Smooth Approximations of Global in Time Solutions to Scalar Conservation Laws

Abstract

We construct global smooth approximate solution to a scalar conservation law with arbitrary smooth monotonic initial data. Different kinds of singularities interactions which arise during the evolution of the initial data are described as well. In order to solve the problem, we use and further develop the weak asymptotic method, recently introduced technique for investigating nonlinear waves interactions.

1. Introduction

In the current paper we present an approach for constructing uniform and global in time approximate solutions to a Cauchy problem for one dimensional scalar conservation law with arbitrary smooth nonlinear convex flux. This approximating solution (see Definition 2.1) is continuous, piecewise smooth for 𝜀≥0 (𝜀 is a regularization parameter) and tends to an admissible weak solution [1] of the corresponding Cauchy problem. More precisely, we consider the problem𝜕𝑢+𝜕𝑡𝜕𝑓(𝑢)𝜕𝑥=0,𝑓^′′>0,(1.1)  𝑢|_𝑡=0=𝑣(𝑥)∈𝐶(𝐑),(1.2) where we also assume that 𝑣 is piecewise monotonic. It is well known that, since 𝑓^′′>0, in the interval where the initial data 𝑣(𝑥) decreases, the shock wave will arise sooner or later (see Figure 3). In order to be more precise in our considerations, we will assume that the initial data are decreasing everywhere. The same procedure can be applied if the initial data are continuous piecewise monotonic functions.

Also, for the initial data we will assume that they are such that for every fixed 𝑡∈𝐑⁺ there exists at most finite number of points of the gradient catastrophe (for more precise explanation see beginning of the next section).

Note that a smooth approximating solution to the problem under consideration was firstly constructed by Il'in [2] (with initial data which are such that there exists exactly one point of the gradient catastrophe along entire time axis; see (3.4), (3.5), (3.6) and corresponding assumptions). The starting point of his construction is the viscosity regularization of the considered conservation law. Using this regularization, the author in [3] constructs a global approximating solution via a set of functional series which are defined in appropriate domains in 𝐑⁺×𝐑. Then, he shows that every two such series match in the domains where they are both defined. His method is known as the matching method.

We mention also two very famous methods for a construction of approximate solution to conservation laws based on the piecewise constant approximations—Glimm scheme [4] and the wave front tracking [5].

Here, we use different technique, so-called the weak asymptotic method [6–14]. In the framework of our approach, the process of shock wave formation is considered as interaction of weak discontinuities, that is, nonlinear waves whose derivatives are the Heaviside type functions.

We stress that the formulas which we will give here are much simpler than the ones obtained by using the matching method. More precisely, our approximate solution is found almost in the same way as the classical solution of a Cauchy problem, by using the method of characteristics.

However, unlike the standard characteristics, we use so-called “new characteristics” which do not intersect in the moment of bifurcation, but they remain on the distance 𝒪(𝜀), where 𝜀 is a parameter of approximation.

The difference between standard and new characteristics is plotted on Figure 2 (one of possible scenarios). As one can see from there, the new characteristics do not intersect each other except ones emanating from the intervals (𝑎₁,𝑎₁+𝜎) and (𝑎₂−𝜎,𝑎₂). More precisely, we will allow the intersection to happen only when characteristics bear the same information. In the case plotted on Figure 2, this means that initial data are constant on the intervals (𝑎₁,𝑎₁+𝜎) and (𝑎₂−𝜎,𝑎₂).

Therefore, we can go back along the trajectory given by a “new characteristics” and thus obtain value of the approximate solution at every point (𝑡,𝑥)∈𝐑⁺×𝐑^𝑑. Of course, the main problem is how to find analytic expression for the new characteristics (see [6, 7, 10–12]).

Still, unlike the situation that we had in [6, 7, 10–12] with simple initial data providing the shock wave to be the same once it is formed (more precisely, to be of constant strength; see Figure 1), here we have more complicated situation. Namely, in the case of initial data (1.2), the shock wave increases the strength after its formation. The increase will not start immediately since in the left neighborhood of 𝑎₂ and right neighborhood of 𝑎₁ the initial data are constant. Therefore, unlike the situation from [6, 7, 10–12], we also need to modify characteristics emanating out of the interval (𝑎₂−𝜎,𝑎₁+𝜎), and this issue is far from being trivial.

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Figure 1 

Evolution of the initial data considered in the previous works.

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Figure 2 

Standard characteristics are plotted on the left coordinate system. The new characteristics are plotted on the right coordinate system. The discontinuity line of an admissible weak solution is dashed on both plots. The points $a_2-\sigma <a_1+\sigma$ are hollowed on the $x$-axis.

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Figure 3 

Evolution of a decreasing initial data.

Next important moment of the approach presented here is the following. In one-dimensional case, the solution 𝑢 for quasilinear equation (1.1) is an impulse for the corresponding Hamilton-Jacobi equation (see Example 5.2) and the shock wave formation (or the gradient catastrophe) denotes appearance of singularity for the projection mapping of the corresponding Lagrange manifold on the 𝑥-axis. In the linear hyperbolic case this situation is described by using Maslov's canonical operator (integral Fourier operator) [15–18] and entire Lagrange manifold.

In the linear parabolic case (corresponding to the flux 𝑓(𝑢)=𝑢²), it is also possible to apply the method of tunnel canonical Maslov's operator [15–17] and for the construction of the solution only “essential” part of the Lagrange manifold (not the entire one) is used (see [15–18]). This essential part exactly coincides with the shock wave profile in one-dimensional case. Approximative analytic description of that “essential” part of the Lagrange manifold is given in this paper.

In the sequel we will often use the notion of smooth description in 𝑡∈𝐑⁺ of a process. Therefore, we give formal definition of the notion. Definition 1.1. By smooth in 𝑡∈𝐑⁺ description of a process one implies function which is smooth in 𝑡∈𝐑⁺ and approximately (in the weak sense) solves an equation that governs the process.

The paper is organized as follows.

In Section 2, we give basic notions and notations of the weak asymptotic method and describe two problematic situations arising when solving the problem. In Section 3, we resolve the first problematic situation that arises in the construction of global approximating solution to the considered problem—we describe smoothly in 𝑡∈𝐑⁺ the shock wave formation process from continuous initial data. If we would assume that we have only one shock wave formed from initial data like in [2] (i.e., if we would have situation similar to one plotted in Figure 2, this section would be enough). However, since we can have two shock waves formed on different places on 𝑥 axis, we need to describe smoothly in 𝑡∈𝐑⁺ interaction of the two shock waves. This is done in Section 4. In Section 5, we use results from previous two sections to describe global approximating solution to the considered problem.

2. Notions, Notations, and Further Explanations

We give basic definitions and fundamental theorem of the method that we are going to use—the weak asymptotic method.Definition 2.1. By 𝑂_𝒟′(𝜀^𝛼)⊂𝒟^′(𝐑), 𝛼∈𝐑, one denotes the family of distributions depending on 𝜀∈(0,1) and 𝑡∈𝐑⁺ such that for any test function 𝜂(𝑥)∈𝐶¹₀(𝐑), the estimate⟨𝑂_𝒟′(𝜀^𝛼),𝜂(𝑥)⟩=𝑂(𝜀^𝛼),𝜀→0,(2.1)holds, where the estimate on the right-hand side is understood in the usual Landau sense and locally uniformly in 𝑡, that is, |𝑂(𝜀^𝛼)|≤𝐶_𝑇𝜀^𝛼 for 𝑡∈[0,𝑇].Definition 2.2. The family of functions (𝑢_𝜀)_𝜀=(𝑢_𝜀(𝑥,𝑡))_𝜀 is called a weak asymptotic solution of (1.1) and (1.2) if there exists 𝛼>0 such that in the space 𝐶¹(𝐑⁺;𝒟^′(𝐑)), one has𝑢_𝜀𝑡+(𝑓(𝑢_𝜀))_𝑥=𝒪_𝒟′(𝜀^𝛼),𝑢_𝜀|_𝑡=0−𝑣=𝒪_𝒟′(𝜀^𝛼),𝜀→0.(2.2)The following theorem is the basic theorem of the method. We also call it the nonlinear superposition law.Theorem 2.3 (see [10]). Suppose that the functions 𝜔_𝑖∈𝐶^∞(𝐑), 𝑖=1,2, satisfy lim_𝑧→+∞𝜔_𝑖(𝑧)=1, lim_{𝑧→−∞}𝜔_𝑖(𝑧)=0 and 𝑑𝜔_𝑖(𝑧)/𝑑𝑧∈𝒮(𝐑) where 𝒮(𝐑) is the Schwartz space of rapidly decreasing functions. For the bounded functions 𝑎,𝑏,𝑐 defined on 𝐑⁺×𝐑, one has 𝑓(𝑎+𝑏𝜔₁(𝜑₁−𝑥𝜀)+𝑐𝜔₂(𝜑₂−𝑥𝜀))=𝑓(𝑎)+𝜃(𝜑₁−𝑥)(𝑓(𝑎+𝑏+𝑐)𝐵₁+𝑓(𝑎+𝑏)𝐵₂−𝑓(𝑎+𝑐)𝐵₁−𝑓(𝑎)𝐵₂)+𝜃(𝜑₂−𝑥)(𝑓(𝑎+𝑏+𝑐)𝐵₂−𝑓(𝑎+𝑏)𝐵₂+𝑓(𝑎+𝑐)𝐵₁−𝑓(𝑎)𝐵₁)+𝒪_𝒟′(𝜀),(2.3)where 𝜃 is the Heaviside function and 𝐵_𝑖=𝐵_𝑖((𝜑₂−𝜑₁)/𝜀), 𝑖=1,2, and for 𝜌∈𝐑 one has𝐵₁(𝜌)=∫𝜔̇₁(𝑧)𝜔₂(𝑧+𝜌)𝑑𝑧,𝐵₂(𝜌)=∫𝜔̇₂(𝑧)𝜔₁𝐵(𝑧−𝜌)𝑑𝑧,₁(𝜌)+𝐵₂(𝜌)=1.(2.4)

At the beginning we make some remarks. Consider the point 𝑥₀∈𝐑 such that𝑡^∗=min_𝑥∈𝐑−1𝑓^′′(𝑢₀(𝑥))𝑢^′₀1(𝑥)=−𝑓^′′(𝑢₀(𝑥₀))𝑢^′₀(𝑥₀).(2.5)Assume for the simplicity that such 𝑥₀∈𝐑 is unique. In that case the shock wave will firstly arise on the characteristics along which the point 𝑥₀ moves. More precisely, blow up of the classical solution happens on the height 𝑢₀(𝑥₀) in the moment 𝑡^∗ and in the point 𝑥^∗ (i.e., initial data changes as plotted on Figure 3). The point (𝑥^∗,𝑡^∗) we call the point of the gradient catastrophe.

In order to describe phenomenon of the gradient catastrophe smoothly in 𝑡∈𝐑⁺, on the first step we define the function 𝑢₁(𝑥) such that𝑓′(𝑢₁(𝑥))=−𝐾𝑥+𝑏,𝑥∈𝐑,(2.6)where 𝐾 and 𝑏 are constants determined from the conditions𝑢₁(𝑥₀−𝜀^𝜇)=𝑣(𝑥₀−2𝜀^𝜇),𝑢₁(𝑥₀+𝜀^𝜇)=𝑣(𝑥₀+2𝜀^𝜇),0<𝜇<1.(2.7)After that, we replace the piece of original initial data 𝑣(𝑥) in a small neighborhood (𝑥₀−2𝜀^𝜇,𝑥₀+2𝜀^𝜇) of the point 𝑥₀ by the function 𝑢₁(𝑥), 𝑥∈(𝑥₀−𝜀^𝜇,𝑥₀+𝜀^𝜇) and by constants in the intervals (𝑥₀−2𝜀^𝜇,𝑥₀−𝜀^𝜇) and (𝑥₀+𝜀^𝜇,𝑥₀+2𝜀^𝜇) so that the continuity is preserved (see Figure 4). This change of initial condition provides the shock wave to arise from the interval (𝑥₀−𝜀^𝜇,𝑥₀+𝜀^𝜇) in the moment 𝑡^∗=1/𝐾. The amplitude of the shock wave is going to be |𝑣(𝑥₀−2𝜀)−𝑣(𝑥₀+2𝜀)|. On the other hand, for 𝑡<𝑡^∗, the solution 𝑢_𝜀 to our problem (with transformed initial data) will be continuous function (see Figure 4 again).

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Figure 4 

Evolution of the approximated initial data. The points $x_0-{\epsilon }^{\mu }$ and $x_0+{\epsilon }^{\mu }$ are dotted on the first plot.

As we will see, with such new initial data it is much easier to find global smooth approximating solution. Namely, it appears that the most difficult issue here is to describe evolution of the initial data in the moment of blowing up of the classical solution, and that it is much easier to do it if we have “line” of gradient catastrophe (in this case it is the line (𝑥₀−𝜀^𝜇,𝑥₀+𝜀^𝜇); see Figure 4) than the point of gradient catastrophe (in the case plotted in Figure 3 it is the point 𝑥₀). We address the reader to [2, 3] in order to understand difficulties which are caused if we have only a point of gradient catastrophe.

The second problematic situation which we meet is shock waves interaction. This is simpler case and we can describe this process smoothly in 𝑡∈𝐑⁺ by direct substitution of an ansatz into the equation, and then applying the weak asymptotic formulas (see Section 4).

3. Formation of the Shock Wave with Nonconstant Amplitude

We return to (1.1) and (1.2). Before we begin, we introduce the notations that we will use. First of all, we imply everywhere 𝑡∈ℝ⁺, 𝑥∈ℝ. Next,𝑢₁=𝑢₁(𝑥,𝑡,𝜀),𝑈=𝑈(𝑥,𝑡),𝑢₀=𝑢₀𝐵(𝑥,𝑡),_𝑖=𝐵_𝑖(𝜌),𝜙_𝑖=𝜙_𝑖𝑥(𝑡,𝜀),𝑖=1,2,_0,𝑅=𝑥_0,𝑅(𝑡,𝜀),𝑥_0,𝐿=𝑥_0,𝐿𝜃(𝑡,𝜀),₁=𝜃(𝜙₁+𝑥_0,𝑅−𝑥),𝜃₂=𝜃(𝜙₂−𝑥_0,𝐿𝛿−𝑥),_𝑖=−(𝜃_𝑖)^′_𝑥𝑡,𝑖=1,2,_∗=𝑎₁−𝑎₂𝑓′(𝑈⁰)−𝑓′(𝑢⁰₀)=1𝐾,𝑥^∗=𝑓′(𝑈⁰)𝑡^∗+𝑎₂=𝑓′(𝑢⁰₀)𝑡^∗+𝑎₁=𝑓′(𝑈⁰)𝑎₁−𝑓′(𝑢⁰₀)𝑎₂𝑓′(𝑈⁰)−𝑓′(𝑢⁰₀),(3.1)where 𝜃 is the Heaviside function and 𝐵_𝑖, 𝑖=1,2, are from Theorem 2.3. The remaining functions will be defined in what follows.

On the first step, we assume that the function 𝑣(𝑥) from (1.2) has the form (see Figure 4 with 𝜎=𝜀^𝜇)𝑈𝑣(𝑥)=⁰(𝑥),𝑥<𝑎₂𝑈−𝜎,₀(𝑎₂−𝜎),𝑎₂−𝜎<𝑥<𝑎₂,𝑢₁₀(𝑥),𝑎₂<𝑥<𝑎₁,𝑢₀₀(𝑎₁+𝜎),𝑎₁<𝑥<𝑎₁𝑢+𝜎,₀₀(𝑥),𝑥>𝑎₁+𝜎,(3.2)where 𝜎 is a positive constant. Furthermore, the functions 𝑢₀₀,𝑢₁₀, and 𝑈₀ are nonincreasing and smooth, and they satisfy𝑢₀₀(𝑥)=𝑢₀₀(𝑎₁+𝜎),𝑥>𝑎₁𝑈+𝜎,₀(𝑥)=𝑈₀(𝑎₂−𝜎),𝑥<𝑎₂−𝜎,𝑓′(𝑢₁₀(𝑥))=−𝐾𝑥+𝑏,𝑥∈[𝑎₂,𝑎₁𝑢],₁₀(𝑎₂)=𝑢₀₀(𝑎₂+𝜎)=𝑢⁰₀,𝑢₁₀(𝑎₁)=𝑈₀(𝑎₁−𝜎)=𝑈⁰,𝑎₁>𝑎₂,−1𝑓^′′(𝑢₀₀(𝑥))𝑢^′₀₀(𝑥₀)1≥𝐶>𝐾,𝑥<𝑎₂,−1𝑓^′′(𝑈₀(𝑥))𝑈^′₀(𝑥₀)1≥𝐶>𝐾,𝑥>𝑎₁.(3.3)This assumption is here in order to obtain the situation such that on characteristics emanating from the intervals (−∞,𝑎₂) and (𝑎₁,∞) we can have gradient catastrophe only for 𝑡≥𝐶>1/𝐾 (i.e., after the straightening the curve connecting the points (𝑎₁,𝑣(𝑎₁)) and (𝑎₂,𝑣(𝑎₂))).

In order to clarify as much as possible the presentation, in this section we will assume more then this. Namely, it is well known that for 𝑡>𝑡^∗=1/𝐾 the solution of (1.1) and (3.2) will admit the shock wave moving along the line 𝑥=𝑐(𝑡) given by the Cauchy problem defining the Rankine-Hugoniot conditions𝑐′(𝑡)=𝑓(𝑈(𝑐(𝑡),𝑡))−𝑓(𝑢₀(𝑐(𝑡),𝑡))𝑈(𝑐(𝑡),𝑡)−𝑢₀(𝑐(𝑡),𝑡),𝑡≥𝑡^∗,𝑐(𝑡^∗)=𝑥^∗=𝑓′(𝑈⁰)𝑡^∗+𝑎₂=𝑓′(𝑢⁰₀)𝑡^∗+𝑎₁,𝑡≤𝑡^∗.(3.4)Here, 𝑈 and 𝑢₀ are classical solutions to the following Cauchy problems, respectively (in a subdomain of 𝐑⁺×𝐑 where they exist)𝜕_𝑡𝑢₀+𝜕_𝑥𝑓(𝑢₀)=0,𝑢|_𝑡=0=𝑢₀₀(𝑥),(3.5) 𝜕_𝑡𝑈+𝜕_𝑥𝑓(𝑈)=0,𝑢|_𝑡=0=𝑈₀(𝑥).(3.6)

So, in the sequel, we will assume that the functions 𝑢₀,𝑈, and 𝑐 satisfying (3.4), (3.5), and (3.6) are well defined (the same is done in [2]).

Before we globally define the “new characteristics” (denoted 𝑥(𝑥₀,𝑡,𝜀)), we need to define extremal “new characteristics,” that is, the ones emanating from 𝑎₂+𝒪(𝜀) and 𝑎₁+𝒪(𝜀). The proof is relatively simple and it relies on the basic ODE theory.Lemma 3.1. The curves 𝜙₂=𝑥(𝑥₀₂,𝑡,𝜀) and 𝜙₁=𝑥(𝑥₀₁,𝑡,𝜀) for 𝑥₀₂=𝑎₂−𝜀((𝑎₁−𝑎₂)/2) and 𝑥₀₁=𝑎₁+𝜀((𝑎₁−𝑎₂)/2) are given by the following Cauchy problems:𝑑𝜙₁𝑑𝑡=(𝐵₂(𝜌)−𝐵₁(𝜌))𝑓′(𝑢₀(𝜙₁,𝑡))+𝑐(𝑡)𝐵₁(𝜌),𝜙₁(0,𝜀)=𝑎₁𝑎+𝜀𝐴¹−𝑎₂2,𝑑𝜙₂𝑑𝑡=(𝐵₂(𝜌)−𝐵₁(𝜌))𝑓′(𝑈(𝜙₂,𝑡))+𝑐(𝑡)𝐵₁(𝜌),𝜙₂(0,𝜀)=𝑎₂𝑎−𝜀𝐴¹−𝑎₂2.(3.7)Introduce the function𝜏=𝑓′(𝑈⁰)𝑡+𝑎₂−𝑓′(𝑢⁰₀)𝑡−𝑎₁𝜀=𝜓₀(𝑡)𝜀,(3.8)describing the relation between standard characteristics emanating from 𝑎₂ and 𝑎₁, respectively.
The function𝜌 given by𝜙𝜌(𝑡,𝜏)=²(𝑡,𝜀)−𝜙₁(𝑡,𝜀)𝜀,(3.9)describing the distance between two nonintersecting curves 𝜙₂ and 𝜙₁, satisfies the following Cauchy problem𝜌̇=1−2𝐵₁(𝜌),lim_{𝜏→−∞}𝜌𝜏=1,(3.10)where 𝜌̇=𝜕_𝜏𝜌 and it tends to a constant 𝜌₀ as 𝜏→∞.Proof. The proof follows from the definition of the curves 𝑥(𝑥₀₁,𝑡) and 𝑥(𝑥₀₂,𝑡), and standard ODE theory (see Figure 5 and put 𝐹(𝜌,⋅)=1−2𝐵₁(𝜌)). For details one can consult [11].

Figure 5 

The curve represents solution of (4.17). Dot on the $\rho$ axis, denoted by ${\rho }_0$, is the minimal zero of the function $F(\rho \mathrm{,}\cdot )$.

After inessential changes caused by replacing the constant 𝑐 by the function 𝑐(𝑡), the proof of the following lemma can be found in the frame of [11, Theorem 5].Lemma 3.2. Consider the set of solutions 𝑥=𝑥(𝑥₀,𝑡), 𝑥₀∈[𝑎₂,𝑎₁], to the following Cauchy problem𝑥̇=(𝐵₂(𝜌)−𝐵₁(𝜌))𝑓′(𝑢₁)+𝑐(𝑡)𝐵₁(𝜌),𝑢̇₁𝑢=0,₁(0)=𝑢₁₀(𝑥₀),𝑥(0)=𝑥₀+𝜀𝐴(𝑥₀−𝑎₁+𝑎₂2),𝑥₀∈[𝑎₂,𝑎₁],(3.11)where 𝐴 is a constant large enough (which provides global solvability of the implicit equation 𝑥=𝑥(𝑥₀,𝑡,𝜀), 𝑡>0, 𝜀>0; see [11, Theorem 5]). The curves 𝑥(𝑥₀,𝑡,𝜀), 𝑥₀∈[𝑎₂,𝑎₁] one calls the “new characteristics” emanating from the interval (𝑎₂−𝜀𝐴((𝑎₁−𝑎₂)/2),𝑎₁+𝜀𝐴((𝑎₁−𝑎₂)/2)).
Then, for arbitrary two𝑥₀₁,𝑥₀₂∈[𝑎₂,𝑎₁], the curves 𝑥(𝑥₀₁,𝑡,𝜀) and 𝑥(𝑥₀₂,𝑡,𝜀) are nonintersecting.Note that we extended a little bit the interval [𝑎₂,𝑎₁]. This is necessary in order to prove that the “new characteristics” do not mutually intersect (see [11]). Also note that this does not affect the weak asymptotic solution of the problem since we perturbed initial function for 𝒪(𝜀).

The previous lemma gives a formula for determining the “new characteristics” emanating from the interval [𝑎₂,𝑎₁], and states that they do not intersect each other along entire time axis. Remark that the “new characteristics” emanate from the interval (𝑎₂−𝜀𝐴((𝑎₁−𝑎₂)/2),𝑎₁+𝜀𝐴((𝑎₁−𝑎₂)/2)) and that we state that only them are non-intersecting. The characteristics emanating out of the interval (𝑎₂−𝜀𝐴((𝑎₁−𝑎₂)/2),𝑎₁+𝜀𝐴((𝑎₁−𝑎₂)/2)) are standard and they can intersect each other. Still, it will happen “late” enough (see (3.4)–(3.6)).

In order to define the new characteristics along entire 𝑥-axis, we introduce the following notationsΨ_𝐿(𝑥₀,𝑡,𝜀)=𝑓′(𝑈₀(𝑥₀))𝑡+𝑥₀+𝜀(𝑥₀−𝑎₁−𝑎₂2),𝑥₀<𝑎₂Ψ−𝜎,_𝑅(𝑥₀,𝑡,𝜀)=𝑓′(𝑢₀₀(𝑥₀))𝑡+𝑥₀+𝜀(𝑥₀−𝑎₁−𝑎₂2),𝑥₀>𝑎₁+𝜎.(3.12)Note that Ψ_𝐿 and Ψ_𝑅 are standard characteristics emanating from the intervals (𝑎₂−𝜎,𝑎₂) and (𝑎₁,𝑎₁+𝜎), respectively. Now, we can define the functions representing the “new characteristics” emanating out of (𝑎₂−𝜎,𝑎₁+𝜎).

For 𝑥₀<𝑎₂−𝜎 we put𝑥_𝐿(𝑡,𝑥₀Ψ,𝜀)=^𝐿(𝑥₀,𝑡,𝜀),𝜙₂(𝑡,𝑥₀,𝜀)−Ψ_𝐿(𝑡,𝑥₀,𝜀)>𝜀(𝑎₂−𝜎−𝑥₀𝜙),₂(𝑡,𝑥₀,𝜀)−𝜀(𝑎₂−𝑥₀),𝜙₂(𝑡,𝑥₀,𝜀)−Ψ_𝐿(𝑡,𝑥₀,𝜀)≤𝜀(𝑎₂−𝜎−𝑥₀).(3.13)Similarly, for 𝑥₀>𝑎₁+𝜎,𝑥_𝑅(𝑡,𝑥₀Ψ,𝜀)=^𝑅(𝑥₀,𝑡,𝜀),Ψ_𝑅(𝑡,𝑥₀,𝜀)−𝜙₁(𝑡,𝑥₀,𝜀)>𝜀(𝑥₀−𝑎₁𝜙−𝜎),₁(𝑡,𝑥₀,𝜀)−𝜀(𝑎₂−𝑥₀),Ψ_𝑅(𝑡,𝑥₀,𝜀)−𝜙₁(𝑡,𝑥₀,𝜀)≤𝜀(𝑥₀−𝑎₁−𝜎).(3.14)For better understanding of the previous formulas see Figure 2. Actually, 𝑥_𝐿 as well as 𝑥_𝑅 are equal to the standard characteristics before they come “close” to the shock curve. After that, they are parallel to the shock curve.

Finally, we can write formula for the “new characteristics” which holds for 𝑥₀∈𝐑⧵((𝑎₂−𝜎,𝑎₂)∪(𝑎₁,𝑎₁+𝜎)) (and this is set for which we need the “new characteristics”):𝑋(𝑡,𝑥₀𝑥,𝜀)=^𝑅(𝑡,𝑥₀,𝜀),𝑥₀≤𝑎₂−𝜎,𝑥(𝑡,𝑥₀,𝜀),𝑎₂≤𝑥₀≤𝑎₁,𝑥_𝐿(𝑡,𝑥₀,𝜀),𝑥₀>𝑎₁,(3.15)where 𝑥(𝑥₀,𝑡,𝜀) is defined in Lemma 3.2. Denote by ^0𝑥0𝑥^02𝑑𝑐₀=𝑥^{0𝑥002𝑑𝑐}₀(𝑥,𝑡,𝜀) the inverse function to the function 𝑥=𝑋(𝑥₀,𝑡,𝜀), 𝑡>0, 𝜀>0, of the “new characteristics” defined by (3.15). Clearly, the function ^0𝑥0𝑥^02𝑑𝑐₀ is not defined in the regionsΓ_𝐿={(𝑥,𝑡)∶0<𝑡<𝑡^∗+𝑀𝜎,𝑓′(𝑈⁰₀)𝑡+𝑎₂−𝜎<𝑥<𝜙₂Γ},_𝑅={(𝑥,𝑡)∶0<𝑡<𝑡^∗+𝑀𝜎,𝑓′(𝑢⁰₀₀)𝑡+𝑎₁+𝜎>𝑥>𝜙₁}.(3.16)Therefore, we introduce the following extension of ^0𝑥0𝑥^02𝑑𝑐₀:𝑥₀(𝑥,𝑡,𝜀)=^{0𝑥002𝑑𝑐}𝑥₀(𝑥,𝑡,𝜀),(𝑥,𝑡)∉Γ_𝐿∪Γ_𝑅,𝑓′(𝑈⁰₀)𝑡+𝑎₂−𝜎,(𝑥,𝑡)∈Γ_𝐿,𝑓′(𝑢⁰₀₀)𝑡+𝑎₁+𝜎,(𝑥,𝑡)∈Γ_𝑅.(3.17)

To continue, assume that the increase of strength of the shock wave starts in the moment 𝑡^∗+𝑀𝜎 for a constant 𝑀>0 (actually, this is the moment when standard characteristics emanating out of the interval [𝑎₂−𝜎,𝑎₁+𝜎] start to intersect with the shock curve). Then, we introduce the functions 𝑥_0,𝐿(𝑡,𝜀) and 𝑥_0,𝑅(𝑡,𝜀) so that they are equal to zero for 𝑡<𝑡^∗+𝑀𝜎 and defined for 𝑡≥𝑡^∗+𝑀𝜎 as follows:

(i)𝑥_0,𝐿(𝑡,𝜀)=𝜀(𝑎₂−𝑥₀) for 𝑥₀<𝑎₂−𝜎 such that 𝜙₂(𝑡,𝑥₀,𝜀)−Ψ_𝐿(𝑡,𝑥₀,𝜀)=𝜀(𝑎₂−𝑥₀),(ii)𝑥_0,𝑅(𝑡,𝜀)=𝜀(𝑥₀−𝑎₁) for 𝑥₀>𝑎₁+𝜎 such that Ψ_𝑅(𝑡,𝑥₀,𝜀)−𝜙₁(𝑡,𝑥₀,𝜀)=𝜀(𝑥₀−𝑎₁).

Note that the functions 𝑥_0,𝐿 and 𝑥_0,𝑅 have the same regularity as the functions 𝑈₀ and 𝑢₀₀, respectively. Therefore,𝜕𝑥_0,𝑅=𝜕𝑡𝜕𝑥_0,𝐿𝜕𝑡=𝒪(𝜀).(3.18)

The following theorem is a slight modification of Theorem 5 from [11]. The motivation for the procedure can be found in [11].

Theorem 3.3. The weak asymptotic solution of problem (1.1) and (1.2) has the form𝑢_𝜀(𝑥,𝑡)=𝑢₀(𝑥,𝑡)+(𝑢₁(𝑥,𝑡,𝜀)−𝑢₀(𝑥,𝑡))𝜔₁(𝜙₁(𝑡,𝜀)+𝑥_0,𝑅(𝑡,𝜀)−𝑥𝜀)+(𝑈(𝑥,𝑡)−𝑢₁(𝑥,𝑡,𝜀))𝜔₂(𝜙₂(𝑡,𝜀)−𝑥_0,𝐿(𝑡,𝜀)−𝑥𝜀),(3.19)where 𝜔_𝑖, 𝑖=1,2, satisfy the conditions from Theorem 2.3 and the functions 𝜙_𝑖=𝜙_𝑖(𝑡,𝜀), 𝑡∈𝐑⁺, 𝑖=1,2, are given in Lemma 3.1.
The function 𝑢₁(𝑥,𝑡,𝜀) is given by𝑢₁(𝑥,𝑡,𝜀)=𝑈(𝑥,𝑡),𝑥<𝜙₂−𝑥_0,𝐿𝑢(𝑡,𝜀),₀(𝑥₀(𝑥,𝑡,𝜀)),𝜙₂−𝑥_0,𝐿(𝑡,𝜀)≤𝑥≤𝜙₁+𝑥_0,𝑅𝑢(𝑡,𝜀),₀(𝑥,𝑡),𝑥>𝜙₁+𝑥_0,𝑅(𝑡,𝜀),(3.20)where 𝑥₀(𝑥,𝑡,𝜀) is defined by (3.17).
The functions 𝑈 and 𝑢₀ are classical solutions to Cauchy problems (3.5) and (3.6), respectively.Proof. The proof completely follows the construction from [11]:
(a)we substitute (3.19) in (1.1);(b)we use formula (2.3);(c)we divide the real line on three intervals (−∞,𝜙₂−𝑥_0,𝐿], (𝜙₂−𝑥_0,𝐿,𝜙₁+𝑥_0,𝑅), and [𝜙₁+𝑥_0,𝑅,+∞). In that way we get three equations in each of the intervals, and we solve them separately in each of them.
We substitute the function 𝑢_𝜀(𝑥,𝑡) from (3.19) in (1.1). Using Theorem 2.3 we get[𝜕𝑢₀+𝜕𝑡𝜕𝑓(𝑢₀)𝜕𝑥](1−𝜃₁)+(𝜕𝑢₁𝜕𝑡+[(𝐵₂−𝐵₁)𝑓′(𝑢₁)+2𝑐′(𝑡)𝐵₁]𝜕𝑢₁𝜕𝑥)(𝜃₁−𝜃₂)+𝐵₁[𝑑𝑑𝑥(𝑓(𝑈+𝑢⁰₀−𝑢₁)+𝑓(𝑢₁)−2𝑐′(𝑡)𝑢₁)](𝜃₁−𝜃₂)+[𝜕𝑈+𝜕𝑡𝜕𝑓(𝑈)𝜕𝑥]𝜃₂+((𝑢₁−𝑢₀)𝜙_1𝑡−𝐵₂(𝑓(𝑢₁)−𝑓(𝑢₀))−𝐵₁(𝑓(𝑈)−𝑓(𝑈+𝑢₀−𝑢₁)))𝛿₁+((𝑈−𝑢₁)𝜑_2𝑡−𝐵₂(𝑓(𝑈)−𝑓(𝑢₁))−𝐵₁(𝑓(𝑈+𝑢₀−𝑢₁)−𝑓(𝑢₀)))𝛿₂=𝒪_𝒟′(𝜀).(3.21)For more detailed computation see [11].
Considering the previous expression for 𝑥∈(−∞,𝜙₂−𝑥_0,𝐿) and 𝑥∈(𝜙₁+𝑥_0,𝑅,+∞) we get the following equations:𝜕𝑈𝜕𝑡+𝑓′(𝑈)𝜕𝑈𝜕𝑥=𝒪_𝒟′(𝜀),𝑥∈(−∞,𝜙₂−𝑥_0,𝐿),𝜕𝑢₀𝜕𝑡+𝑓′(𝑢₀)𝜕𝑢₀𝜕𝑥=𝒪_𝒟′(𝜀),𝑥∈(𝜙₁+𝑥_0,𝑅,∞),(3.22)which is true by definition of the functions 𝑈 and 𝑢₀ (see (3.5) and (3.6)). Thus, (3.21) reduces to(𝜕𝑢₁𝜕𝑡+[(𝐵₂−𝐵₁)𝑓′(𝑢₁)+2𝑐′(𝑡)𝐵₁]𝜕𝑢₁𝜕𝑥)(𝜃₁−𝜃₂)+𝐵₁[𝑑𝑑𝑥(𝑓(𝑈+𝑢₀−𝑢₁)+𝑓(𝑢₁)−2𝑐′(𝑡)𝑢₁)](𝜃₁−𝜃₂)+((𝑢₁−𝑢₀)𝜕_𝑡(𝜙₁+𝑥_0,𝐿)−𝐵₂(𝑓(𝑢₁)−𝑓(𝑢₀))−𝐵₁(𝑓(𝑈)−𝑓(𝑈+𝑢₀−𝑢₁)))𝛿₁+((𝑈−𝑢₁)𝜕_𝑡(𝜙₂−𝑥_0,𝑅)−𝐵₂(𝑓(𝑈)−𝑓(𝑢₁))−𝐵₁(𝑓(𝑈+𝑢₀−𝑢₁)−𝑓(𝑢₀)))𝛿₂=𝒪_𝒟′(𝜀).(3.23)
We pass to the interval (𝜙₂−𝑥_0,𝐿,𝜙₁+𝑥_0,𝑅).
Notice that for 𝑡<𝑡^∗+𝜎𝑀 we have 𝑥_0,𝑅(𝑡,𝜀)=𝑥_0,𝐿(𝑡,𝜀)=0. Therefore, the situation is the same as in [11] (see proof of Theorem 5 from [11]).
Consider the interval (𝑡^∗+𝜎𝑀/2,𝑇) for a 𝑇>𝑡^∗+𝜎𝑀/2. In that interval we have (see again [10, 11])𝐵₂−𝐵₁=𝒪(𝜀^𝑁),𝑁∈𝐍,(3.24)and from here, since 𝐵₂+𝐵₁=1, we also have𝐵_𝑖=12+𝒪(𝜀^𝑁),𝑖=1,2,𝑁∈𝐍.(3.25)Having this in mind, from (3.23) we get(𝜕𝑢₁𝜕𝑡+𝑐′(𝑡)𝜕𝑢₁𝜕𝑥)(𝜃₁−𝜃₂)+12[𝑑𝑑𝑥(𝑓(𝑈+𝑢₀−𝑢₁)+𝑓(𝑢₁)−2𝑐′(𝑡)𝑢₁)](𝜃₁−𝜃₂)+((𝑢₁−𝑢₀)𝜕_𝑡(𝜙₁+𝑥_0,𝑅1)−2(𝑓(𝑢₁)−𝑓(𝑢₀1))−2(𝑓(𝑈)−𝑓(𝑈+𝑢₀−𝑢_𝐿)))𝛿₁+((𝑈−𝑢₁)𝜕_𝑡(𝜙₂−𝑥_0,𝐿1)−2(𝑓(𝑈)−𝑓(𝑢₁1))−2(𝑓(𝑈+𝑢₀−𝑢₁)−𝑓(𝑢₀)))𝛿₂=𝒪_𝒟′(𝜀).(3.26)Multiplying the last relation by a test function 𝜂∈𝐶¹₀(𝐑) and integrating over 𝐑 we get (recall that 𝑡 is fixed)(𝑢₁(𝜙₁+𝑥_0,𝑅,𝑡,𝜀)(𝜕_𝑡𝜙₁+𝜕_𝑡𝑥_0,𝐿)𝜂(𝜙₁+𝑥_0,𝑅,𝑡)−𝑢₁(𝜙₂−𝑥_0,𝐿,𝑡,𝜀)(𝜕_𝑡𝜙₂−𝜕_𝑡𝑥_0,𝑅)𝜂(𝜙₂−𝑥_0,𝐿,𝑡)−𝑐′(𝑡)𝑢₁(𝜙₁+𝑥_0,𝑅,𝑡,𝜀)𝜂(𝜙₁+𝑥_0,𝑅,𝑡)+𝑐′(𝑡)𝑢₁(𝜙₂−𝑥_0,𝐿,𝑡,𝜀)𝜂(𝜙₂−𝑥_0,𝐿+,𝑡))_{𝜙1+𝑥0,𝑅𝜙2−𝑥0,𝐿}12[𝑑𝑑𝑥(𝑓(𝑈+𝑢₀−𝑢₁)+𝑓(𝑢₁)−2𝑐′(𝑡)𝑢₁+)]𝜂(𝑥)𝑑𝑥_𝐑((𝑢₁−𝑢₀)𝜕_𝑡(𝜙₁+𝑥_0,𝑅1)−2(𝑓(𝑢₁)−𝑓(𝑢₀1))+2(𝑓(𝑢₁)+𝑓(𝑢₀)−2𝑐′(𝑡)𝑢₁))𝛿₁+⋅𝜂(𝑥)𝑑𝑥_𝐑((𝑈−𝑢₁)𝜕_𝑡(𝜙₂−𝑥_0,𝐿1)−2(𝑓(𝑈)−𝑓(𝑢₁1))+2(𝑓(𝑢₁)+𝑓(𝑢)−2𝑐′(𝑡)𝑢₁))𝛿₂⋅𝜂(𝑥)𝑑𝑥=𝒪(𝜀).(3.27)Since for 𝑡>𝑡^∗+𝑀𝜎/2 we have𝑢₁(𝜙₁+𝑥_0,𝑅,𝑡,𝜀)=𝑢₀(𝑥₀(𝜙₁+𝑥_0,𝑅𝑢,𝑡,𝜀)),₁(𝜙₂−𝑥_0,𝐿,𝑡,𝜀)=𝑈(𝑥₀(𝜙₂−𝑥_0,𝐿𝜙,𝑡,𝜀)),₁+𝑥_0,𝑅−𝜙₂+𝑥_0,𝐿𝜙=𝒪(𝜀),₁+𝑥_0,𝑅−𝑐(𝑡)=𝜙₂+𝑥_0,𝐿𝜕−𝑐(𝑡)=𝒪(𝜀),_𝑡(𝜙₁+𝑥_0,𝐿)−𝑐′(𝑡)=𝜕_𝑡(𝜙₂−𝑥_0,𝑅𝜕)−𝑐′(𝑡)=𝒪(𝜀),_𝑡𝑥_0,𝐿=𝜕_𝑡𝑥_0,𝑅+𝒪(𝜀)=𝒪(𝜀),(3.28)and since the functions 𝑈 and 𝑢₀ are continuous, from the definition of 𝛿 distribution we see that (3.27) is fulfilled.
Details of the construction for 𝑡<𝑡^∗+𝑀𝜎 can be found in [11].
Since 𝑢_𝜀 given by (3.2) is the weak asymptotic solution in the intervals 𝐼₁=[0,𝑡^∗+𝜎𝑀) and 𝐼₂=(𝑡^∗+𝜎𝑀/2,𝑇) and common part of the intervals is large enough (more precisely, it is enough to be |𝐼₁∩𝐼₂|>𝒪(𝜀^𝛼) for an 𝛼<1), we see that 𝑢_𝜀 is the weak asymptotic solution to (1.1) and (3.2) along entire time axis.

4. Interaction of Shock Waves with a Nonconstant Amplitude

In this section we construct the weak asymptotic solution to equation (1.1) with the following initial condition:𝑢(𝑥,0)=𝑢₀(𝑥)=𝑢₁₀(𝑥)+(𝑢₂₀(𝑥)−𝑢₁₀(𝑥))𝜃(𝑎₁−𝑥)+(𝑢₃₀(𝑥)−𝑢₂₀(𝑥))𝜃(𝑎₂−𝑥),(4.1)where 𝑢_𝑖(𝑥), 𝑥∈𝐑, are continuous decreasing functions. Furthermore, we assume that 𝑢_0𝑖, 𝑖=1,2,3, satisfy𝑢₁(𝑎₁)<𝑢₂(𝑎₁),𝑢₂(𝑎₂)<𝑢₃(𝑎₂).(4.2)Clearly, the function 𝑢₀(𝑥) has two admissible jumps at the points 𝑎₁ and 𝑎₂. Those jumps start to move according to the Rankine-Hugoniot conditions until they merge at the moment 𝑡^∗. Furthermore, since we want to investigate interaction of shock waves appearing in the initial data, we assume that−1𝑓^′′(𝑢_𝑖(𝑥))𝑢^′_𝑖(𝑥)≥𝑇>𝑡^∗,𝑖=1,2,3,𝑥∈𝐑.(4.3)Such conditions provide that the gradient catastrophe will not happen before the two shock waves collide.

By 𝑢_𝑖(𝑥,𝑡), 𝑖=1,2,3, (𝑥,𝑡)∈𝐑×𝐑⁺, respectively, we denote classical solutions of the following Cauchy problems (in the regions of 𝐑⁺×𝐑 where they exist)𝑢_𝑡+(𝑓(𝑢))_𝑥=0,𝑢|_𝑡=0=𝑢_𝑖0(𝑥),𝑖=1,2,3,(4.4)where the flux 𝑓 is the same as the one from (1.1). Since the initial functions 𝑢_𝑖0(𝑥), 𝑖=1,2,3, are decreasing, the solutions to (4.4) will be also decreasing.

By 𝜙_𝑖0(𝑡), 𝑖=1,2, 𝑡∈[0,𝑡^∗), and 𝜑(𝑡), 𝑡∈[𝑡^∗,𝑇], we denote the solutions to the following Cauchy problems𝜙_𝑖0𝑡(𝑡)=𝑓(𝑢_𝑖+1(𝜙_𝑖0(𝑡),𝑡))−𝑓(𝑢_𝑖(𝜙_𝑖0(𝑡),𝑡)𝑢_𝑖+1(𝜙_𝑖0(𝑡),𝑡)−𝑢_𝑖(𝜙_𝑖0(𝑡),𝑡),𝜙_𝑖0(0)=𝑎_𝑖,𝜑_𝑡(𝑡)=𝑓(𝑢₃(𝜑(𝑡)))−𝑓(𝑢₁(𝜑(𝑡)))𝑢₃(𝜑(𝑡))−𝑢₁(𝜑(𝑡)),𝜑(𝑡^∗)=𝜙₁₀(𝑡^∗)=𝜙₂₀(𝑡^∗).(4.5)Note that continuous solutions to those Cauchy problems always exist since those are actually Rankine-Hugoniot conditions for the admissible shock placed at 𝜙_𝑖0, 𝑖=1,2, and corresponding to Cauchy problem (1.1) with initial conditions, respectively (below we imply 𝑢_𝑖,0=𝑢_𝑖0)𝑢|_𝑡=0=𝑢_𝑖0(𝑥)+(𝑢_𝑖+1,0(𝑥)−𝑢_𝑖0(𝑥))𝜃(𝑎_𝑖−𝑥),𝑖=1,2,𝑢|_𝑡=𝑡∗=𝑢₁(𝑥,𝑡^∗)+(𝑢₃(𝑥,𝑡^∗)−𝑢₁(𝑥,𝑡^∗))𝜃(𝜑(𝑡^∗)−𝑥).(4.6)

Furthermore, since the function 𝑢₀(𝑥) decreases and 𝑓 is convex it is clear that (𝜙_20𝑡(𝑡)−𝜙_10𝑡(𝑡))^′>0.

We will prove the following theorem.

Theorem 4.1. The weak asymptotic solution to Cauchy problem (1.1) and (4.1) one has in the form𝑢_𝜀(𝑥,𝑡)=𝑢₁(𝑥,𝑡)+(𝑢₂(𝑥,𝑡)−𝑢₁(𝑥,𝑡))𝜔₁(𝜙₁(𝜏,𝑡)−𝑥𝜀)+(𝑢₃(𝑥,𝑡)−𝑢₂(𝑥,𝑡))𝜔₂(𝜙₂(𝜏,𝑡)−𝑥𝜀),(4.7)where 𝑥∈𝐑, 𝑡∈[0,𝑡^∗+𝜎)⊂[0,𝑇), and the functions 𝜔_𝑖, 𝑖=1,2, are the ones from Theorem 2.3.
Here,𝜙𝜏=²⁰(𝑡)−𝜙₁₀(𝑡)𝜀=𝜓₀(𝑡)𝜀.(4.8)The function 𝜌=𝜌(𝜏,𝑡) is given by the relation𝜙𝜌(𝜏,𝑡)=²(𝜏,𝑡)−𝜙₁(𝜏,𝑡)𝜀,(4.9)and the functions 𝜙_𝑖(𝑡,𝜏), 𝑖=1,2, are given by the formulas𝜙₁(𝑡,𝜏)=𝜙₁₀(𝑡)+𝜓₀(𝑡)Ω₁(𝑡,𝜏)𝜙⁺₁₁(𝑡),(4.10) 𝜙₂(𝑡,𝜏)=𝜙₂₀(𝑡)+𝜓₀(𝑡)Ω₂(𝑡,𝜏)𝜙⁺₂₁(𝑡),(4.11)where𝜙⁺_𝑖1(𝑡)=𝜑(𝑡)−𝜙_𝑖0(𝑡)𝜓₀(𝑡),𝑖=1,2,(4.12)and Ω_𝑖, 𝑖=1,2, are given by (4.26).

Proof. In the sequel we use the following notation.
By 𝜃 and 𝛿 the Heaviside and Dirac distributions, respectively, and𝛿_𝑖=𝛿(𝑥−𝜙_𝑖(𝑡)),𝜃_𝑖=𝜃(𝑥−𝜙_𝑖𝑢(𝑡)),𝑖=1,2,_𝑗(𝜙_𝑖)=𝑢_𝑗(𝜙_𝑖(𝑡),𝑡),𝑖=1,2,𝑗=1,2,3.(4.13)We start by substituting (4.7) in (1.1) and applying Theorem 2.3. We obtain𝑢_1𝑡+(𝑢_2𝑡−𝑢_1𝑡)𝜃₁+(𝑢_3𝑡−𝑢_2𝑡)𝜃₂+𝜙_1𝑡(𝑢₂(𝜙₁)−𝑢₁(𝜙₁))𝛿₁+𝜙_2𝑡(𝑢₃(𝜙₂)−𝑢₂(𝜙₂))𝛿₂+𝑢_1𝑥𝑓′(𝑢₁)+𝜃₁[(𝑢_3𝑥𝑓′(𝑢_3𝑥)−𝑓′(𝑢₁+𝑢₃−𝑢₂)(𝑢_1𝑥+𝑢_3𝑥−𝑢_2𝑥))𝐵₁+(𝑓′(𝑢₂)𝑢_2𝑥−𝑓′(𝑢₁)𝑢_1𝑥)𝐵₂]+𝜃₂[(𝑢_3𝑥𝑓′(𝑢₃)−𝑢_2𝑥𝑓′(𝑢₂))𝐵₂+(𝑓′(𝑢₁+𝑢₃−𝑢₂)(𝑢_1𝑥+𝑢_3𝑥−𝑢₂)−𝑓′(𝑢₁)𝑢_1𝑥)𝐵₁]−[𝐵₁(𝑓(𝑢₃(𝜙₁))−𝑓(𝑢₁+𝑢₃−𝑢₂)(𝜙₁)))+𝐵₂(𝑓(𝑢₂(𝜙₁))−𝑓(𝑢₁(𝜙₁)))]𝛿₁−[𝐵₂(𝑓(𝑢₃(𝜙₂))−𝑓(𝑢₂(𝜙₂)))+𝐵₁(𝑓((𝑢₁+𝑢₃−𝑢₂)(𝜙₂))−𝑓(𝑢₁(𝜙₂)))]𝛿₂=𝒪_𝒟′(𝜀),(4.14)where, as usual, 𝐵_𝑖=𝐵_𝑖((𝜙₂−𝜙₁)/𝜀), 𝑖=1,2. If we equalize by zero the sum of the coefficients multiplying 𝛿₁ and 𝛿₂, respectively, we get𝜙_1𝑡(𝑢₂(𝜙₁)−𝑢₁(𝜙₁))−𝐵₁(𝑓(𝑢₃(𝜙₁))−𝑓(𝑢₁+𝑢₃−𝑢₂)(𝜙₁))−𝐵₂(𝑓(𝑢₂(𝜙₁))−𝑓(𝑢₁(𝜙₁)))=0,(4.15)   𝜙_2𝑡(𝑢₃(𝜙₂)−𝑢₂(𝜙₂))−𝐵₂(𝑓(𝑢₃(𝜙₂))−𝑓(𝑢₂(𝜙₂)))+𝐵₁(𝑓((𝑢₁+𝑢₃−𝑢₂)(𝜙₂))−𝑓(𝑢₁(𝜙₂)))=0.(4.16)Now, we subtract (4.15) and (4.16). Using (4.10) and (4.11), and passing from the variable 𝑡 to the variable 𝜏, we obtain the following Cauchy problem (see also [19, pages 108–110])𝑑𝜌=𝜓̇𝑑𝜏⁰−𝑏₁(𝛼𝑒₁|_𝑥=𝜙1+𝛼𝑒₂|_𝑥=𝜙2𝜌)=𝑓(𝜌,𝑡),𝜏|_{𝜏→−∞}=1,(4.17)where𝛼=𝑓(𝑢₃)−𝑓(𝑢₂)−𝑓(𝑢₁+𝑢₃−𝑢₂)+𝑓(𝑢₁𝑒),₁=𝑢₂−𝑢₁,𝑒₂=𝑢₃−𝑢₂.(4.18)We can rewrite the function 𝐹(𝜌,𝑡) in the following manner:𝑓(𝜌,𝑡)=[𝑓(𝑢₃)−𝑓(𝑢₁+𝑢₃−𝑢₂)𝑢₂−𝑢₁|_𝑥=𝜙1−𝑓(𝑢₁+𝑢₃−𝑢₂)−𝑓(𝑢₁)𝑢₃−𝑢₂|_𝑥=𝜙2𝛼]+(𝑒₁|_𝑥=𝜙1+𝛼𝑒₂|_𝑥=𝜙1).(4.19)
Now, we return to (4.17). From the standard ODE theory we see that the solution 𝜌 of (4.17) tends to the stationary solution 𝜌₀ of (4.17). Clearly, 𝜌₀=𝜌₀(𝑡) is the minimal root to the equation 𝐹(𝜌,𝑡)=0 in 𝜌 (Cauchy theorem for existence and uniqueness of the solution to an ODE with an initial condition; see Figure 5). Since 𝜌=(𝜙₂−𝜙₁)/𝜀 after the interaction we have 𝜙₁=𝜙₂+𝒪(𝜀).
It remained to determine the functions Ω_𝑖, 𝑖=1,2. In the sequel, we letΩ̇_𝑖=𝜕Ω_𝑖𝜕𝜏,𝑖=1,2.(4.20)
Substituting expressions (4.10) and (4.11) in (4.15) and (4.16), respectively, we obtain the equations𝜕Ω_𝑖𝜓𝜕𝑡⁰𝜙⁺_𝑖1𝜏+Ω̇_𝑖+Ω_𝑖(𝜓₀𝜙⁺_𝑖1)_𝑡𝜏𝜙⁺_𝑖1=(−1)^𝑖𝐵₁𝜏𝑓(𝑢₃)−𝑓(𝑢₂)−𝑓(𝑢₁+𝑢₃−𝑢₂)+𝑓(𝑢₁)𝑢_𝑖+1−𝑢_𝑖|_{𝑥=𝜙𝑖},𝑖=1,2.(4.21)
Furthermore, notice that if |𝜕Ω_𝑖/𝜕𝑡|<∞, we have𝜕Ω_𝑖𝜓𝜕𝑡⁰𝜙⁺_𝑖1𝜏=𝜀𝜕Ω_𝑖𝜙𝜕𝑡^+𝑖1=𝒪(𝜀).(4.22)Therefore, we will determine Ω_𝑖, 𝑖=1,2, so that they satisfy the following differential equations in 𝜏∈𝐼𝑅 (compare with (4.21))Ω̇_𝑖+Ω_𝑖(𝜓₀𝜙⁺_𝑖1)_𝑡𝜏𝜙⁺_𝑖1=(−1)^𝑖𝐵₁𝜏𝑓(𝑢₃)−𝑓(𝑢₂)−𝑓(𝑢₁+𝑢₃−𝑢₂)+𝑓(𝑢₁)𝑢₂−𝑢₁|_{𝑥=𝜙𝑖},𝑖=1,2,(4.23)and then we will prove that|𝜕Ω_𝑖𝜕𝑡|<∞.(4.24)Notice that differential equation (4.23) is defined only on the interval [0,𝑡^∗]. Therefore, it is necessary to extend continuously the functions 𝐹_𝑖(𝜌,𝑡) after the moment 𝑡=𝑡^∗. Put for 𝑡>𝑡^∗,𝐹_𝑖(𝑡)=𝑓(𝑢₃)−𝑓(𝑢₂)−𝑓(𝑢₁+𝑢₃−𝑢₂)+𝑓(𝑢₁)𝑢_𝑖+1−𝑢_𝑖|_{𝑥=𝜑(𝑡)},𝑖=1,2.(4.25)Since for 𝑡=𝑡^∗ we have 𝜙₁(𝑡^∗)=𝜙₂(𝑡^∗)=𝜑(𝑡^∗), it follows that the extension is well defined. It is not difficult to find the solution to (4.23)Ω_𝑖1(𝑡,𝜏)=𝜏^{(𝜓0𝜙+𝑖1)𝑡/𝜙+𝑖1𝜏0}(−1)^𝑖𝐵₁(𝜌)𝐹_𝑖(𝑡)𝜏^{0𝑥002𝑑𝑐(𝜓0𝜙+𝑖1)𝑡/𝜙+𝑖1}𝑑𝜏^{0𝑥002𝑑𝑐}.(4.26)Using the l'Hospital theorem for limits we get from here|lim_𝜏→±∞Ω_𝑖(𝜏,𝑡)|<∞,|lim_𝜏→±∞𝜕Ω_𝑖(𝜏,𝑡)𝜕𝑡|<∞,(4.27)proving (4.24) and thus (4.22). This, in turn, proves that Ω_𝑖, 𝑖=1,2, are approximate solutions to (4.21), and thus, (4.10) and (4.11) are fulfilled.
In that way, we have eliminated the coefficients multiplying 𝛿 distribution in (4.14). Now, we have to annulate the coefficients multiplying 𝜃 distribution in (4.14), more precisely, we need to get𝑢_1𝑡+𝑢_1𝑥𝑓′(𝑢₁)+[𝑢_3𝑡+𝑓(𝑢₃)𝑢_3𝑥−𝑢_1𝑡−𝑓(𝑢₁)𝑢_1𝑥]𝜃₁+[𝑢_3𝑡−𝑢_2𝑡+𝑢_3𝑥𝑓(𝑢₃)−𝑢_2𝑥𝑓(𝑢₂)+𝐵₁(−𝑢_3𝑥𝑓′(𝑢_3𝑥)+𝑢_2𝑥𝑓(𝑢₂)−𝑓′(𝑢₁+𝑢₃−𝑢₂)(𝑢_1𝑥+𝑢_3𝑥−𝑢_2𝑥)+𝑓(𝑢₁)𝑢_1𝑥)](𝜃₂−𝜃₁)=𝒪_𝒟′(𝜀).(4.28)
Since the functions 𝑢_𝑖, 𝑖=1,2,3 satisfy (4.4), we can rewrite the previous expression in the form𝐵₁[−𝑢_3𝑥𝑓′(𝑢_3𝑥)+𝑢_2𝑥𝑓(𝑢₂)−𝑓′(𝑢₁+𝑢₃−𝑢₂)(𝑢_1𝑥+𝑢_3𝑥−𝑢_2𝑥)+𝑓(𝑢₁)𝑢_1𝑥](𝜃₂−𝜃₁)=𝒪_𝒟′(𝜀).(4.29)If we multiply this by 𝜂∈𝐶¹₀(𝐑), we have𝜀𝜌𝐵₁1(𝜌)𝜙₁−𝜙₂^𝜙1_𝜙2[(−𝑢_3𝑥𝑓′(𝑢_3𝑥)+𝑢_2𝑥𝑓(𝑢₂)−𝑓′(𝑢₁+𝑢₃−𝑢₂)(𝑢_1𝑥+𝑢_3𝑥−𝑢_2𝑥)+𝑓(𝑢₁)𝑢_1𝑥)]𝜂(𝑥)𝑑𝑥=𝒪(𝜀),(4.30)since |𝜌𝐵₁(𝜌)|<∞ (it is clear from the definition but one can check in [11]). In that way, we see that (4.14) is correct which proves the theorem.
Finally, notice that since 𝑢_𝜀 represents the weak asymptotic solution to the considered problem we have 𝑤−lim_𝜀→0𝑢_𝜀=𝑢 where 𝑢 is the weak solution to the considered problem. Therefore, the functions 𝜙_𝑖, 𝑖=1,2, up to 𝒪(𝜀), satisfy the Rankine-Hugoniot conditions. In turn, from there it follows that for 𝑡>𝑡^∗ we havelim_𝜏→+∞Ω(𝜏,𝑡)=1,(4.31)and for 𝑡<𝑡^∗ we havelim_{𝜏→−∞}Ω(𝜏,𝑡)=0.(4.32)

5. Scalar Conservation Law with Decreasing Initial Data

In this section, we consider (1.1) with the following initial condition:𝑢|_𝑡=0=𝑣(𝑥)∈𝐶(𝐑).(5.1)We assume that the function 𝑣(𝑥) decreases and that it has finite number of points in which the function−1𝑓^′′(𝑣(𝑥))𝑣′(𝑥),𝑥∈𝐑,(5.2)reaches maximum. Denote this set of points by 𝑆={𝑥₁,𝑥₂,…,𝑥_𝑛}. Assume also that 𝑥₁<𝑥₂<⋯<𝑥_𝑛.

Now, we continue in the following way. Around every 𝑥_𝑖∈𝑆, 𝑖=1,…,𝑛, we allocate 𝜀-neighborhoods of the form (𝑥_𝑖−2𝜀^𝜇,𝑥_𝑖+2𝜀^𝜇), 𝜇∈(0,1). Then, we transform the initial data 𝑣, replacing it by the function 𝑢_𝑖(𝑥), 𝑥∈(𝑥_𝑖−2𝜀^𝜇,𝑥_𝑖+2𝜀^𝜇), such that for every 𝑖=1,…,𝑛, we have the following (compare Figures 3 and 4)𝑓′(𝑢_𝑖(𝑥))=−𝐾_𝑖𝑥+𝑏_𝑖,𝑥∈(𝑥_𝑖−𝜀^𝜇,𝑥_𝑖+𝜀^𝜇𝑢),_𝑖(𝑥)=𝑣(𝑥_𝑖−2𝜀^𝜇),𝑥∈(𝑥_𝑖−2𝜀^𝜇,𝑥_𝑖−𝜀^𝜇𝑢),_𝑖(𝑥)=𝑣(𝑥_𝑖−2𝜀^𝜇),𝑥∈(𝑥_𝑖+𝜀^𝜇,𝑥_𝑖+2𝜀^𝜇),(5.3)where the constants 𝐾_𝑖 and 𝑏_𝑖 are determined by the conditions𝑢_𝑖(𝑥_𝑖−𝜀^𝜇)=𝑣(𝑥_𝑖−2𝜀^𝜇),𝑢_𝑖(𝑥_𝑖+𝜀^𝜇)=𝑣(𝑥_𝑖+2𝜀^𝜇),𝑖=1,…,𝑛.(5.4)So, we have replaced the original initial data 𝑣 by the function𝑣(𝑥)=𝑣(𝑥),𝑥∈(−∞,𝑥₁−2𝜀^𝜇)∪(𝑥₁+2𝜀^𝜇,𝑥₂−2𝜀^𝜇)∪⋯∪(𝑥_𝑛−1+2𝜀^𝜇,𝑥_𝑛−2𝜀^𝜇)∪(𝑥_𝑛+2𝜀^𝜇𝑢,∞),_𝑖(𝑥),𝑥∈(𝑥_𝑖−2𝜀^𝜇,𝑥_𝑖+2𝜀^𝜇),𝑖=1,…,𝑛.(5.5)Obviously, we have the following estimate fulfilled 𝑣(𝑥)−𝑣(𝑥)=𝒪_𝒟′(𝜀^𝜇). Beside that, since the function 𝑓^′′(𝑣(𝑥))𝑣^′(𝑥) reaches its maximum at the points 𝑥_𝑖∈𝑆, then there exist neighborhoods 𝒰(𝑥_𝑖), 𝑖=1,…,𝑛, such that for every 𝑥∈𝒰(𝑥_𝑖) we have𝑓^′′(𝑣(𝑥))𝑣′(𝑥)>𝑓^′′(𝑣(𝑦))𝑣′(𝑦)where𝑦∉^𝑛_𝑖=1𝒰(𝑥_𝑖).(5.6)The moment of “straightening” of the curves connecting the points (𝑥_𝑖−𝜀^𝜇,𝑣(𝑥_𝑖−𝜀^𝜇)) and (𝑥_𝑖+𝜀^𝜇,𝑣(𝑥_𝑖+𝜀^𝜇)), according to Section 2 and the Lagrange mean value theorem is given by𝑡^∗₁=2𝜀^𝜇𝑓′(𝑣(𝑥_𝑖−𝜀^𝜇))−𝑓′(𝑣(𝑥_𝑖+𝜀^𝜇1))=−(𝑓′∘𝑣)′(𝑥_𝑖)1=−(𝑓^′′(𝑣(𝑥_𝑖))𝑣′(𝑥_𝑖)<max_𝑦∈𝑈−1𝑓^′′,𝑥(𝑣(𝑥))𝑣′(𝑥)^𝑖∈(𝑥_𝑖−𝜀^𝜇,𝑥_𝑖+𝜀^𝜇).(5.7)

From here we see that for every 𝑖=1,…,𝑛 we have 𝑡^∗₁>−1/𝑓^′′(𝑣(𝑦))𝑣′(𝑦), where 𝑦∉⋃^𝑛_𝑖=1𝒰(𝑥_𝑖). This actually means that a gradient catastrophe will not happen before straightening of the lines connecting the points 𝑥_𝑖−𝜀^𝜇 and 𝑥_𝑖+𝜀^𝜇, at least for 𝜀 small enough. Knowing that, we can describe behavior of the weak asymptotic solution of the considered problem relying on the previous sections.

We track evolution of the new initial data 𝑣_𝜀(𝑥) in (𝑥,𝑢) space. In the interval [0,𝑡^∗₁−𝑔(𝜀)), for appropriate function 𝑔(𝜀)=𝒪(𝜀^𝜇), the solution will be continuous function. In the interval [𝑡^∗₁−𝑠(𝜀),𝑡^∗+𝑠(𝜀)], for some function 𝑠(𝜀)=𝒪(𝜀^𝜇), the curves connecting the points (𝑥_𝑖−𝜀^𝜇,𝑣(𝑥_𝑖−𝜀^𝜇)) and (𝑥_𝑖+𝜀^𝜇,𝑣(𝑥_𝑖+𝜀^𝜇)), 𝑖=1,…,𝑛, will straighten in the shock waves of increasing amplitude. After that, two general cases as well as their combinations can happen:(a) in the intervals between some pair of shock waves, the gradient catastrophe will happen in the moment 𝑡^∗₂=𝑡^∗₁+𝜉, but before the collision of the two shock waves, or(b)two shock waves will collide in the moment 𝑡^∗₂=𝑡^∗₁+𝜉, but before a gradient catastrophe happens in the interval between them.

In case (a), we repeat the procedure from the beginning. More precisely, we take the weak limit in 𝑥∈𝐑 of the weak asymptotic solution in the moment 𝑡^∗₁. Denote it by ^0𝑥0𝑣^02𝑑𝑐(𝑥,𝑡^∗₁). Then, we replace the part of the function ^0𝑥0𝑣^02𝑑𝑐(𝑥,𝑡^∗₁) around the point 𝑥₀ from which emanates the characteristics along which the gradient catastrophe will happen. We replace it completely analogically as we did for the initial condition 𝑣. Namely, we take a smooth function 𝑟(𝑥,𝑡^∗₁) such that 𝑓^′(𝑟(𝑥,𝑡^∗₁))=−𝐾𝑥+𝑏, for some constants 𝐾 and 𝑏, in the interval (𝑥₀−𝜀^𝜇,𝑥₀+𝜀^𝜇). Then, instead of the part of ^0𝑥0𝑣^02𝑑𝑐(𝑥,𝑡^∗₁) in the interval (𝑥₀−2𝜀^𝜇,𝑥₀+2𝜀^𝜇) we put the function 𝑟. In that way we get the function 𝑣^∗(𝑥). More precisely, we take𝑣(𝑥,𝑡^∗₁)=^{0𝑥002𝑑𝑐}𝑣(𝑥,𝑡^∗₁),𝑥∉(𝑥₀−2𝜀^𝜇,𝑥₀+2𝜀^𝜇),𝑟(𝑥,𝑡^∗₁),𝑥∈(𝑥₀−𝜀^𝜇,𝑥₀+𝜀^𝜇),_{0𝑥002𝑑𝑐}𝑣(𝑥₀−𝜀^𝜇,𝑡^∗₁),𝑥∈(𝑥₀−2𝜀^𝜇,𝑥₀−𝜀^𝜇),_{0𝑥002𝑑𝑐}𝑣(𝑥₀+𝜀^𝜇,𝑡^∗₁),𝑥∈(𝑥₀+𝜀^𝜇,𝑥₀+2𝜀^𝜇).(5.8)

Then, using Sections 3 and 4, we find the weak asymptotic solution 𝑢^∗_𝜀(𝑥,𝑡) to (1.1) with initial data 𝑣^∗(𝑥) in the strip (𝑡^∗₁,𝑡^∗₂+(𝑡^∗₃−𝑡^∗₂)/2)×𝐑 where 𝑡^∗₃ is the moment of the next situation (a) or (b) or their combination. Then, using the partition of unity, we connect the weak asymptotic solutions in the intervals [0,𝑡^∗₁+(𝑡^∗₂−𝑡^∗₁)/2) and [𝑡^∗₁,𝑡^∗₂+(𝑡^∗₃−𝑡^∗₂)/2).

In case (b), we use the results of Section 4 on (1.1) with initial condition ^0𝑥0𝑣^02𝑑𝑐(𝑥,𝑡^∗₁). Then, like in case (a) we use the partition of unity to connect solutions on the intervals [0,𝑡^∗₁+(𝑡^∗₂−𝑡^∗₁)/2) and [𝑡^∗₁,𝑡^∗₂+(𝑡^∗₃−𝑡^∗₂)/2), where 𝑡^∗₃ is the moment of the next situation (a) or (b) or their combination.

To detail the previous analysis, we formulate the following theorem. Theorem 5.1. Fix arbitrary𝑇>0 and denote by 𝑡^∗_𝑖, 𝑖=1,…,𝑛, moments of nonlinear wave interactions in the interval [0,𝑇] (more precisely, situations (a) and/or (b)) corresponding to Cauchy problem (1.1) and (4.1).
Global weak asymptotic solution𝑢_𝜀(𝑥,𝑡)∈𝐶¹(𝐑⁺×𝐑) of (1.1) and (4.1) has the form𝑢_𝜀(𝑥,𝑡)=𝜂₁(𝑡)𝑢_𝜀^[1](𝑥,𝑡)+𝜂₂(𝑡)𝑢_𝜀^[2](𝑥,𝑡)+⋯+𝜂_𝑛(𝑡)𝑢_𝜀^[𝑛](𝑥,𝑡),(5.9)where 𝜂_𝑖, 𝑖=1,…,𝑛 is partition of unity of the interval [0,𝑇] such that (we take 𝑡^∗₀=0, 𝑡^∗₋₁=−1 and 𝑡^∗_𝑛+1=𝑇+1)𝜂_𝑖(𝑡)≡1,𝑡∈(𝑡^∗_𝑖−1,𝑡^∗_𝑖),𝜂_𝑖(𝑡)≡0,𝑡∉(𝑡^∗_𝑖−2,𝑡^∗_𝑖+1).(5.10)The function 𝑢_𝜀^[1](𝑥,𝑡) is given by (5.11) while the functions 𝑢_𝜀^[𝑘](𝑥,𝑡), 𝑘=2,…,𝑛, are of form (5.17) (with difference in indexing depending on time and place of singularities interactions).Proof. Denote by {𝑥₁,…,𝑥_𝑛} the set such that for every 𝑥₀∈{𝑥₁,…,𝑥_𝑛} we have (2.5) satisfied. According to the previous analysis, in the interval [0,𝑡^∗₁+(𝑡^∗₂−𝑡^∗₁)/2) the weak asymptotic solution to (1.1) and (5.1) we have in the form𝑢_𝜀^[1](𝑥,𝑡)=𝑣₀(𝑥,𝑡)+(𝑢₁(𝑥¹₀(𝑥,𝑡,𝜀))−𝑣₀(𝑥,𝑡))𝜔₁(𝜙¹₁(𝑡,𝜏¹)−𝑥𝜀)+(𝑣₁(𝑥,𝑡)−𝑢₁(𝑥¹₀(𝑥,𝑡,𝜀)))𝜔₂(𝜙²₁(𝑡,𝜏¹)−𝑥𝜀)+⋯+(𝑢_𝑛(𝑥^𝑛₀(𝑥,𝑡,𝜀),𝑡)−𝑣_𝑛−1(𝑥,𝑡))𝜔₁(𝜙¹_𝑛(𝑡,𝜏^𝑛)−𝑥𝜀)+(𝑣_𝑛(𝑥,𝑡)−𝑢_𝑛(𝑥^𝑛₀(𝑥,𝑡,𝜀),𝑡))𝜔₂(𝜙²_𝑛(𝑡,𝜏^𝑛)−𝑥𝜀),(5.11)where 𝜔_𝑖, 𝑖=1,2, are the ones from Theorem 3.3.
Using the last two sections we can describe all unknown functions appearing in (5.11). We have the following
(i)The functions 𝑥^𝑖₀=𝑥^𝑖₀(𝑥,𝑡,𝜀), 𝑖=1,…,𝑛−1, are inverse functions to the “new characteristics”. The “new characteristics” we obtain if in (3.15) we replace 𝜌 by 𝜌_𝑖, 𝑢₁ by 𝑢_𝑖, and 𝑐 by 𝑐_𝑖. The functions 𝑐_𝑖 we obtain when in (3.4) we replace 𝑈 by 𝑣_𝑖−1, and 𝑢₀ by 𝑣_𝑖.(ii)The functions 𝜙^𝑗_𝑖=𝜙^𝑗_𝑖(𝑡,𝜏^𝑖), 𝑖=0,1,…,𝑛, 𝑗=1,2, are the “new characteristics” emanating from the points 𝑥_𝑖+(−1)^𝑗𝜀^𝜇, and 𝜏^𝑖=𝜑_𝑖¹⁰(𝑡)−𝜑_𝑖²⁰(𝑡)𝜀,𝜌_𝑖=𝜌_𝑖(𝑡,𝜏_𝑖𝜙)=^1𝑖(𝑡,𝜏^𝑖)−𝜙²_𝑖(𝑡,𝜏^𝑖)𝜀,𝜑_𝑖²⁰(𝑡)=𝑓′(𝑣(𝑥_𝑖−𝜀^𝜇))𝑡+𝑥_𝑖−𝜀^𝜇,𝜑_𝑖¹⁰(𝑡)=𝑓′(𝑣(𝑥_𝑖+𝜀^𝜇))𝑡+𝑥_𝑖+𝜀^𝜇,𝑖=1,…,𝑛.(5.12) Also, notice that according to Theorem 3.3 the functions 𝜌_𝑖 are the solutions to the following Cauchy problems (analogical to Cauchy problem (3.10)):𝜌̇_𝑖=1−2𝐵₁(𝜌_𝑖),lim_{𝜏→−∞}𝜌_𝑖𝜏_𝑖=1,(5.13)where 𝜌̇_𝑖=𝜕_𝜏𝑖𝜌_𝑖.
The functions 𝑣_𝑖(𝑥,𝑡), 𝑖=1,…,𝑛, are classical solutions to the following Cauchy problems in the regions 𝑡∈[0,𝑡^∗+(𝑡^∗₂−𝑡^∗₁)/2), 𝑥∈𝐑,𝑣_𝑖𝑡+(𝑓(𝑣_𝑖)𝑣_𝑖)_𝑥=0,𝑣_𝑖(𝑥,0)=𝑣(𝑥).(5.14)As we have already explained, in the moment 𝑡^∗₁+𝜉=𝑡^∗₂ we will have situation (a) or (b) or we will have situation (a) in an interval between two shock waves and, at the same time, situation (b) between two other shock waves.
Due to locality of the singularity formation process, without loosing on the generality, we can assume that exactly one collision of shock waves and exactly one gradient catastrophe happened at the moment 𝑡^∗₂ (that means that we have combination of situations (a) and (b)). It only remains to impose initial data at the moment 𝑡=𝑡^∗₁ for (1.1). We simply put 𝑡=𝑡^∗₁ in (1.1) and use the fact that up to 𝒪(𝜀), we have 𝜙¹_𝑖(𝑡^∗₁,0)=𝜙²_𝑖(𝑡^∗₁,0), 𝑖=1,...,𝑛. Thus, we obtain the following initial data:𝑣_[2](𝑥,𝑡^∗₁)=𝑣₀(𝑥,𝑡^∗₁)+(𝑣₁(𝑥,𝑡^∗₁)−𝑣₀(𝑥,𝑡^∗₁))𝜔₁(𝜙¹₁(𝑡^∗₁,0)−𝑥𝜀)+(𝑣₁(𝑥,𝑡^∗₁)−𝑣₂(𝑥,𝑡^∗₁))𝜔₂(𝜙¹₂(𝑡^∗₁,0)−𝑥𝜀)+⋯+(𝑣_𝑛−1(𝑥,𝑡^∗₁)−𝑣_𝑛−2(𝑥,𝑡^∗₁))𝜔₁(𝜙¹_𝑛−1(𝑡^∗₁,0)−𝑥𝜀)+(𝑣_𝑛(𝑥,𝑡^∗₁)−𝑣_𝑛−1(𝑥,𝑡^∗₁))𝜔₂(𝜙¹_𝑛(𝑡^∗₁,0)−𝑥𝜀).(5.15)
Then, we can assume that case (a) happened at the point 𝑥₀=𝑥_[2] placed between shock waves concentrated at the points 𝜙¹₁(𝑡^∗₁,0) and 𝜙¹₂(𝑡^∗₁,0), and, at the same time, case (b) happened between shock waves concentrated in the points 𝜙¹₂(𝑡^∗₁) and 𝜙¹₃(𝑡^∗₁).
For the part of the function 𝑣_[2](𝑥,𝑡^∗₁) disposed between 𝜙¹₁(𝑡^∗₁,0) and 𝜙¹₂(𝑡^∗₁,0), we repeat the procedure from the beginning of the section. Namely, we approximate the function 𝑣₁(𝑥,𝑡^∗₁) in the interval (𝑥_[2]−2𝜀^𝜇,𝑥_[2]+2𝜀^𝜇) by the function 𝑢_[2](𝑥) such that𝑓′(𝑢_[2](𝑥))=−𝐾𝑥+𝑏,𝑥∈(𝑥_[2]−𝜀^𝜇,𝑥_[2]+𝜀^𝜇𝑢),_[2](𝑥)=𝑣₁(𝑥_[2]−2𝜀^𝜇,𝑡^∗₁),𝑥∈(𝑥_[2]−2𝜀^𝜇,𝑥_[2]−𝜀^𝜇𝑢),_[2](𝑥)=𝑣₁(𝑥_[2]+2𝜀^𝜇,𝑡^∗₁),𝑥∈(𝑥_[2]+𝜀^𝜇,𝑥_[2]+2𝜀^𝜇),(5.16)where 𝐾 and 𝑏 are appropriate constants.
Denote this new function by ^0𝑥0𝑣^02𝑑𝑐_[2](𝑥). Now, we solve (1.1) with the function ^0𝑥0𝑣^02𝑑𝑐_[2] as initial data but starting from 𝑡=𝑡^∗₁. According to Sections 3 and 4, the weak asymptotic solution to (1.1) with initial condition 𝑢(𝑥,𝑡^∗₁)=𝑣^{0𝑥002𝑑𝑐}_[2](𝑥) in the interval [𝑡^∗₁,𝑡^∗₂+(𝑡^∗₃−𝑡^∗₂)/2), where 𝑡^∗₃ is the moment of the third set of interactions (more precisely, the third set of gradient catastrophe or shock wave collisions), we have in the form𝑢_𝜀^[2](𝑥,𝑡)=𝑣₀(𝑥,𝑡)+(𝑣_0[2](𝑥,𝑡)−𝑣₀𝜙(𝑥,𝑡))𝜔(¹(𝑡,𝜏¹)−𝑥𝜀)+(𝑢_1[2](𝑥¹₀(𝑥,𝑡,𝜀))−𝑣_0[2]𝜙(𝑥,𝑡))𝜔(^1[2](𝑡,𝜏_[2])−𝑥𝜀)+(𝑣_1[2](𝑥,𝑡)−𝑢_1[2](𝑥¹₀𝜙(𝑥,𝑡,𝜀)))𝜔(^2[2](𝑡,𝜏_[2])−𝑥𝜀)+(𝑣₂(𝑥,𝑡)−𝑣_1[2]𝜙(𝑥,𝑡))𝜔(²(𝑡,𝜏²)−𝑥𝜀)+(𝑣₃(𝑥,𝑡)−𝑣₂𝜙(𝑥,𝑡,𝜀))𝜔(³(𝑡,𝜏²)−𝑥𝜀)+⋯+(𝑣_𝑛−1(𝑥,𝑡)−𝑣_𝑛−2𝜙(𝑥,𝑡))𝜔(^𝑛−1(𝑡,𝜏^𝑛−1)−𝑥𝜀)+(𝑣_𝑛(𝑥,𝑡)−𝑣_𝑛−1𝜙(𝑥,𝑡))𝜔(^𝑛(𝑡,𝜏^𝑛)−𝑥𝜀).(5.17)Here, the functions 𝜙_𝑖, 𝑖=1,4,…,𝑛, are independent on 𝜀 and they are given by the following Cauchy problems𝜙_1𝑡=𝑓(𝑣_0[2](𝜙₁,𝑡))−𝑓(𝑣₀(𝜙₁,𝑡))𝑣_0[2](𝜙₁,𝑡)−𝑣₀(𝜙₁,𝑡),𝜙₁(𝑡^∗₁)=𝜙¹₁(𝑡^∗₁𝜙,0),_𝑖𝑡=𝑓(𝑣_𝑖−1(𝜙_𝑖,𝑡))−𝑓(𝑣_𝑖(𝜙_𝑖,𝑡))𝑣_𝑖−1(𝜙_𝑖,𝑡)−𝑣_𝑖(𝜙_𝑖,𝑡),𝜙_𝑖(𝑡^∗₁)=𝜙¹_𝑖(𝑡^∗₁,0),(5.18)where 𝑖=4,…,𝑛, 𝑡∈[𝑡^∗₁,𝑡^∗₂+(𝑡^∗₃−𝑡^∗₂)/2).
If by 𝜙₂₀(𝑡) and 𝜙₃₀(𝑡) we denote global solutions of the following Cauchy problems (see corresponding equations (4.7)):𝜙_20𝑡=𝑓(𝑣₂(𝜙₂₀,𝑡))−𝑓(𝑣_1[2](𝜙₂₀,𝑡))𝑣₂(𝜙₂₀,𝑡)−𝑣_1[2](𝜙₂₀,𝑡),𝜙₂₀(𝑡^∗₁)=𝜙¹₂(𝑡^∗₁𝜙,0),_30𝑡=𝑓(𝑣₃(𝜙₂₀,𝑡))−𝑓(𝑣₂(𝜙₂₀,𝑡))𝑣₃(𝜙₃₀,𝑡)−𝑣₂(𝜙₃₀,𝑡),𝜙₃₀(𝑡^∗₁)=𝜙¹₃(𝑡^∗₁,0),𝑡∈[𝑡^∗₁,𝑡^∗₂+(𝑡^∗₃−𝑡^∗₂)2),(5.19)we have the quantity 𝜏₂ defined by𝜏₂=𝜙₂₀(𝑡)−𝜙₃₀(𝑡)𝜀.(5.20)If we also put𝜙¹⁰_[2](𝑡)=𝑓′(𝑣(𝑥_[2]+𝜀^𝜇))𝑡+𝑥_[2]+𝜀^𝜇,𝜙²⁰_[2](𝑡)=𝑓′(𝑣(𝑥_[2]−𝜀^𝜇))𝑡+𝑥_[2]−𝜀^𝜇,𝑡∈[𝑡^∗₁,𝑡^∗₂+(𝑡^∗₃−𝑡^∗₂)2),(5.21)then we have the quantity 𝜏_[2] given by𝜏_[2]=𝜙¹⁰_[2](𝑡)−𝜙²⁰_[2](𝑡)𝜀.(5.22)Note that 𝜙^𝑖0_[2], 𝑖=1,2, are standard characteristics emanating from 𝑥_[2]+𝜀^𝜇 and 𝑥_[2]−𝜀^𝜇.
The functions 𝑣_𝑖(𝑥,𝑡), 𝑖=0,2,3,…,𝑛 are classical solutions to Cauchy problems (5.14) in the regions 𝑡∈[𝑡^∗₁,𝑡^∗₂+(𝑡^∗₃−𝑡^∗₂)/2), 𝑥∈𝐑, while 𝑣_0[2](𝑥,𝑡) and 𝑣_1[2](𝑥,𝑡) satisfy, respectively, the following equations:𝑣_0[2]𝑡+𝑓′(𝑣_0[2])𝑣_0[2]𝑥=0,𝑣_0[2](𝑥,0)=𝑣(𝑥),𝑡∈[𝑡^∗₁,𝑡^∗₂+(𝑡^∗₃−𝑡^∗₂)2𝑣),𝑥∈𝐑,_1[2]𝑡+𝑓′(𝑣_1[2])𝑣_1[2]𝑥=0,𝑣_1[2](𝑥,0)=𝑣(𝑥).(5.23)The function 𝑢_1[2](𝑥¹₀(𝑥,𝑡,𝜀)) is analogical to the function 𝑢₁(𝑥,𝑡,𝜀) from (5.11) with an obvious difference in indexing.
The functions 𝜙_𝑖, 𝑖=1,4,5,…,𝑛, are independent on 𝜀 and they are given by Rankine-Hugoniot conditions𝜙_𝑖𝑡=𝑓(𝑣_𝑖(𝜙_𝑖,𝑡))−𝑓(𝑣_𝑖−1(𝜙_𝑖,𝑡))𝑣_𝑖(𝜙_𝑖,𝑡)−𝑣_𝑖−1(𝜙_𝑖,𝜙,𝑡)_1𝑡=𝑓(𝑣_0[1](𝜙₁,𝑡))−𝑓(𝑣₀(𝜙₁))𝑣_0[1](𝜙₁,𝑡)−𝑣₀(𝜙₁,𝜙,𝑡)^𝑖(𝑡^∗₁)=𝜙¹_𝑖(𝑡^∗₁,0),𝑖=1,4,5,…,𝑛.(5.24)
Thus, we have described the weak asymptotic solution to (1.1), (5.1) in the intervals [0,𝑡^∗₁+(𝑡^∗₂−𝑡^∗₁)/2) and [𝑡^∗₁,𝑡^∗₂+(𝑡^∗₃−𝑡^∗₂)/2) through the functions 𝑢_𝜀^[1] and 𝑢_𝜀^[2], respectively. Notice, then, the partition of unity 𝜂(𝑡)∈𝐶¹₀(𝐑⁺) of the time axis satisfying𝜂(𝑡)≡1,𝑡≤𝑡^∗₁,𝜂(𝑡)≡0,𝑡≥𝑡^∗₁+(𝑡^∗₃−𝑡^∗₂)2.(5.25)If we put𝑢_𝜀(𝑥,𝑡)=𝜂(𝑡)𝑢_𝜀^[1](𝑥,𝑡)+(1−𝜂(𝑡))𝑢_𝜀^[2](𝑥,𝑡),𝑥∈𝐑,𝑡∈[0,𝑡^∗₃),(5.26)and use the fact that in the interval (𝑡^∗₁,𝑡^∗₁+(𝑡^∗₂−𝑡^∗₁)/2) we have𝑢_𝜀^[1]=𝑢_𝜀^[2]+𝒪_𝒟′(𝜀),(5.27)we see that the function 𝑢_𝜀 gives the weak asymptotic solution to (1.1) and (5.1) in the interval [0,𝑡^∗₃). Repeating described procedure for 𝑡>𝑡^∗₃, we obtain weak asymptotic solution in an arbitrary interval [0,𝑇]⊂𝐑⁺ (we have in mind the assumption on discreteness of set of moments in which interactions of nonlinear waves happen).
This concludes the proof.Example 5.2. Consider the Hamilton-Jacobi equation𝑢_𝑡+𝑓(𝑢_𝑥)=0,𝑥∈𝐑,𝑡∈𝐑⁺.(5.28)By introducing the following change of the unknown function 𝑢=∫^𝑥_𝑥0𝑤(𝑥^′)𝑑𝑥^′ and differentiating with respect to 𝑥∈𝐑 we get the equation𝑤_𝑡+(𝑓(𝑤))_𝑥=0,𝑥∈𝐑,𝑡∈𝐑⁺.(5.29)Thus, using the weak asymptotic method, we have global in time solution to the Hamilton-Jacobi equation providing approximate analytic description of the “essential” part of the Lagrange manifold (see the Section 1).

Acknowledgments

The work of the first author is supported by RFFI grant 05-01-00912, DFG Project 436 RUS 113/895/0-1. The work of the second author is supported in part by the local government of the municipality Budva.