In order to solve the Cauchy problem in this nonclassical situation, it is necessary to introduce a new generalized solutions of the Cauchy problem called δ-shocks. In particular, system 1.1 with the initial data u 0 x = u 0 + u 1 H x, v 0 x = v 0 + v 1 H x, 1.3 where u 0, u 1, v 0, v 1 are constants and Hξ is the Heaviside function, may admit a δ-shock wave type solution: ux, t = u 0 + u 1 H x + ct, vx, t = v 0 + v 1 H x + ct + etδ x + ct, 1.4 where e0 = 0 and δξ is the Dirac δ function.

3 V.G. Danilov, V.M. Shelkovich / J. Differential Equations Several approaches to constructing δ-shock type solutions are known. An apparent difficulty in defining such solutions arises due to the fact that, to introduce a definition of the δ-shock type solution, we needto define the singular superpositions of distributions for example, the product of the Heaviside function and the δ-function. We also needto define in which sense a distributional solution for example, 1.4 satisfies a nonlinear system of form 1.1. In what follows, we present a short review of well-known methods used to solve problems close to those studied in this paper. In [14], a δ-shock wave type solution of the system u t + u 2 /2 x = 0, v t + uv x = 0 here F u, v = u 2 /2, Gu, v = vu with the initial data 1.3, is defined as the weak limit of the solution ux, t,, vx, t, of the parabolic regularization u t + u 2 /2 x = u xx, v t + uv x = v xx with the initial data 1.3, as +0. In [12], in order to obtain a δ-shock wave type solution of system L 1 [u, v] =u t + f u x = 0, L 2[u, v] =v t + guv = 0, 1.5 x here F u, v = f u, Gu, v = vgu, this system is reduced to a system of Hamilton Jacobi equations, andthen the Lax formula is used. In [10], δ-shock wave type solution of this system is constructedas self-similar viscosity limits. In [17], to construct a δ-shock wave type solution of system 1.5 for the case gu = f u, the problem of multiplication of distributions is solved by using the definition of Volpert s averagedsuperposition [33]. A general framework for nonconservative products of this type was introduced in [25]. In the framework of this approach the Cauchy problems for nonlinear hyperbolic systems in nonconservative form can be considered, but the notion of generalizedsolution does depend on the specific family of paths, which cannot be derived from the hyperbolic system only. The system 1 u t + u 2 v x = 0, v t + 3 u3 u = 0, 1.6 x here F u, v = u 2 v, Gu, v = u 3 u with the initial data 1.3 is studied in [16]. In order to construct approximate δ-shock type solution the Colombeau theory approach, as well as the Dafermos DiPerna regularization under the assumption that these profiles exist, andthe box approximations are used. But the notion of a singular solution of system 1.6 has not been defined. In [26] in the framework of the Colombeau theory approach, for particular cases of system 1.1 approximate δ-shock type solutions were constructed.

4 336 V.G. Danilov, V.M. Shelkovich / J. Differential Equations In [32] for the system L 01 [u] =u t + u 2 x = 0, L 02 [u, v] =v t + uv x = 0, 1.7 in [1,18] for the zero-pressure gas dynamics system v t + vu x = 0, vu t + vu 2 = 0, 1.8 x here v 0 is the density, u is the velocity, andin [36] for the system v t + vf u x = 0, vu t + vuf u = 0, 1.9 x with the initial data 1.3, the δ-shock wave type solutions are defined as measurevalued solutions see also [31]. Recall a definition of a measure-valued solution. Let BMR be the space of bounded Borel measures. A pair u, v, where ux, t L L R, [0,, vx, t C BMR, [0,, and u is measurable with respect to v at almost all t 0, is said to be a measure-valuedsolution of the Cauchy problem 1.9, 1.3 if the integral identities φ t + f uφ x vdx, t = 0, 0 0 u φ t + f uφ x vdx, t = 0, 1.10 holdfor all φx, t DR [0,. Within the framework of this definition in [1,32,36] for systems 1.7, 1.8, and 1.9, respectively, the following formulas for δ-shock waves were derived u,v, x < t, ux, t, vx, t = uδ, wtδx t, x = t, 1.11 u+,v +, x > t. Here u, u + and u δ are the velocities before the discontinuity, after the discontinuity, andat the point of discontinuity, respectively, and t = σ δ t is the equation for the discontinuity line. In [34], for system 1.8 the global δ-shock wave type solution in the sense of Radon measures was obtained. In [13], the interaction of two δ-shocks for system 1.9 is considered. In [9] the weak asymptotics methodwas usedto study the propagation of δ-shock waves for systems 1.5, 1.6, 1.8. A short review of our results on the propagation andinteraction of δ-shock waves for system 1.5 was presentedin [6 8]. In[29,30]

5 V.G. Danilov, V.M. Shelkovich / J. Differential Equations exact δ-shock wave type solutions for the well-known Keyfitz Kranzer system 1.6 andits generalization u t + f u v x = 0, v t + gu x = 0, were first constructedin the framework of the weak asymptotics method, where f u and gu are polynomials of degree n and n + 1, respectively, n is an even integer. In the papers [8,9] a new definition of a δ-shock wave type solution for systems 1.1, 1.8 was introduced. This definition is close to the standard definition of the shock type solutions 1.2 andrelevant to the structure of δ-shocks. The study of systems 1.1, 1.9, which admit δ-shock wave type solutions is very important in applications, because systems of this type often arise in modelling physical processes in gas dynamics, magnetohydrodynamics, filtration theory, and cosmology [11,15,34]. In order to describe the formation of large-scale structures of the universe the inviscidburgers equation was usedin [37], andthe whole system of zero-pressure gas dynamics was used in [27]. These models are used to describe the motion of free particles which stick under collision. In the present paper the weak asymptotics method is usedto investigate both propagation and interaction of δ-shock waves for system 1.5, i.e., to solve the Cauchy problem for system 1.5 with the initial data of the form u 0 x = u 0 0 x + 2 k=1 u0 k xh x + x0 k, v 0 x = v0 0 x + 2 k=1 v 0 k xh x + x0 k + e0 k δ x + x0 k, 1.12 where u 0 0 x, u0 k x, v0 0 x, and v0 k x are smooth functions, u0 k x0 k > 0, e0 k are constants, k = 1, 2, and x1 0 <x0 2. If we study only the dynamics of propagation of δ-shocks, then we set u 0 2 x = v0 2 x = e0 2 = 0, e0 1 = e0, and x1 0 = 0 andsolve the Cauchy problem for system 1.5 with the initial data u 0 x = u 0 0 x + u0 1 xh x, v 0 x = v0 0 x + v0 1 xh x + e0 δ x, 1.13 where u >0. The initial data 1.12, 1.13 can contain a δ-function, while in most papers on δ-shocks, initial data without a δ-function are considered, because the technical base of these papers is connectedwith self-similar solutions. As in [10,16,32], we shall use the overcompression condition for details, see [19] λ 1 u +,v + t λ 1 u,v, λ 2 u +,v + t λ 2 u,v, 1.14 as the admissibility condition for the δ-shocks. Here λ 1 u, v, λ 2 u, v are eigenvalues of the characteristic matrix of a hyperbolic system of conservation laws, t is the velocity of motion of the δ-shock wave, and u, v and u +, v + are the respective

6 338 V.G. Danilov, V.M. Shelkovich / J. Differential Equations left- andright-handvalues of u, v on the discontinuity curve. Condition 1.14 means that all characteristics on both sides of the discontinuity are in-coming. Since for system 1.5 λ 1 u, v = f u, λ 2 u, v = gu, we assume that f u > 0, g u > 0, f u gu The weak asymptotics method and the main results 2.1. δ-shock wave type solution In what follows, we introduce a definition of a generalized solution [8,9] for system 1.1. Suppose that Γ ={γ i : i I} is a graph in the upper half-plane {x, t : x R, t [0, } R 2 containing smooth arcs γ i, i I, and I is a finite set. By I 0 we denote a subset of I such that an arc γ k for k I 0 starts from the points of the x-axis; Γ 0 ={xk 0 : k I 0} is the set of initial points of arcs γ k, k I 0. Consider δ-shock wave type initial data u 0 x, v 0 x, where v 0 x = V 0 x + e 0 δγ 0, where u 0,V 0 L R; R, and e 0 δγ 0 def = k I 0 ek 0δx x0 k, e0 k k I 0. are constants, Definition 1. A pair of distributions ux, t, vx, t andgraph Γ, where vx, t is representedin form of the sum vx, t = Vx,t+ ex, tδγ, u, V L R 0, ; R, ex, tδγ def = i I e ix, tδγ i, e i x, t C 1 Γ, i I, is calleda generalizedδ-shock wave type solution of system 1.1 with the initial data u 0 x, v 0 x if the integral identities uφ t + F u, V φ x dx dt + u 0 xφx, 0dx = 0, 0 V φ t + Gu, V φ x dx dt 0 + i I + φx, t e i x, t dl γ i l V 0 xφx, 0dx+ ek 0 φx0 k, 0 = 0, 2.1 k I 0

7 V.G. Danilov, V.M. Shelkovich / J. Differential Equations holdfor all test functions φx, t DR [0,, where φx,t l derivative on the graph Γ, γ dl is a line integral over the arc γ i i. is the tangential For instance, the graph Γ containing only one arc {x, t : x = ct}, 0 = 0 corresponds to solution 1.4. Remark 2. The system of δ-shocks integral identities 2.1 is natural generalization of the usual system of integral identities 1.2 which is the definition of a weak L - solution. The integral identities 2.1 differ from integral identities 1.2 by an additional term φx, t ex, t dl = φx, t e i x, t dl Γ l i I γ i l in the second identity which appears due to the Rankine Hugoniot deficit. Namely, if Γ ={x, t : x = t} then the Rankine Hugoniot conditions for δ-shock can be representedas the pair of equations t = [F u, v] [u], ė t, t = x= t [Gu, v] [v] [F u, v], [u] x= t where [ ] is the jump in the corresponding function on the discontinuity curve x = t, = dt d. Here the first equation is the standard Rankine Hugoniot condition, and the right-handside of the secondequation calledthe Rankine Hugoniot deficit. For system 1.5 the Rankine Hugoniot conditions are given by the fifth and sixth equations of According to Definition 1 a generalized δ-shock wave type solution is a pair of distributions u, v, while in [1,32,36] measure-valuedsolution is definedas a pair u, v, where vdx, t is a measure and ux, t is understood as a measurable function with respect to vdx, t Weak asymptotic solution Now we introduce the notion of a weak asymptotic solution, which is one of the most important in the weak asymptotics method. We shall write fx,t,= O D α,iffx,t, D R is a distribution such that for any test function ψx DR x we have fx,t,,ψx =O α, where O α denotes a function continuous in t that admits the usual estimate O α const α uniform in t. Relations of the form o D α are understood in the same way.

8 340 V.G. Danilov, V.M. Shelkovich / J. Differential Equations Definition 3. A pair of functions ux, t,, vx, t, smooth as >0 is calleda weak asymptotic solution of system 1.1 with the initial data u 0,v 0 if L1 [ux, t,, vx, t, ]ψx dx = o1, L2 [ux, t,, vx, t, ]ψx dx = o1, ux, 0, u 0 x ψx dx = o1, vx, 0, v 0 x ψx dx = o1, +0, for all ψx DR. The last relations can be rewritten as L 1 [ux, t,, vx, t, ] =o D 1, L 2 [ux, t,, vx, t, ] =o D 1, where the first two estimates are uniform in t The outline of the weak asymptotics method ux, 0, = u 0 x + o D 1, vx, 0, = v 0 x + o D 1, 2.2 Now for the case of δ-shocks we will describe the typical technique of our approach without paying attention to the algebraic aspects given in detail in [2,3,28]. a According to our method, we will seek a δ-shock wave type solution of the Cauchy problem 1.5, 1.13 in the form of the singular ansatz ux, t = u 0 x, t + u 1 x, th x + t, vx, t = v 0 x, t + v 1 x, th x + t + etδ x + t, 2.3 where u k x, t, v k x, t, k = 0, 1, et, t are the desired functions. We will seek a δ-shock wave type solution of the Cauchy problem 1.5, 1.12 in the form of the singular ansatz ux, t = u 0 x, t + 2 u kx, th x + k=1 k t, vx, t = v 0 x, t + 2 v k x, th x + k=1 k t +e k tδ x + k t, 2.4 where u 0 x, t, u k x, t, v 0 x, t, v k x, t, e k t, k t are the desired functions, k = 1, 2.

9 V.G. Danilov, V.M. Shelkovich / J. Differential Equations The singular ansatzs 2.3 and2.4 correspond to the structure of the initial data 1.13 and1.12, respectively. b Next, we construct a weak asymptotic solution of the problem in the form of the smooth ansatz: ux,t,= ũx,t,+ R u x,t,, vx,t,= ṽx, t, + R v x,t,, where a pair of functions ũx, t,, ṽx, t, is a regularization of the singular ansatz 2.3 or2.4 with respect to singularities Hx, δx, with respect to phases k t, and with respect to amplitudes of δ-functions e k t, k = 1, 2. Here the so-called corrections R u x,t,, R v x,t, are desired functions which admit the estimates: R j x,t,= o D 1, R j x,t, = o D 1, +0, 2.5 t j = u, v. In order to construct a regularization fx, of the distribution fx D R we use the representation fx,= fx 1 ω x, > 0, 2.6 where is a convolution, anda mollifier ωη has the following properties: a ωη C R, b ωη has a compact support or decreases sufficiently rapidly, as η, c ωηdη = 1, d ωη 0. It is known that fx,,φx = fx,φx lim +0 for all φx DR. Thus we will seek a weak asymptotic solution of the Cauchy problem 1.5, 1.12 in the form ux,t, = u 0 x, t + 2 u kx, th uk x + k t,, + R u x,t,, k=1 vx, t, = v 0 x, t + 2 v k x, th vk x + k t,, k=1 +e k t, δ vk x + k t,, + R v x,t,, 2.7 where k t,, e k t, are desired functions such that k t = lim k t,, e +0 kt = lim +0 e kt,. According to 2.6 δ vk ξ,= 1 ω δk ξ 2.8

10 342 V.G. Danilov, V.M. Shelkovich / J. Differential Equations are regularizations of the δ-function, and ξ H jk ξ,= ω 0jk = x/ ω jk ηdη 2.9 are regularizations of the Heaviside function Hξ, where ω 0jk z C R, and lim z + ω 0jk z = 1, lim z ω 0jk z = 0, j = u, v, k = 1, 2. A weak asymptotic solution of the Cauchy problem 1.5, 1.13 is constructedin form 2.7, where u 2 x, t v 2 x, t e 2 t, 0, and 1 t, t, e 1 t, et. Thus we will seek a weak asymptotic solution of the Cauchy problem 1.5, 1.12 in the form ux,t, = u 0 x, t + u 1 x, th u1 x + t, + R u x,t,, vx, t, = v 0 x, t + v 1 x, th v1 x + t, +etδ v1 x + t, + Rv x,t, The next step is to substitute the smooth ansatz 2.10 or2.7 into the quasilinear system L[u, v] =0 andto calculate the weak asymptotics in the sense of the space of distributions D R x of the left-handside of L[ux, t,, vx, t, ] up to o D 1, as +0. We stress that in the framework of the weak asymptotics method, the discrepancy is assumed to be small in the sense of the space of functionals D x over test functions depending only on the space variable x. As we shall see below, this trivial trick allows us to reduce the problem of describing interaction of nonlinear waves to solving some system of ordinary differential equations instead of solving partial differential equations. In order to construct the weak asymptotics of L[ux, t,, vx, t, ], we needto calculate some weak asymptotics of the type f ux, t,, vx, t,, where f u, v is a smooth function. These calculations are given at the endof the paper in Appendix A. According to results of Appendix A, the weak asymptotics of L[ux, t,, vx, t, ] can be representedas linear combinations of the singularities H x + k t, δ x + k t, δ x + k t, k = 1, 2 with smooth coefficients. That is why we can separate singularities andfinda system of equations in particular, the Rankine Hugoniot type conditions, which describes the dynamics of singularities and defines the desired functions u 0 x, t, u k x, t, v 0 x, t, v k x, t, e k t,, k t,, k = 1, 2, and R u x,t,, R v x,t,. In this way a weak asymptotic solution of the problem is constructed. c To describe the dynamics of interaction, we shall seek the phases of a weak asymptotic solution k t, = k τ,t as functions of the fast variable τ = ψ 0 t/ andthe slow variable t, where k0 t is the distance between the solitary wave fronts before the instant of interaction. Next, we obtain systems of equations for k τ,t

11 V.G. Danilov, V.M. Shelkovich / J. Differential Equations and the differential equation with the boundary condition: dρ dτ = Fρ,t, ρτ,t τ = 1, 2.11 τ + where ρ = ψt, /, ψt, = 2 t, 1 t,. Here the boundary condition shows that, before the interaction, the singularities propagate independently. Finding the solution of the boundary value problem 2.11 andfinding the limit values ρτ τ and k t = lim τ k τ,t, we can describe the dynamics of propagation andinteraction of nonlinear waves andthus define the result of the interaction for details, see Section 4. The autonomous ODE 2.11 is typical for our approach see 4.15, d Within the framework of the weak asymptotics method, we find the generalized solution ux, t, vx, t of the Cauchy problem 1.5, 1.12 as the weak limit in the sense of the space of distributions D R [0, ux, t = lim ux, t,, +0 vx, t = lim vx, t,, where ux, t,, vx, t, is a weak asymptotic solution of this problem. Multiplying the first two relations 2.2 by a test function φx, t DR [0,, integrating these relations by parts andthen passing to the limit as +0, we see that the pair of distributions 2.12 satisfy integral identities 2.1. Thus, we will prove that the limit of a weak asymptotic solutions 2.12 is a generalized δ-shock wave type solution of the Cauchy problem 1.5, Since the generalized δ-shock wave type solution 2.4 is defined as a weak limit of 2.12, in view of estimates 2.5, the corrections R u x,t,, R v x,t, do not make a contribution to the generalizedsolution of the problem. However, according to formulas A.6, A.7 from Appendix A, these terms make a contribution to the weak asymptotics of the superposition f ux, t,, vx, t,, andhence play a role in the construction of the generalizedsolution of the problem. Generally speaking, without introducing these terms, we cannot solve the Cauchy problem with arbitrary initial data see also Remark 8 below. Remark 4. To study the interaction of two shock waves for the scalar conservation law u t + f u x = 0, we seek a weak asymptotic solution of the problem in the form of the first relation 2.7, where we set R u x,t,= 0 [3 5]. To study the interaction of two infinitely narrow δ-solitons relatedto KdV type equation v t + f v x + 2 v xxx = 0, we seek a weak asymptotic solution of the problem in the form of the secondrelation 2.7, where we set R v x,t, = 0 and replace H vk ξ, by H vk ξ,, and δ vk ξ, by δ vk ξ,, k = 1, 2 [3]. On infinitely narrow δ-solitons see [24].

12 344 V.G. Danilov, V.M. Shelkovich / J. Differential Equations One example We illustrate our approach by constructing a solitary δ-shock wave type solution to the Cauchy problem 1.7, 1.3. A We seek a weak asymptotic solution of the Cauchy problem 1.7, 1.3 inthe form x + t ux,t, = u 0 + u 1 H u1 x + t, + QΩ, vx, t, = v 0 + v 1 H v1 x + t, + etδ v1 x + t,, 2.13 where u k, v k are constants, k = 0, 1. We choose corrections in the form R u x,t,= QΩ x+ t, Rv x,t, = 0, where 1 Ω ξ/ is regularization 2.8 of the δ- function, Q is a constant. It is clear that x Ω ψx dx = ψ0 Ωξdξ + O 2, +0, for all ψx DR, i.e., estimates 2.5 hold. Note that in the pointwise limit we have { x + t QΩ0, x = t, lim QΩ = +0 0, x = t, 2.14 i.e., the correction R u x,t, is a regularization of the characteristic function of the point. Thus, if we set QΩ0 = u δ, then, in the pointwise limit, our regularization ux, t, converges to the expression for the component ux, t in solution 1.11 obtainedin [32]. For our purposes, this similarity is not necessary. Moreover, in what follows, we shall construct another weak asymptotic solution, which does not possess this property, but satisfies the integral identities 2.1 in the limit. This weak asymptotic solution turns out to be more preferable for describing the δ-shock wave interaction studied in Section 4. So we show how the weak asymptotic solution is constructedin our example. According to formulas of Appendix A, with accuracy O D, wehave ux,t, 2 = u [ u 2] H x + t + O D, +0, 2.15 ux,t,vx,t, = u 0 v 0 + [ uv ] H x + t + u 0 + au 1 + bq etδ x + t + O D, +0, 2.16

13 V.G. Danilov, V.M. Shelkovich / J. Differential Equations where [ ] is the jump in the corresponding function on the discontinuity curve x = t, a = ω 0u1 ξω δ1 ξdξ, b = Ωξω δ1 ξdξ. Substituting regularization 2.13 andrelations 2.15, 2.16 into the left-handside of system 1.7, we see that L 01 [ux,t,]=o D, L 02 [ux, t,, vx, t, ] =O D 2.17 if andonly if [u] t [u 2 ] δ x + t = 0, [v] t +ėt [uv] δ x + t + t u 0 + au 1 + bq etδ x + t = 0. From this system we findthe functions andthe relation t = [u2 ] [u] t = 2u 0 + u 1 t, et = [uv] [u2 ] [u] [v] t = u 1 v 0 u 0 v 1 t, 2.18 Q = u au 1, 2.19 b which determines the constant Q. The pair of functions 2.18 is the solution of the Rankine Hugoniot conditions see Remark 2. Thus, the weak asymptotic solution of the Cauchy problem 1.7, 1.3 is constructed. Defining the generalizedsolution of our problem 1.7, 1.3 as the weak limit of regularizations 2.12, we obtain 1.4, where c = t, t and et are determined by system Relations 2.18 are the same as in [32]. We show that the weak limit 1.4 of the weak asymptotic solution 2.13 satisfies the integral identities 2.1. The integral identities 2.1 are derived in the same way as it is provedin [5] that the weak limit of the weak asymptotic solution satisfies the integral identity. Since ux,t, and vx, t, are smooth functions as >0, applying the leftandright-handsides of relations 2.17 to φx, t DR [0, andintegrating by parts the expression obtainedin the left-handside, we obtain relations 3.13, 3.14, where f ux, t, = u 2 x,t,, gux, t, = ux,t,, T =. Next, passing to the limit in the last relations, as +0, andtaking into account relations 2.13,

14 346 V.G. Danilov, V.M. Shelkovich / J. Differential Equations , 2.16, we obtain the following integral identities ux, tφ t + u 2 x, tφ x dx dt + Vx,tφ t + ux, tv x, tφ x dx dt u 0 xφx, 0dx = 0, + et φ t t, t + tφ x t, t dt 0 + V 0 xφx, 0dx+ e 0 φ0, 0 = 0, 2.20 for all test functions φx, t DR [0,. Here, according to the notation of Definition 1, vx, t = Vx,t+ etδ x + t, Vx,t= v 0 + v 1 H x + t. B Now we will construct the solution of the Cauchy problem 1.7, 1.3, using the weak asymptotic solution of a different structure ux,t, = u 0 + u 1 H u1 x + t,, vx, t, = v 0 + v 1 H v1 x + t, 1 +etδ v1 x + t, + Rt Ω x + t In this case corrections are chosen in the form R u x,t,= 0, R v x,t,= Rt 1 Ω x+ t, where 1 Ω ξ 3 is a regularization of the distribution δ ξ. Since, for all ψx DR, wehave 1 Ω x φx dx = 2 φ 0 Ωξdξ + O 3, +0, it is clear that estimates 2.5 hold. Here, in contrast to 2.13, in the pointwise limit, as +0, the component ux, t does not contain the characteristic function of the curve x = t. As above, substituting 2.21 into the left-handside of system 1.7, we see that 2.17 holds if and only if [u ] t [ u 2] δ x + t = 0, [v ] t +ėt [ uv ] δ x + t + et t u 0 + au 1 crt δ x + t = 0,

15 V.G. Danilov, V.M. Shelkovich / J. Differential Equations where the constant a = ω 0u1 ξω δ1 ξdξ is the same as in 2.16, andthe constant c = ω 0u1 ξω ξdξ. This allows us to findfrom above system the functions t and et, which, as before, are determined by relations 2.18, andto findthe relation Rt = et u au 1, 2.22 c which determines the function Rt. Obviously, the weak limit of the weak asymptotic solution 2.21 is the same, i.e., it is 1.4. As in the preceding case, it is easy to show that the weak limit 1.4 satisfies the integral identities 2.1. In this paper we shall use the correction of the secondkindsee 3.1 and4.1, because, from the analytic viewpoint, this simplifies describing the interaction of δ- shocks Contents of the paper In Sections 2.1 and 2.2 we have defineda generalizedsolution anda weak asymptotic solution of the δ-shock wave type to the Cauchy problem, respectively. In Section 2.3, the technique of the weak asymptotics methodin the case of δ- shock waves is given. Here we constructedthe singular ansatz andthe smooth ansatz, which are usedto solve the Cauchy problems 1.5, 1.13 and1.5, In Section 2.4, to demonstrate our technique, we construct a δ-shock wave type solution for the simplest case of the Cauchy problem 1.7, 1.3. The problem of constructing singular superpositions products of distributions in connection with the problem of defining δ-shock wave type solutions to the Cauchy problems is discussed in Section 2.6. In Section 3, Theorem 5, we construct a weak asymptotic solution of the Cauchy problem 1.5, 1.13 in the form of a solitary δ-shock wave. Theorem 6 gives a generalizedsolution of our problem. We show that our solution satisfying the integral identities 2.1. Corollary 7 gives the same results in the case of piecewise constant initial data. Moreover, in the case of the piecewise constant initial data our identity coincides with the similar expression [32, 3.5], [36] treatedas an element of the space D R 2 see 1.10 in Section 1. In Section 4, in Theorem 9, we construct a weak asymptotic solution of the Cauchy problem 1.5, 1.12 with pointwise constant initial values. The limiting properties of the weak asymptotic solution are described in Corollary 10. Next, using Corollary 10, in Theorem 11 we construct a generalizedsolution of this Cauchy problem, which describes the dynamics of propagation and interaction of two δ-shock waves. Formulas 4.42 describing the propagation and interaction of two δ-shock waves are constructed. Here the velocities k t andthe Rankine Hugoniot deficit ė k t of δ-shocks have jumps Systems with the boundary conditions 4.9, 4.10, obtainedin the proof of Theorem 9, uptoo D, describe the process of merging two δ-shock waves into one. In Section 4.3, to illustrate Theorem 11, we consider the dynamics of interaction of two δ-shocks for the simplest case of the system u t + u 2 x = 0, v t + 2uv x = 0.

16 348 V.G. Danilov, V.M. Shelkovich / J. Differential Equations As an example, we consider the above system instead of system 1.7, since in system 1.7 two δ-shocks cannot exist for details, see Section 4.3. Special Appendixes A and B at the endof the paper contain some auxiliary results of the weak asymptotics method. In Appendix A, we prove the main lemmas about the asymptotic expansions, which can be usedfor constructing the weak asymptotic solution. In Appendix B, we prove a lemma from the theory of ordinary autonomous differential equations, which will be used for analyzing the process of interaction of δ-shock waves Singular superpositions products of distributions As mentioned in the Introduction, the problem of defining δ-shock wave type solution of the Cauchy problem is connectedwith the construction of singular superpositions products of distributions. We stress that the right singular superpositions of distributions can be obtained only in the context of constructing weak asymptotic solution to the Cauchy problem. 1 Let ux, t, vx, t be a generalized δ-shock wave type solution 2.3 ofthe Cauchy problem 1.5, 1.13 given by Theorem 6. Then a pair of functions 2.10 is a regularization of the pair of distributions 2.3. According to relation A.6 from Appendix A, we can introduce singular superpositions as the following a weak limit: {}}{ vx, tg ux, t def = lim +0 vx, t, g ux,t, = v + gu + + [ guv ] H x + t { } + etat + Rtct δ x + t, 2.23 where at, ct are defined by 3.5. Here singular superposition 2.23 depends on the correction function Rt, coefficients at, ct, and, consequently, depends on mollifiers ω 0u1 η, ω δ1 η, Ω η. This fact means that the above introduced singular superpositions are not unique. However, in the context constructing of weak asymptotic solutions of the Cauchy problems we can define explicit formulas for the right singular superpositions. Namely, substituting correction function Rt given by3.4, into 2.23, we obtain vx, tg ux, t def = lim +0 vx, t, g ux,t, = v + gu + + [ vgu ] x= t H x + t +et [f u] δ x + t [u] x= t

17 V.G. Danilov, V.M. Shelkovich / J. Differential Equations According to relation A.1 from Appendix A, we obtain f ux, t def = lim +0 f ux,t, = fu 0 + [ f u ] H x + t In contrast to 2.23, where functions ux,t,, vx, t, are regularizations of distributions 2.3, in 2.24, 2.25, functions ux,t,, vx, t, give the weak asymptotic solution of the Cauchy problem 1.5, 1.13 given by Theorem 5. 2 Let ux, t, vx, t be a generalized δ-shock wave type solution 2.4 ofthe Cauchy problem 1.5, 1.12 given by Theorem 11. Similarly to 2.24, 2.25, using formulas A.2, A.7, 4.45, andthe limit properties of interaction switches B k 1 k 1 ρ, B 2 1 k 1 ρ, k = 1, 2 given by A.4, A.9, we introduce the right singular superpositions: f ux, t def = lim +0 f ux,t, = fu 0 + [ f u ] 1 H x + 1 t + [ f u ] 2 H x + 2 t, vx, tg ux, t def = lim vx, t, g ux,t, +0 = v 0 gu 0 + [ vgu ] 1 H x + 1 t + [ vgu ] 2 H x + 2 t +e 1 t [f u] 1 δ x + [u] 1 t + e 2 t [f u] 2 δ x + 1 [u] 2 t, 2 where k t, e k t are given by 4.42, andthe jumps [hu, v] k in function hu, v are defined in Section 4.1, k = 1, 2. It is clear that, in general, the weak limits of the functions f ux,t, and vx, t, g ux,t, depend on the regularizations of the Heaviside function and δ function. But the above unique right singular superpositions can be obtained only in the context constructing of weak asymptotic solution. If we knew the above right singular superpositions in advance then Theorems 6, 11 couldbe provedexplicitly by substituting these superpositions into system Propagation of δ-shocks 3.1. Construction of a weak asymptotic solution To study the propagation of a solitary δ-shock wave in system 1.5, we solve the Cauchy problem 1.5, 1.13.

18 350 V.G. Danilov, V.M. Shelkovich / J. Differential Equations In order to construct the weak asymptotic solution 2.10 of the problem we choose corrections in the form as in the above example B, in Section 2.4 R u x,t,= 0, R v x,t,= Rt 1 Ω x + t, 3.1 where Rt is a continuous function, 3 Ω x/ is a regularization of the distribution δ x, Ωη has the properties a c see Section 2.3. Since for any test function ψx DR x we have x relations 2.5 hold. 1 Ω x ψx dx = 2 ψ 0 1 Ω x ψx dx = 2 ψ 0 Ωξdξ + O 3, Ωξdξ + O 3, 3.2 Theorem 5. Let conditions 1.15 be satisfied. Then there exists T>0 such that, for t [0,T, the Cauchy problem 1.5, 1.13 has a weak asymptotic solution 2.10, 3.1 if and only if L 1 [u + ]=0, L 1 [u ]=0, L 2 [u +,v + ]=0, x > t, x < t, x > t, L 2 [u,v ]=0, x < t, 3.3 [f u] t =, [u] x= t [f u] ėt = [vgu] [v] [u] x= t, Rt = et [f u] at, 3.4 ct [u] x= t where u + = u 0, v + = v 0, u = u 0 + u 1, v = v 0 + v 1, [ hux, t, vx, t ] x= t = hu x, t, v x, t hu + x, t, v + x, t x= t

19 V.G. Danilov, V.M. Shelkovich / J. Differential Equations is a jump in function hux, t, vx, t across the discontinuity curve x = t, at = g u x, tω 0u1 η + u + x, t1 ω 0u1 η x= t ω δ1 ηdη, ct = g u x, tω 0u1 η + u + x, t1 ω 0u1 η x= t Ω ηdη = 0. The initial data for system 3.3, 3.4 are defined from 1.13, and 0 = 0, R0 = e0 [fu 0 ] c0 [u 0 a0. ] x=0 3.5 Proof. Let us substitute ansatz 2.10, 3.1, andasymptotics f ux, t, and gu x,t,vx,t, given by formulas A.1 anda.6 from Appendix A, respectively, into system 1.5. As in the above example B, in Section 2.4, taking into account estimates 3.2, 2.5, we obtain up to O D the following relations { [u] L 1 [ux,t,]=l 1 [u + ]+ t + [ ] } f u H x + t x { + [u] t [ f u ]} δ x + t + O D, 3.6 L 2 [ux, t,, vx, t, ] =L 2 [u +,v + ] { [v] + t + [ ] } vgu H x + t x { + [v] t +ėt [ vgu ]} δ x + t { + et t } etat ctrt δ x + t +O D, 3.7 where at, ct are defined by formula 3.5. It is clear that mollifiers ω 0u1 ξ, Ωξ can be chosen such that ω u1 ηω ηdη > 0. Consequently, taking into account that g u > 0, u 0 1 x > 0 andintegrating by parts, we obtain ct = g u 0 x, t + u 1 x, tω 0u1 η u 1 x, t ω u1 ηω ηdη = 0. x= t Setting the right-handside of 3.6, 3.7 equal to zero, we obtain the necessary andsufficient conditions for the equalities L 1 [ux,t,]=o D and L 2 [ux, t,, vx, t, ] =O D, i.e. system 3.3, 3.4.

20 352 V.G. Danilov, V.M. Shelkovich / J. Differential Equations Now we consider the Cauchy problem L 11 [u] =u t + f u x = 0, ux, 0 = u0 x. Since, according to 1.15, f u is convex and u >0, according to the results [20, Chapter 4.2.], we extend u 0 + x = u0 0 x u0 x = u0 0 x + u0 1 x tox 0 x 0 in a bounded C 1 fashion and continue to denote the extended functions by u 0 ± x. By u ± x, t we denote the C 1 solutions of the problems L 11 [u] =u t + f u x = 0, u ±x, 0 = u 0 ± x which exist for small enough time interval [0,T 1 ] andare determinedby integration along characteristics. The functions u ± x, t determine a two-sheeted covering of the plane x, t. Next, we define the discontinuity curve x = t as a solution of the problem t = f u + x, t f u x, t u + x, t u x, t, 0 = 0. x= t It is clear that there exists a unique function t for sufficiently short times [0, T 2 ]. To this end, for T = mint 1,T 2 we define the shock solution by { u+ x, t, x > t, ux, t = u x, t, x < t. Thus the first, secondandfifth equations of system 3.3 define a unique solution of the Cauchy problem L 11 [u] =u t + f u x = 0, ux, 0 = u0 x for t [0,T. Solving this problem, we obtain ux, t, t. Then substituting these functions into system 3.3, we obtain Vx,t= v 0 x, t + v 1 x, th x + t, et, and vx, t = Vx,t+ etδ x + t. Moreover, for any functions u 0 x, t, u 1 x, t, et, t, t [0, T, there exists a function Rt, which is defined by relation Construction of a generalized solution We obtain a generalizedsolution of the Cauchy problem 1.5, 1.13 as a weak limit 2.12 of a weak asymptotic solution constructedby Theorem 5. Theorem 6. Assume that conditions 1.15 are satisfied. Then, for t [0,T, where T>0 is given by Theorem 5, the Cauchy problem 1.5, 1.13, has a unique generalized solution 2.3: ux, t = u + x, t +[ux, t]h x + t, vx, t = v + x, t +[vx, t]h x + t + etδ x + t, 3.8

21 V.G. Danilov, V.M. Shelkovich / J. Differential Equations which satisfies the integral identities cf. 2.1: T uφ t + f uφ x dx dt + u 0 xφx, 0dx = 0, 0 T φ t + guφ x Vdxdt+ V 0 xφx, 0dx 0 Γ φx, t ex, t dl + e 0 φ0, 0 = 0, 3.9 l for all φx, t DR [0,T, where Γ ={x, t : x = t, t [0,T}, φx, t T ex, t dl = et φ t t, t + tφ x t, t dt, Γ l 0 Vx,t = v + +[v]h x + t. Here functions u + = u 0, v + = v 0, u = u 0 + u 1, v = v 0 + v 1, and t, et are defined by the system L 1 [u + ]=0, L 1 [u ]=0, L 2 [u +,v + ]=0, x > t, x < t, x > t, L 2 [u,v ]=0, x < t, [f u] t =, [u] x= t [f u] ėt = [vgu] [v] [u] x= t 3.10 with the initial data defined from 1.13, 0 = 0. Proof. According to relation A.1 from Appendix A, f ux,t, = fu + + [ f ux, t ] H x + + O D, +0, 3.11 By substituting relation 3.4, which determines Rt, into the relation A.6 from Appendix A, we obtain vx, t, g ux,t, = v + gu + + [ vgu ] x= t H x + t [f u] +et δ x + t + O D, [u] x= t +0, 3.12

22 354 V.G. Danilov, V.M. Shelkovich / J. Differential Equations By Theorem 5 we have the following estimates: L 1 [ux,t,]=o D, L 2 [ux, t,, vx, t, ] =O D. Let us apply the left- andright-handsides of these relations to an arbitrary test function φx, t DR [0, T. Then integrating by parts, we obtain T ux,t,φ t x, t + f ux, t, φ x x, t dx dt 0 + ux, 0, φx, 0 dx = O, 3.13 T 0 vx, t, φ t x, t + vx, t, gux, t, φ x x, t dx dt + vx, 0,φx, 0dx = O, Now let us substitute the asymptotics of ux,t,, vx, t,, andthe asymptotics of f ux, t, gux, t, vx, t,, given by 3.11 and3.12, respectively, into the last relations. Now by passing to the limit as +0 in each of the integrals 3.13, 3.14, andtaking into account that T lim +0 0 T etδ v1 x + t, φx,tdxdt = etφ t, t dt, 3.15 lim +0 e0δ v1 x, φx, 0dx = e0φ0, 0, we obtain the integral identities 3.9. In view of the above remark in Theorem 5, the Cauchy problem has a unique generalizedsolution. Let us consider the piecewise constant case of initial data 1.13, where u 0 0 = u 0, u 0 1 = u 1 > 0, v 0 0 = v 0, v 0 1 = v 1 are constants. Corollary 7. Assume that conditions 1.15 are satisfied. Then for t [0,, the Cauchy problem 1.5 with the piecewise constant initial data 1.13 has a unique generalized solution ux, t = u + +[u]h x + t, vx, t = v + +[v]h x + t + etδ x + t,

23 V.G. Danilov, V.M. Shelkovich / J. Differential Equations where t = [f u] t, et = e 0 + [guv] [u] [f u] [v] t. [u] Remark 8. To finda generalizedsolution of the Cauchy problem 1.5, 1.13, we construct a weak asymptotic solution 2.10, 3.1 of the problem, where the functions u ± x, t, v ± x, t, et, t are determined by system 3.3 andthe functions ω 0u1, ω δ1, Ω and Rt are determined by relation 3.4 andsystem 3.5. According to 3.4, 3.5, without introducing the corrections 3.1 we can construct a weak asymptotic solution of the Cauchy problem only if the following relation [f ux, t] [ux, t] = g ω 0u1 ηu t, t + 1 ω 0u1 ηu + t, t ω δ1 ηdη x= t holds. In the general case, this relation makes the Cauchy problem 1.5, 1.13 overdetermined. Thus we cannot solve the Cauchy problem with an arbitrary jump. Nevertheless, in the case of the piecewise constant initial data 1.13 andin the case of system 1.5, in view of condition 1.15, one can choose the mollifiers ω u1, ω δ1 in 3.5 such that the above relation holds, i.e., g ω 0u1 ηu + 1 ω 0u1 ηu + ωδ1 ηdη = fu fu + u u Interaction of δ shocks 4.1. Construction of a weak asymptotic solution We describe the dynamics of propagation and interaction of two δ-shock waves for system 1.5 with the piecewise constant initial data 1.12, where u 0 0 = u 0, u 0 k = u k > 0, v 0 0 = v 0, v 0 k = v k are constants, k = 1, 2. In order to construct a weak asymptotic solution 2.7 of the problem we choose corrections in the form R u x,t,= 0, R v x,t,= 2 k=1 R k t, 1 Ω k x + k t,, 4.1 where R k t, are the desired functions, 3 Ω k x/ are regularizations of the distribution δ x, Ω k η has the properties a c see Section 2.3, k = 1, 2. Relations 3.2 imply 2.5. Thus, according to our approach, for problem 1.5, 1.12 we present a two-δ-shocks weak asymptotic solution in form 2.7: ux,t, = u u kh uk x + k t,,, k=1

24 356 V.G. Danilov, V.M. Shelkovich / J. Differential Equations vx, t, = v k=1 v k H vk x + k t,, +e k t, δ vk x + k t,, +R k t, 1 x + k t, Ω k. 4.2 Let t = t > 0 be the time instant of interaction. In the interval t [0,t we have two δ-shock waves propagating without interaction. By Corollary 7, their phase functions k0 t andthe amplitudes of δ-functions e k0 t are defined by the system of equations k0 t = k0 0 + [f u] k t, e k0 t = ek 0 [u] + [f u] k [guv] k [v] k t, 4.3 k [u] k where by [hu, v] 1 = hu 0 + u 1 + u 2,v 0 + v 1 + v 2 hu 0 + u 2,v 0 + v 2, [hu, v] 2 = hu 0 + u 2,v 0 + v 2 hu 0,v 0 we denote jumps in function hu, v across the discontinuity curves x = 10 t, x = 20 t, respectively, k0 0 = x 0 k are initial positions of singularities, e k00 = e 0 k are initial amplitudes of δ-functions, k = 1, 2. By 4.3, before interaction, two δ-shock waves propagate across the lines x 1 = 10 t, x 2 = 20 t which intersect at the point with the coordinates: t x2 0 = u 1 u x0 1 2 u 2 fu 0 + u 1 + u 2 u 1 + u 2 f u 0 + u 2 + u 1 fu 0, x = fu0 + u 1 + u 2 fu 0 + u 2 u 2 x2 0 fu 0 + u 2 fu 0 u 1 x1 0. u 2 fu 0 + u 1 + u 2 u 1 + u 2 f u 0 + u 2 + u 1 fu We define the time instant of interaction t = t > 0 as a root of the equation ψ 0 t = 0, where ψ 0 t = 20 t 10 t is the distance between the fronts of noninteracting δ-shock waves. We write the weak asymptotic solution 4.2 in the form that potentially describes different scenarios of the processes that occur in the confluence of two free δ-shocks. Therefore, summarizing the above remarks, in order to describe the interaction dynamics, we will seek phases of δ-shocks andamplitudes of δ-functions as functions of

25 V.G. Danilov, V.M. Shelkovich / J. Differential Equations the fast variable fast time τ = ψ 0 t R andthe slow variable t 0: k t, def = k τ,t = k0 t + ψ 0 t k1 τ e k t, def = ê k τ,t = e k0 t + ψ 0 te k1 τ τ= ψ 0 t τ= ψ 0 t,, 4.5 where the functions k0 t, e k0 t are defined by Eqs. 4.3 for t [0, t ; for t [t, + these functions are defined by the same equations continuously extended for t t. The desired functions k1 τ, e k1 τ are corrections to the phases andthe amplitudes, respectively, rapidly varying during the time of interaction. We assume k1 τ, e k1 τ to be differentiable with respect to τ, k = 1, 2. Analogously to 4.5, we will seek the corrections R k t, as functions of the fast variable τ andthe slow variable t: R k t, def = R k τ,t= R k0 t + R k1 τ,t τ= ψ 0 t, 4.6 where, according to Theorem 5, the terms R k0 t are determined by the relations R k0 t = e k0t [f u]k a k, 4.7 c k [u] k and R k1 τ,t are desired functions, k = 1, 2. Here by 3.5, we have a 1 = gu 0 + u 1 ω 0u1 η + u 2 ω δ1 ηdη, c 1 = gu 0 + u 1 ω 0u1 η + u 2 Ω 1 ηdη = 0, a 2 = gu 0 + u 2 ω 0u2 η ω δ2 ηdη, c 2 = gu 0 + u 2 ω 0u2 η Ω 2 ηdη = Before interaction, as t < t, we have 10 t < 20 t and τ = ψ 0 t > 0, after interaction, as t > t, we have 10 t > 20 t and τ = ψ 0 t < 0. We set the following boundary conditions for the corrections to the phases and the amplitudes: k1 τ = 0, e k1 τ = 0, τ + τ + d k1 τ dτ = oτ 1, τ de k1 τ = oτ dτ τ

26 358 V.G. Danilov, V.M. Shelkovich / J. Differential Equations This means that the derivatives of the phases and amplitudes with respect to the fast variable τ tendto zero as τ, while the phases tendto zero as τ, i.e. before interaction. We assume that, analogously to 4.9, the following boundary conditions hold: R k1 τ,t = 0, R k1 τ,t = R k1, t, 4.10 τ + τ R k1 τ,t and R k1 τ,t, are bounded functions for all t 0, k = 1, 2. τ Finding the limit values of the corrections to the phases and the amplitudes, as τ for t>t k1 τ = k1,, τ e k1 τ = e k1,, τ we findthe limit values of the phases k t, andamplitudes e k t, : k, t = k τ,t = k0 t + ψ 0 t k1,, τ ê k, t = ê k τ,t = e k0 t + ψ 0 te k1, τ Thus, in fact, we determine the result of the interaction of δ-shocks as t>t. Theorem 9. Assume that conditions 1.15 are satisfied. Then for t [0,, the Cauchy problem 1.5 with the piecewise constant initial data 1.12, has a weak asymptotic solution 4.2, 4.5, 4.6, where functions k0 t, e k0 t, R k0 t are determined by the system of Eqs. 4.3, 4.7, and desired corrections are defined by the system: k1 τ = 1 k u k [f u]2 [u] 2 [f u] 1 [u] 1 τ τ B 2 ρτ dτ, k τ e k1 τ = B 2 ρτ [f u]2 [u] 2 [f u] 1 [u] 1 τ dτ v k k1 τ, R k1 τ,t = ê k τ,t C Rk 1 k 1 ρ fu0 + u k fu 0 + B k 1 k 1 ρ u k A k 1 k 1 ρ R k0 t, 4.14

27 V.G. Danilov, V.M. Shelkovich / J. Differential Equations where B 2 ρ and B k 1 k 1 ρ, A k 1 k 1 ρ, C Rk 1 k 1 ρ are the so-called interaction switch functions whose explicit form are given by formulas A.3 and A.8 from Appendix A, respectively, k = 1, 2. Here ρ = ρτ is a solution of the differential equation with the boundary condition: where dρ dτ = Fρ, Fρ = 1 + ρ = 1, 4.15 τ τ + 1u1 + 1 u2 B 2 ρ [f u] 2 [u] 2 [f u] 1 [u] Proof. 1 Ansatz substitution. Let us substitute the smooth ansatz 4.2 andthe weak asymptotics f ux,t,, g ux, t, vx, t,, which are given by relations A.2, A.7 from Appendix A, respectively, into system 1.5. Obviously, we obtain up to O D the following relations: L 1 [ux,t,]= 2 k=1 { u k k t, fu 0 + u k fu 0 B k 1 k 1 ρ } δ x + k t, + O D, 4.17 L 2 [ux, t,, vx, t, ] = 2 k=1 { v k k t, +ė k t, gu 0 + u k v 0 + v k gu 0 v 0 + B k 1 k 1 ρ δ x + k t, + e k t, k t, e k t, A k 1 k 1 ρ +R k t, C Rk 1 k 1 ρ } δ x + k t, + O D, 4.18 where ρ = ψt,, ψt, = 2 t, 1 t, is the distance between regularizations of the δ-shocks fronts 2 t, and 1 t,. Here the estimate O D is uniform with respect to ψt,. By equating the coefficients of δ, δ with zero in the right-handside of 4.17, 4.18, we obtain the necessary andsufficient conditions for the relations L 1 [ux,t,]=o D, L 2 [ux, t,, vx, t, ] =O D,

28 360 V.G. Danilov, V.M. Shelkovich / J. Differential Equations i.e. the generalizedrankine Hugoniot type conditions andthe system u k k t, = fu 0 + u k fu 0 + B k 1 k 1 ρ, 4.19 v k k t, +ė k t, = v 0 + v k gu0 + u k v 0 gu 0 + B k 1 k 1 ρ, e k t, k t, = e k t, A k 1 k 1 ρ + R k t, C Rk 1 k 1 ρ, 4.20 k = 1, 2. Systems describe functions k t,, e k t,, R k t,, which determine the weak asymptotics solution 4.2. According to our assumption, we will seek functions k t,, e k t,, R k t,, in form 4.5, 4.6 by introducing the dependence on into them, k = 1, 2. Form 4.5, 4.6 also reflects the structure of argument of interaction switch functions ρ = ψt, andthe structure of Eqs. 4.19, Let ψ 1 τ = 21 τ 11 τ, then the full phase difference is ψt, = ψ 0 t 1 + ψ 1 τ, and ρτ = ψt, = τ 1 + ψ 1 τ The derivatives of the phases and amplitudes with respect to time are given by the following equalities: d k t, dt de k t, dt def = d k τ,t dt def = dê kτ,t dt = k0 t + ψ 0 t d dτ [τ k1 τ], = ė k0 t + ψ 0 t d dτ [τe k1τ] Taking into account the boundary conditions 4.9, we findthe limit values of the phases andtheir derivatives with respect to time as τ for t>t : d k τ,t dt = d k, t, dt d êk τ,t dt = d ê k, t, 4.23 dt where k, t, ê k, t are defined by By substituting 4.5, 4.22, 4.6 into 4.19 and4.20, we obtain for all t 0 and τ R the generalizedrankine Hugoniot type conditions: 10 t + ψ 0 t d dτ τ 11 τ = fu 0 + u 1 fu 0 + B 1 ρ, u 1

29 V.G. Danilov, V.M. Shelkovich / J. Differential Equations t + ψ 0 t d dτ andthe following systems of equations: ė 10 t + ψ 0 t d dτ τ 21 τ = fu 0 + u 2 fu 0 + B 2 ρ, 4.24 u 2 τe 11 τ = v 0 + v 1 gu0 + u 1 v 0 gu 0 ė 20 t + ψ 0 t d dτ fu0 + u 1 fu 0 + B 1 ρ + B 1 ρ v 1, u 1 τe 21 τ = v 0 + v 2 gu0 + u 2 v 0 gu 0 fu0 + u 2 fu 0 + B 2 ρ + B 2 ρ v 2, 4.25 u 2 fu0 + u 1 fu 0 + B 1 ρ R 10 t + R 11 τ,t C R1 ρ = ê 1 τ,t A 1 ρ, u 1 R 20 t + R 21 τ,t C R2 ρ fu0 + u 2 fu 0 + B 2 ρ = ê 2 τ,t u 2 A 2 ρ, 4.26 u 2 with the boundary conditions 4.9, Here k0 t, e k0 t, R k0 t for all t 0 are defined by systems 4.3, 4.7. Subtracting the one Rankine Hugoniot type condition 4.24 from the other, we reduce system 4.24 to the differential equation with the boundary condition 4.15 dρ dτ = Fρ, ρ = 1, τ τ + where Fρ = 1 fu0 + u 2 fu 0 + B 2 ρ fu 0 + u 1 fu 0 + B 1 ρ, ψ 0 t u 2 u 1 andaccording to 4.3 ψ 0 t = fu 0 + u 2 fu 0 u 2 fu 0 + u 1 + u 2 fu 0 + u 2 u