Smooth Approximations of Global in Time Solutions to Scalar Conservation Laws

and Applied Analysis 3


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 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 a 1 , a 1 σ and a 2 − σ, a 2 .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 a 1 , a 1 σ and a 2 − σ, a 2 .
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 a 2 and right neighborhood of a 1 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 a 2 − σ, a 1 σ , and this issue is far from being trivial.
Next important moment of the approach presented here is the following.In onedimensional 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 u 2 , 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.

Notions, Notations, and Further Explanations
We give basic definitions and fundamental theorem of the method that we are going to usethe 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 ∈ C 1 0 R , the estimate holds, where the estimate on the right-hand side is understood in the usual Landau sense and locally uniformly in t, that is, 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 C 1 R ; D R , one has 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
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 where θ is the Heaviside function and , and for ρ ∈ R one has

2.4
At the beginning we make some remarks.Consider the point x 0 ∈ R such that Assume for the simplicity that such x 0 ∈ R is unique.In that case the shock wave will firstly arise on the characteristics along which the point x 0 moves.More precisely, blow up of the classical solution happens on the height u 0 x 0 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 u 1 x such that where K and b are constants determined from the conditions After that, we replace the piece of original initial data v x in a small neighborhood x 0 − 2ε μ , x 0 2ε μ of the point x 0 by the function u 1 x , x ∈ x 0 − ε μ , x 0 ε μ and by constants in the intervals x 0 − 2ε μ , x 0 − ε μ and x 0 ε μ , x 0 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 x 0 − ε μ , x 0 ε μ in the moment t * 1/K.The amplitude of the shock wave is going to be |v x 0 − 2ε − v x 0 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 .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 x 0 − ε μ , x 0 ε μ ; see Figure 4 than the point of gradient catastrophe in the case plotted in Figure 3 it is the point x 0 .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 .

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, 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 where σ is a positive constant.Furthermore, the functions u 00 , u 10 , and U 0 are nonincreasing and smooth, and they satisfy

3.3
This assumption is here in order to obtain the situation such that on characteristics emanating from the intervals −∞, a 2 and a 1 , ∞ we can have gradient catastrophe only for t ≥ C > 1/K i.e., after the straightening the curve connecting the points a 1 , v a 1 and a 2 , v a 2 .
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 conditions

3.4
Here, U and u 0 are classical solutions to the following Cauchy problems, respectively in a subdomain of R × R where they exist So, in the sequel, we will assume that the functions u 0 , 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 x 0 , t, ε , we need to define extremal "new characteristics," that is, the ones emanating from a 2 O ε and a 1 O ε .The proof is relatively simple and it relies on the basic ODE theory.

3.7
Introduce the function describing the relation between standard characteristics emanating from a 2 and a 1 , respectively.The function ρ given by describing the distance between two nonintersecting curves φ 2 and φ 1 , satisfies the following Cauchy problem where ρ ∂ τ ρ and it tends to a constant ρ 0 as τ → ∞.
Proof.The proof follows from the definition of the curves x x 01 , t and x x 02 , t , and standard ODE theory see Figure 5 and put F ρ, • 1 − 2B 1 ρ .For details one can consult 11 .
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 .where A is a constant large enough (which provides global solvability of the implicit equation x x x 0 , t, ε , t > 0, ε > 0; see [11,Theorem 5]).The curves x x 0 , t, ε , x 0 ∈ a 2 , a 1 one calls the "new characteristics" emanating from the interval
Note that we extended a little bit the interval a 2 , a 1 .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 a 2 , a 1 , and states that they do not intersect each other along entire time axis.Remark that the "new characteristics" emanate from the interval a 2 − εA a 1 − a 2 /2 , a 1 εA a 1 − a 2 /2 and that we state that only them are non-intersecting.The characteristics emanating out of the interval a 2 −εA a 1 −a 2 /2 , a 1 εA a 1 −a 2 /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 notations

3.12
Note that Ψ L and Ψ R are standard characteristics emanating from the intervals a 2 − σ, a 2 and a 1 , a 1 σ , respectively.Now, we can define the functions representing the "new characteristics" emanating out of a 2 − σ, a 1 σ .For x 0 < a 2 − σ we put 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 and this is set for which we need the "new characteristics" :

3.15
where x x 0 , t, ε is defined in Lemma 3.2.Denote by x 0 x 0 x, t, ε the inverse function to the function x X x 0 , t, ε , t > 0, ε > 0, of the "new characteristics" defined by 3.15 .Clearly, the function x 0 is not defined in the regions

3.16
Therefore, we introduce the following extension of x 0 :

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 a 2 − σ, a 1 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: Note that the functions x 0,L and x 0,R have the same regularity as the functions U 0 and u 00 , respectively.Therefore, 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 form

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 u 1 x, t, ε is given by

3.20
where x 0 x, t, ε is defined by 3.17 c we divide the real line on three intervals −∞, φ 2 − x 0,L , φ 2 − x 0,L , φ 1 x 0,R , and φ 1 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

3.21
For more detailed computation see 11 .
Considering the previous expression for x ∈ −∞, φ 2 − x 0,L and x ∈ φ 1 x 0,R , ∞ we get the following equations: which is true by definition of the functions U and u 0 see 3.5 and 3.6 .Thus, 3.21 reduces to

3.23
We pass to the interval φ 2 − x 0,L , φ 1 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 and from here, since B 2 B 1 1, we also have

3.25
Having this in mind, from 3.23 we get

3.26
Abstract and Applied Analysis 13 Multiplying the last relation by a test function η ∈ C 1 0 R and integrating over R we get recall that t is fixed

3.27
Since for t > t * Mσ/2 we have and since the functions U and u 0 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 I 1 0, t * σM and I 2 t * σM/2, T and common part of the intervals is large enough more precisely, it is enough to be |I 1 ∩ I 2 | > 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.

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: where u i x , x ∈ R, are continuous decreasing functions.Furthermore, we assume that u 0i , i 1, 2, 3, satisfy Clearly, the function u 0 x has two admissible jumps at the points a 1 and a 2 .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 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 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 problems

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, Furthermore, since the function u 0 x decreases and f is convex it is clear that φ 20t t − φ 10t t > 0.
We will prove the following theorem.Proof.In the sequel we use the following notation.By θ and δ the Heaviside and Dirac distributions, respectively, and u 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 obtain where, as usual, B i B i φ 2 − φ 1 /ε , i 1, 2. If we equalize by zero the sum of the coefficients multiplying δ 1 and δ 2 , respectively, we get 4.17 where

4.18
We can rewrite the function F ρ, t in the following manner:

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 ρ 0 of 4.17 .Clearly, ρ 0 ρ 0 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

4.21
Furthermore, notice that if |∂Ω i /∂t| < ∞, we have Therefore, we will determine Ω i , i 1, 2, so that they satisfy the following differential equations in τ ∈ IR compare with 4.21 and then we will prove that

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 * ,

4.25
Since for t t * we have φ 1 t * φ 2 t * ϕ t * , it follows that the extension is well defined.It is not difficult to find the solution to 4.23

4.26
Using the l'Hospital theorem for limits we get from here 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 get

4.28
Since the functions u i , i 1, 2, 3 satisfy 4.4 , we can rewrite the previous expression in the form If we multiply this by η ∈ C 1 0 R , we have 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 → 0 u 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 have

Scalar Conservation Law with Decreasing Initial Data
In this section, we consider 1.1 with the following initial condition: We assume that the function v x decreases and that it has finite number of points in which the function reaches maximum.Denote this set of points by S {x 1 , x 2 , . . ., x n }.Assume also that 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 where the constants K i and b i are determined by the conditions So, we have replaced the original initial data v by the function

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 have 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 by

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 1 U 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 * 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 v x, t * 1 .Then, we replace the part of the function v x, t * 1 around the point x 0 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 x 0 − ε μ , x 0 ε μ .Then, instead of the part of v x, t * 1 in the interval x 0 − 2ε μ , x 0 2ε μ we put the function r.In that way we get the function v * x .More precisely, we take

5.8
Then, using Sections 3 and 4, we find the weak asymptotic solution u * ε x, t to 1.

5.10
The function u 1 ε x, t is given by 5.11 while the functions u k ε 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 {x 1 , . . ., x n } the set such that for every x 0 ∈ {x 1 , . . ., 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 form 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 i 0 x i 0 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 , u 1 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 u 0 by v i .

22
Abstract and Applied Analysis 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 : where ρi ∂ τ i ρ i .
The functions v i x, t , i 1, . . ., n, are classical solutions to the following Cauchy problems in the regions t ∈ 0, t

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 φ 1 i t * 1 , 0 φ 2 i t * 1 , 0 , i 1, ..., n.Thus, we obtain the following initial data:

5.15
Then, we can assume that case a happened at the point x 0 x 2 placed between shock waves concentrated at the points φ 1 1 t * 1 , 0 and φ 1 2 t * 1 , 0 , and, at the same time, case b happened between shock waves concentrated in the points φ 1 2 t * 1 and φ 1 3 t * 1 .For the part of the function v 2 x, t * 1 disposed between φ 1 1 t * 1 , 0 and φ 1 2 t * 1 , 0 , we repeat the procedure from the beginning of the section.Namely, we approximate the function v 1 x, t * 1 in the interval x 2 − 2ε μ , x 2 2ε μ by the function u 2 x such that v n x, t − v n−1 x, t ω φ n t, τ n − x ε .

5.23
The function u 1 2 x 1 0 x, t, ε is analogical to the function u 1 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 conditions 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 equation u t f u x 0, x ∈ R, t ∈ R .

5.28
By introducing the following change of the unknown function u x x 0 w x dx and differentiating with respect to x ∈ R we get the equation w 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 .

Figure 1 :
Figure 1: Evolution of the initial data considered in the previous works.

Figure 2 :
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 − σ < a 1 σ are hollowed on the x-axis.

Figure 3 :
Figure 3: Evolution of a decreasing initial data.

Figure 4 :
Figure 4: Evolution of the approximated initial data.The points x 0 − ε μ and x 0 ε μ are dotted on the first plot.

Figure 5 :
Figure 5: The curve represents solution of 4.17 .Dot on the ρ axis, denoted by ρ 0 , is the minimal zero of the function F ρ, • .
. The functions U and u 0 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 ;

Theorem 4 . 1 .
The weak asymptotic solution to Cauchy problem 1.1 and 4.1 one has in the form

2 t * 1 ξ 2 t * 1 ξ
, but before the collision of the two shock waves, or b two shock waves will collide in the moment t * , but before a gradient catastrophe happens in the interval between them. u 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 . Since ρ φ 2 − φ 1 /ε after the interaction we have φ 1 φ 2 O ε .It remained to determine the functions Ω i , i 1, 2. In the sequel, we let 1 with initial data v * x in the strip t * 1 , t * Fix arbitrary T > 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 solution u ε x, t ∈ C 1 R × R of 1.1 and 4.1 has the form x 2 2ε μ , Denote this new function by v 2 x .Now, we solve 1.1 with the function v 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 * where K and b are appropriate constants.