# Smooth Approximations of Global in Time Solutions to Scalar Conservation Laws

Authors: V. G. Danilov; D. Mitrovic
Journal: Abstract and Applied Analysis
 (2009)

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## 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.


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## Body

## 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 Definition2.1) 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(1.1)∂u∂t+∂f(u)∂x=0,f′′>0,(1.2)u|t=0=v(x)∈C(R), 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 fixedt∈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 distance𝒪(ε),
where ε is a parameter of approximation.The difference between standard and new
characteristics is plotted on Figure2 (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+×Rd.
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.Evolution of
the initial data considered in the previous works.
(a)(b)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 pointsa2−σ<a1+σ are hollowed on the x-axis.
(a)(b)Evolution of a
decreasing initial data.
(a)(b)Next important moment of the approach presented here
is the following. In one-dimensional case, the solutionu 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
fluxf(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 int∈R+ of a process. Therefore, we give formal
definition of the notion.
Definition 1.1.
By smooth int∈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 Section2, 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.
ByO𝒟'(εα)⊂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)∈C01(R),
the estimate(2.1)〈O𝒟'(εα),η(x)〉=O(εα),ε→0,holds, where the estimate on the
right-hand side is understood in the usual Landau sense and locally uniformly
in t,
that is, |O(εα)|≤CTεα for t∈[0,T].Definition 2.2.
The family of functions(uε)ε=(uε(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 has(2.2)uεt+(f(uε))x=𝒪𝒟'(εα),uε|t=0−v=𝒪𝒟'(εα),ε→0.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 functionsωi∈C∞(R), i=1,2,
satisfy limz→+∞ωi(z)=1, limz→−∞ωi(z)=0 and dωi(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
(2.3)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)+𝒪𝒟'(ε),where θ is the Heaviside function and Bi=Bi((φ2−φ1)/ε), i=1,2,
and for ρ∈R one has(2.4)B1(ρ)=∫ω˙1(z)ω2(z+ρ)dz,B2(ρ)=∫ω˙2(z)ω1(z−ρ)dz,B1(ρ)+B2(ρ)=1.At the beginning we make some remarks. Consider the
pointx0∈R such that(2.5)t*=minx∈R−1f′′(u0(x))u0′(x)=−1f′′(u0(x0))u0′(x0).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 int∈R+,
on the first step we define the function u1(x) such that(2.6)f′(u1(x))=−Kx+b,x∈R,where K and  b are constants determined from the conditions(2.7)u1(x0−εμ)=v(x0−2εμ),u1(x0+εμ)=v(x0+2εμ),0<μ<1.After that, we replace the piece
of original initial data v(x) in a small neighborhood (x0−2εμ,x0+2εμ) of the point x0 by the function u1(x), x∈(x0−εμ,x0+εμ) and by constants in the intervals (x0−2εμ,x0−εμ) and (x0+εμ,x0+2εμ) so that the continuity is preserved (see
Figure 4). This change of initial condition provides the shock wave to arise
from the interval (x0−εμ,x0+εμ) 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 uε to our problem (with transformed initial data)
will be continuous function (see Figure 4 again).Evolution of
the approximated initial data. The pointsx0−εμ and x0+εμ are dotted on the first plot.
(a)(b)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−εμ,x0+εμ);
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 int∈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,(3.1)u1=u1(x,t,ε),U=U(x,t),u0=u0(x,t),Bi=Bi(ρ),ϕi=ϕi(t,ε),i=1,2,x0,R=x0,R(t,ε),x0,L=x0,L(t,ε),θ1=θ(ϕ1+x0,R−x),θ2=θ(ϕ2−x0,L−x),δi=−(θi)x′,i=1,2,t*=a1−a2f′(U0)−f′(u00)=1K,x*=f′(U0)t*+a2=f′(u00)t*+a1=f′(U0)a1−f′(u00)a2f′(U0)−f′(u00),where θ is the Heaviside function and Bi, 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 functionv(x) from (1.2) has the form (see Figure 4 with σ=εμ)(3.2)v(x)={U0(x),x<a2−σ,U0(a2−σ),a2−σ<x<a2,u10(x),a2<x<a1,u00(a1+σ),a1<x<a1+σ,u00(x),x>a1+σ,where σ is a positive constant. Furthermore, the
functions u00,u10, and U0 are nonincreasing and smooth, and they
satisfy(3.3)u00(x)=u00(a1+σ),x>a1+σ,U0(x)=U0(a2−σ),x<a2−σ,f′(u10(x))=−Kx+b,x∈[a2,a1],u10(a2)=u00(a2+σ)=u00,u10(a1)=U0(a1−σ)=U0,a1>a2,−1f′′(u00(x))u00'(x0)≥C>1K,x<a2,−1f′′(U0(x))U0′(x0)≥C>1K,x>a1.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 fort>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)c′(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*.Here, U and u0 are classical solutions to the following
Cauchy problems, respectively (in a subdomain of R+×R where they exist)(3.5)∂tu0+∂xf(u0)=0,u|t=0=u00(x),(3.6)∂tU+∂xf(U)=0,u|t=0=U0(x).So, in the sequel, we will assume that the functionsu0,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”
(denotedx(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 curvesϕ2=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:(3.7)dϕ1dt=(B2(ρ)−B1(ρ))f′(u0(ϕ1,t))+c(t)B1(ρ),ϕ1(0,ε)=a1+εAa1−a22,dϕ2dt=(B2(ρ)−B1(ρ))f′(U(ϕ2,t))+c(t)B1(ρ),ϕ2(0,ε)=a2−εAa1−a22.Introduce the function(3.8)τ=f′(U0)t+a2−f′(u00)t−a1ε=ψ0(t)ε,describing the relation between
standard characteristics emanating from a2 and a1,
respectively.
The functionρ given by(3.9)ρ(t,τ)=ϕ2(t,ε)−ϕ1(t,ε)ε,describing the distance between
two nonintersecting curves ϕ2 and ϕ1,
satisfies the following Cauchy problem(3.10)ρ˙=1−2B1(ρ),limτ→−∞ρτ=1,where ρ˙=∂τρ and it tends to a constant ρ0 as τ→∞.Proof.
The
proof follows from the definition of the curvesx(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 ρ axis, denoted by ρ0,
is the minimal zero of the function F(ρ,⋅).After inessential changes caused by replacing the
constantc 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 solutionsx=x(x0,t), x0∈[a2,a1],
to the following Cauchy problem(3.11)x˙=(B2(ρ)−B1(ρ))f′(u1)+c(t)B1(ρ),u˙1=0,u1(0)=u10(x0),x(0)=x0+εA(x0−a1+a22),x0∈[a2,a1],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
entirex-axis, we introduce the following
notations(3.12)ΨL(x0,t,ε)=f′(U0(x0))t+x0+ε(x0−a1−a22),x0<a2−σ,ΨR(x0,t,ε)=f′(u00(x0))t+x0+ε(x0−a1−a22),x0>a1+σ.Note that ΨL and ΨR 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).Forx0<a2−σ we put(3.13)xL(t,x0,ε)={ΨL(x0,t,ε),ϕ2(t,x0,ε)−ΨL(t,x0,ε)>ε(a2−σ−x0),ϕ2(t,x0,ε)−ε(a2−x0),ϕ2(t,x0,ε)−ΨL(t,x0,ε)≤ε(a2−σ−x0).Similarly, for x0>a1+σ,(3.14)xR(t,x0,ε)={ΨR(x0,t,ε),ΨR(t,x0,ε)−ϕ1(t,x0,ε)>ε(x0−a1−σ),ϕ1(t,x0,ε)−ε(a2−x0),ΨR(t,x0,ε)−ϕ1(t,x0,ε)≤ε(x0−a1−σ).For better understanding of the
previous formulas see Figure 2. Actually, xL as well as xR 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 forx0∈R∖((a2−σ,a2)∪(a1,a1+σ)) (and this is set for which we need the “new
characteristics”):(3.15)X(t,x0,ε)={xR(t,x0,ε),x0≤a2−σ,x(t,x0,ε),a2≤x0≤a1,xL(t,x0,ε),x0>a1,where x(x0,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(x0,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)ΓL={(x,t):0<t<t*+Mσ,f′(U00)t+a2−σ<x<ϕ2},ΓR={(x,t):0<t<t*+Mσ,f′(u000)t+a1+σ>x>ϕ1}.Therefore, we introduce the following
extension of x˜0:(3.17)x0(x,t,ε)={x˜0(x,t,ε),(x,t)∉ΓL∪ΓR,f′(U00)t+a2−σ,(x,t)∈ΓL,f′(u000)t+a1+σ,(x,t)∈ΓR.To continue, assume that the increase of strength of
the shock wave starts in the momentt*+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 x0,L(t,ε) and x0,R(t,ε) so that they are equal to zero for t<t*+Mσ and defined for t≥t*+Mσ as follows:(i)
x0,L(t,ε)=ε(a2−x0) for x0<a2−σ such that ϕ2(t,x0,ε)−ΨL(t,x0,ε)=ε(a2−x0),(ii)
x0,R(t,ε)=ε(x0−a1) for x0>a1+σ such that ΨR(t,x0,ε)−ϕ1(t,x0,ε)=ε(x0−a1).Note that the functionsx0,L and x0,R have the same regularity as the functions U0 and u00,
respectively. Therefore,(3.18)∂x0,R∂t=∂x0,L∂t=𝒪(ε).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)uε(x,t)=u0(x,t)+(u1(x,t,ε)−u0(x,t))ω1(ϕ1(t,ε)+x0,R(t,ε)−xε)+(U(x,t)−u1(x,t,ε))ω2(ϕ2(t,ε)−x0,L(t,ε)−xε),where ωi, i=1,2,
satisfy the conditions from Theorem 2.3 and the functions ϕi=ϕi(t,ε), t∈R+, i=1,2,
are given in Lemma 3.1.
The functionu1(x,t,ε) is given by(3.20)u1(x,t,ε)={U(x,t),x<ϕ2−x0,L(t,ε),u0(x0(x,t,ε)),ϕ2−x0,L(t,ε)≤x≤ϕ1+x0,R(t,ε),u0(x,t),x>ϕ1+x0,R(t,ε),where x0(x,t,ε) is defined by (3.17).
The functionsU 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−x0,L], (ϕ2−x0,L,ϕ1+x0,R), and [ϕ1+x0,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 functionuε(x,t) from (3.19) in (1.1). Using Theorem 2.3 we
get(3.21)[∂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)ϕ1t−B2(f(u1)−f(u0))−B1(f(U)−f(U+u0−u1)))δ1+((U−u1)φ2t−B2(f(U)−f(u1))−B1(f(U+u0−u1)−f(u0)))δ2=𝒪𝒟'(ε).For more detailed computation
see [11].
Considering the previous expression forx∈(−∞,ϕ2−x0,L) and x∈(ϕ1+x0,R,+∞) we get the following
equations:(3.22)∂U∂t+f′(U)∂U∂x=𝒪𝒟'(ε),x∈(−∞,ϕ2−x0,L),∂u0∂t+f′(u0)∂u0∂x=𝒪𝒟'(ε),x∈(ϕ1+x0,R,∞),which is true by definition of
the functions U and u0 (see (3.5) and (3.6)). Thus, (3.21) reduces
to(3.23)(∂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+x0,L)−B2(f(u1)−f(u0))−B1(f(U)−f(U+u0−u1)))δ1+((U−u1)∂t(ϕ2−x0,R)−B2(f(U)−f(u1))−B1(f(U+u0−u1)−f(u0)))δ2=𝒪𝒟'(ε).
We pass to the interval(ϕ2−x0,L,ϕ1+x0,R).
Notice that fort<t*+σM we have x0,R(t,ε)=x0,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])(3.24)B2−B1=𝒪(εN),N∈N,and from here, since B2+B1=1,
we also have(3.25)Bi=12+𝒪(εN),i=1,2,N∈N.Having this in mind, from (3.23)
we get(3.26)(∂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+x0,R)−12(f(u1)−f(u0))−12(f(U)−f(U+u0−uL)))δ1+((U−u1)∂t(ϕ2−x0,L)−12(f(U)−f(u1))−12(f(U+u0−u1)−f(u0)))δ2=𝒪𝒟'(ε).Multiplying the last relation by
a test function η∈C01(R) and integrating over R we get (recall that t is fixed)(3.27)(u1(ϕ1+x0,R,t,ε)(∂tϕ1+∂tx0,L)η(ϕ1+x0,R,t)−u1(ϕ2−x0,L,t,ε)(∂tϕ2−∂tx0,R)η(ϕ2−x0,L,t)−c′(t)u1(ϕ1+x0,R,t,ε)η(ϕ1+x0,R,t)+c′(t)u1(ϕ2−x0,L,t,ε)η(ϕ2−x0,L,t))+∫ϕ2−x0,Lϕ1+x0,R12[ddx(f(U+u0−u1)+f(u1)−2c′(t)u1)]η(x)dx+∫R((u1−u0)∂t(ϕ1+x0,R)−12(f(u1)−f(u0))+12(f(u1)+f(u0)−2c′(t)u1))δ1⋅η(x)dx+∫R((U−u1)∂t(ϕ2−x0,L)−12(f(U)−f(u1))+12(f(u1)+f(u)−2c′(t)u1))δ2⋅η(x)dx=𝒪(ε).Since for t>t*+Mσ/2 we have(3.28)u1(ϕ1+x0,R,t,ε)=u0(x0(ϕ1+x0,R,t,ε)),u1(ϕ2−x0,L,t,ε)=U(x0(ϕ2−x0,L,t,ε)),ϕ1+x0,R−ϕ2+x0,L=𝒪(ε),ϕ1+x0,R−c(t)=ϕ2+x0,L−c(t)=𝒪(ε),∂t(ϕ1+x0,L)−c′(t)=∂t(ϕ2−x0,R)−c′(t)=𝒪(ε),∂tx0,L=∂tx0,R+𝒪(ε)=𝒪(ε),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 fort<t*+Mσ can be found in [11].
Sinceuε 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(εα) for an α<1), we see that uε 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:(4.1)u(x,0)=u0(x)=u10(x)+(u20(x)−u10(x))θ(a1−x)+(u30(x)−u20(x))θ(a2−x),where ui(x), x∈R,
are continuous decreasing functions. Furthermore, we assume that u0i, i=1,2,3,
satisfy(4.2)u1(a1)<u2(a1),u2(a2)<u3(a2).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(4.3)−1f′′(ui(x))ui′(x)≥T>t*,i=1,2,3,x∈R.Such conditions provide that the
gradient catastrophe will not happen before the two shock waves collide.Byui(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)(4.4)ut+(f(u))x=0,u|t=0=ui0(x),i=1,2,3,where the flux f is the same as the one from (1.1). Since the
initial functions ui0(x), i=1,2,3,
are decreasing, the solutions to (4.4) will be also decreasing.Byϕi0(t), i=1,2, t∈[0,t*), and φ(t), t∈[t*,T],
we denote the solutions to the following Cauchy problems(4.5)ϕi0t(t)=f(ui+1(ϕi0(t),t))−f(ui(ϕi0(t),t)ui+1(ϕi0(t),t)−ui(ϕi0(t),t),ϕi0(0)=ai,φt(t)=f(u3(φ(t)))−f(u1(φ(t)))u3(φ(t))−u1(φ(t)),φ(t*)=ϕ10(t*)=ϕ20(t*).Note that continuous solutions
to those Cauchy problems always exist since those are actually Rankine-Hugoniot
conditions for the admissible shock placed at ϕi0, i=1,2, and corresponding to Cauchy problem (1.1) with
initial conditions, respectively (below we imply ui,0=ui0)(4.6)u|t=0=ui0(x)+(ui+1,0(x)−ui0(x))θ(ai−x),i=1,2,u|t=t*=u1(x,t*)+(u3(x,t*)−u1(x,t*))θ(φ(t*)−x).Furthermore, since the functionu0(x) decreases and f is convex it is clear that (ϕ20t(t)−ϕ10t(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 form(4.7)uε(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),where x∈R, t∈[0,t*+σ)⊂[0,T),
and the functions ωi, i=1,2,
are the ones from Theorem 2.3.
Here,(4.8)τ=ϕ20(t)−ϕ10(t)ε=ψ0(t)ε.The function ρ=ρ(τ,t) is given by the relation(4.9)ρ(τ,t)=ϕ2(τ,t)−ϕ1(τ,t)ε,and the functions ϕi(t,τ), i=1,2,
are given by the formulas(4.10)ϕ1(t,τ)=ϕ10(t)+ψ0(t)Ω1(t,τ)ϕ11+(t),(4.11)ϕ2(t,τ)=ϕ20(t)+ψ0(t)Ω2(t,τ)ϕ21+(t),where(4.12)ϕi1+(t)=φ(t)−ϕi0(t)ψ0(t),i=1,2,and Ωi, i=1,2,
are given by (4.26).Proof.
In the sequel we use the following
notation.
Byθ and δ the Heaviside and Dirac distributions,
respectively, and(4.13)δi=δ(x−ϕi(t)),θi=θ(x−ϕi(t)),i=1,2,uj(ϕi)=uj(ϕi(t),t),i=1,2,j=1,2,3.We start by substituting (4.7) in
(1.1) and applying Theorem 2.3. We obtain(4.14)u1t+(u2t−u1t)θ1+(u3t−u2t)θ2+ϕ1t(u2(ϕ1)−u1(ϕ1))δ1+ϕ2t(u3(ϕ2)−u2(ϕ2))δ2+u1xf′(u1)+θ1[(u3xf′(u3x)−f′(u1+u3−u2)(u1x+u3x−u2x))B1+(f′(u2)u2x−f′(u1)u1x)B2]+θ2[(u3xf′(u3)−u2xf′(u2))B2+(f′(u1+u3−u2)(u1x+u3x−u2)−f′(u1)u1x)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=𝒪𝒟'(ε),where, as usual, Bi=Bi((ϕ2−ϕ1)/ε), i=1,2.
If we equalize by zero the sum of the coefficients multiplying δ1 and δ2,
respectively, we get(4.15)ϕ1t(u2(ϕ1)−u1(ϕ1))−B1(f(u3(ϕ1))−f(u1+u3−u2)(ϕ1))−B2(f(u2(ϕ1))−f(u1(ϕ1)))=0,(4.16)ϕ2t(u3(ϕ2)−u2(ϕ2))−B2(f(u3(ϕ2))−f(u2(ϕ2)))+B1(f((u1+u3−u2)(ϕ2))−f(u1(ϕ2)))=0.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])(4.17)dρdτ=ψ˙0−b1(αe1|x=ϕ1+αe2|x=ϕ2)=f(ρ,t),ρτ|τ→−∞=1,where(4.18)α=f(u3)−f(u2)−f(u1+u3−u2)+f(u1),e1=u2−u1,e2=u3−u2.We can rewrite the function F(ρ,t) in the following manner:(4.19)f(ρ,t)=[f(u3)−f(u1+u3−u2)u2−u1|x=ϕ1−f(u1+u3−u2)−f(u1)u3−u2|x=ϕ2]+(αe1|x=ϕ1+αe2|x=ϕ1).
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 functionsΩi, i=1,2.
In the sequel, we let(4.20)Ω˙i=∂Ωi∂τ,i=1,2.
Substituting expressions (4.10) and (4.11) in (4.15) and
(4.16), respectively, we obtain the equations(4.21)∂Ωi∂tψ0ϕi1+τ+Ω˙i+Ωi(ψ0ϕi1+)tτϕi1+=(−1)iB1τf(u3)−f(u2)−f(u1+u3−u2)+f(u1)ui+1−ui|x=ϕi,i=1,2.
Furthermore, notice that if|∂Ωi/∂t|<∞, we have(4.22)∂Ωi∂tψ0ϕi1+τ=ε∂Ωi∂tϕi1+=𝒪(ε).Therefore, we will determine Ωi, i=1,2,
so that they satisfy the following differential equations in τ∈IR (compare with (4.21))(4.23)Ω˙i+Ωi(ψ0ϕi1+)tτϕi1+=(−1)iB1τf(u3)−f(u2)−f(u1+u3−u2)+f(u1)u2−u1|x=ϕi,i=1,2,and then we will prove
that(4.24)|∂Ωi∂t|<∞.Notice that differential
equation (4.23) is defined only on the interval [0,t*].
Therefore, it is necessary to extend continuously the functions Fi(ρ,t) after the moment t=t*.
Put for t>t*,(4.25)Fi(t)=f(u3)−f(u2)−f(u1+u3−u2)+f(u1)ui+1−ui|x=φ(t),i=1,2.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)(4.26)Ωi(t,τ)=1τ(ψ0ϕi1+)t/ϕi1+∫0τ(−1)iB1(ρ)Fi(t)τ˜(ψ0ϕi1+)t/ϕi1+dτ˜.Using the l'Hospital theorem for
limits we get from here(4.27)|limτ→±∞Ωi(τ,t)|<∞,|limτ→±∞∂Ωi(τ,t)∂t|<∞,proving (4.24) and thus (4.22).
This, in turn, proves that Ωi, 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
multiplyingδ 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)u1t+u1xf′(u1)+[u3t+f(u3)u3x−u1t−f(u1)u1x]θ1+[u3t−u2t+u3xf(u3)−u2xf(u2)+B1(−u3xf′(u3x)+u2xf(u2)−f′(u1+u3−u2)(u1x+u3x−u2x)+f(u1)u1x)](θ2−θ1)=𝒪𝒟'(ε).
Since the functionsui, i=1,2,3 satisfy (4.4), we can rewrite the
previous expression in the form(4.29)B1[−u3xf′(u3x)+u2xf(u2)−f′(u1+u3−u2)(u1x+u3x−u2x)+f(u1)u1x](θ2−θ1)=𝒪𝒟'(ε).If we multiply this by η∈C01(R),
we have(4.30)ερB1(ρ)1ϕ1−ϕ2∫ϕ2ϕ1[(−u3xf′(u3x)+u2xf(u2)−f′(u1+u3−u2)(u1x+u3x−u2x)+f(u1)u1x)]η(x)dx=𝒪(ε),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 sinceuε represents the weak asymptotic solution to the
considered problem we have w−limε→0uε=u where u is the weak solution to the considered problem.
Therefore, the functions ϕi, i=1,2,
up to 𝒪(ε),
satisfy the Rankine-Hugoniot conditions. In turn, from there it follows that
for t>t* we have(4.31)limτ→+∞Ω(τ,t)=1,and for t<t* we have(4.32)limτ→−∞Ω(τ,t)=0.


## 5. Scalar Conservation Law with Decreasing Initial Data

In this section, we consider (1.1) with the following initial condition:(5.1)u|t=0=v(x)∈C(R).We assume that the function v(x) decreases and that it has finite number of
points in which the function(5.2)−1f′′(v(x))v′(x),x∈R,reaches maximum. Denote this set
of points by S={x1,x2,…,xn}.
Assume also that x1<x2<⋯<xn.Now, we continue in the following way. Around everyxi∈S, i=1,…,n,
we allocate ε-neighborhoods of the form (xi−2εμ,xi+2εμ), μ∈(0,1).
Then, we transform the initial data v,
replacing it by the function ui(x), x∈(xi−2εμ,xi+2εμ),
such that for every i=1,…,n,
we have the following (compare Figures 3 and 4)(5.3)f′(ui(x))=−Kix+bi,x∈(xi−εμ,xi+εμ),ui(x)=v(xi−2εμ),x∈(xi−2εμ,xi−εμ),ui(x)=v(xi−2εμ),x∈(xi+εμ,xi+2εμ),where the constants Ki and bi are determined by the conditions(5.4)ui(xi−εμ)=v(xi−2εμ),ui(xi+εμ)=v(xi+2εμ),i=1,…,n.So, we have replaced the
original initial data v by the function(5.5)v^(x)={v(x),x∈(−∞,x1−2εμ)∪(x1+2εμ,x2−2εμ)∪⋯∪(xn−1+2εμ,xn−2εμ)∪(xn+2εμ,∞),ui(x),x∈(xi−2εμ,xi+2εμ),i=1,…,n.Obviously, we have the following
estimate fulfilled v(x)−v^(x)=𝒪𝒟'(εμ).
Beside that, since the function f′′(v(x))v′(x) reaches its maximum at the points xi∈S,
then there exist neighborhoods 𝒰(xi), i=1,…,n,
such that for every x∈𝒰(xi) we have(5.6)f′′(v(x))v′(x)>f′′(v(y))v′(y)wherey∉⋃i=1n𝒰(xi).The moment of “straightening” of
the curves connecting the points (xi−εμ,v(xi−εμ)) and (xi+εμ,v(xi+εμ)),
according to Section 2 and the Lagrange mean value theorem is given
by(5.7)t1*=2εμf′(v(xi−εμ))−f′(v(xi+εμ))=−1(f′∘v)′(x^i)=−1(f′′(v(x^i))v′(x^i)<maxy∈U−1f′′(v(x))v′(x),x^i∈(xi−εμ,xi+εμ).From here we see that for everyi=1,…,n we have t1*>−1/f′′(v(y))v′(y),
where y∉⋃i=1n𝒰(xi).
This actually means that a gradient catastrophe will not happen before straightening
of the lines connecting the points xi−εμ and xi+εμ,
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 datav^ε(x) in (x,u) space. In the interval [0,t1*−g(ε)),
for appropriate function g(ε)=𝒪(εμ),
the solution will be continuous function. In the interval [t1*−s(ε),t*+s(ε)],
for some function s(ε)=𝒪(εμ),
the curves connecting the points (xi−εμ,v(xi−εμ)) and (xi+εμ,v(xi+εμ)), 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 momentt2*=t1*+ξ,
but before the collision of the two shock waves, or(b)
two shock waves will collide in the momentt2*=t1*+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 inx∈R of the weak asymptotic solution in the moment t1*.
Denote it by v˜(x,t1*).
Then, we replace the part of the function v˜(x,t1*) 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,t1*) such that f′(r(x,t1*))=−Kx+b,
for some constants K and b,
in the interval (x0−εμ,x0+εμ).
Then, instead of the part of v˜(x,t1*) in the interval (x0−2εμ,x0+2εμ) we put the function r.
In that way we get the function v*(x).
More precisely, we take(5.8)v(x,t1*)={v˜(x,t1*),x∉(x0−2εμ,x0+2εμ),r(x,t1*),x∈(x0−εμ,x0+εμ),v˜(x0−εμ,t1*),x∈(x0−2εμ,x0−εμ),v˜(x0+εμ,t1*),x∈(x0+εμ,x0+2εμ).Then, using Sections3 and 4, we find the weak asymptotic solution uε*(x,t) to (1.1) with initial data v*(x) in the strip (t1*,t2*+(t3*−t2*)/2)×R where t3* 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,t1*+(t2*−t1*)/2) and [t1*,t2*+(t3*−t2*)/2).In case (b), we use the results of Section4 on (1.1) with initial condition v˜(x,t1*). Then, like in case (a) we use the partition of unity to
connect solutions on the intervals [0,t1*+(t2*−t1*)/2) and [t1*,t2*+(t3*−t2*)/2),
where t3* 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 ti*, 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 solutionuε(x,t)∈C1(R+×R) of (1.1) and (4.1) has the
form(5.9)uε(x,t)=η1(t)uε[1](x,t)+η2(t)uε[2](x,t)+⋯+ηn(t)uε[n](x,t),where ηi, i=1,…,n is partition of unity of the interval [0,T] such that (we take t0*=0, t−1*=−1 and tn+1*=T+1)(5.10)ηi(t)≡1,t∈(ti−1*,ti*),ηi(t)≡0,t∉(ti−2*,ti+1*).The function uε[1](x,t) is given by (5.11) while the functions uε[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,…,xn} the set such that for every x0∈{x1,…,xn} we have (2.5) satisfied. According to the
previous analysis, in the interval [0,t1*+(t2*−t1*)/2) the weak asymptotic solution to (1.1) and (5.1) we have in the form(5.11)uε[1](x,t)=v0(x,t)+(u1(x01(x,t,ε))−v0(x,t))ω1(ϕ11(t,τ1)−xε)+(v1(x,t)−u1(x01(x,t,ε)))ω2(ϕ12(t,τ1)−xε)+⋯+(un(x0n(x,t,ε),t)−vn−1(x,t))ω1(ϕn1(t,τn)−xε)+(vn(x,t)−un(x0n(x,t,ε),t))ω2(ϕn2(t,τn)−xε),where ωi, 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 functionsx0i=x0i(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 ρi, u1 by ui,
and c by ci.
The functions ci we obtain when in (3.4) we replace U by vi−1,
and u0 by vi.(ii)
The functionsϕij=ϕij(t,τi), i=0,1,…,n, j=1,2,
are the “new characteristics” emanating from the points xi+(−1)jεμ,
and
(5.12)τi=φi10(t)−φi20(t)ε,ρi=ρi(t,τi)=ϕi1(t,τi)−ϕi2(t,τi)ε,φi20(t)=f′(v(xi−εμ))t+xi−εμ,φi10(t)=f′(v(xi+εμ))t+xi+εμ,i=1,…,n.
Also, notice that according to
Theorem 3.3 the functions ρi are the solutions to the following Cauchy
problems (analogical to Cauchy problem (3.10)):(5.13)ρ˙i=1−2B1(ρi),limτ→−∞ρiτi=1,where ρ˙i=∂τiρi.
The functionsvi(x,t), i=1,…,n,
are classical solutions to the following Cauchy problems in the regions t∈[0,t*+(t2*−t1*)/2), x∈R,(5.14)vit+(f(vi)vi)x=0,vi(x,0)=v(x).As we have already explained, in
the moment t1*+ξ=t2* 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 momentt2* (that means that we have combination of
situations (a) and (b)). It only remains to impose initial
data at the moment t=t1* for (1.1). We simply put t=t1* in (1.1) and use the fact that up to 𝒪(ε),
we have ϕi1(t1*,0)=ϕi2(t1*,0), i=1,...,n.
Thus, we obtain the following initial data:(5.15)v[2](x,t1*)=v0(x,t1*)+(v1(x,t1*)−v0(x,t1*))ω1(ϕ11(t1*,0)−xε)+(v1(x,t1*)−v2(x,t1*))ω2(ϕ21(t1*,0)−xε)+⋯+(vn−1(x,t1*)−vn−2(x,t1*))ω1(ϕn−11(t1*,0)−xε)+(vn(x,t1*)−vn−1(x,t1*))ω2(ϕn1(t1*,0)−xε).
Then, we can assume that case (a) happened at the pointx0=x[2] placed between shock waves concentrated at the
points ϕ11(t1*,0) and ϕ21(t1*,0),
and, at the same time, case (b) happened between shock waves concentrated in the points ϕ21(t1*) and ϕ31(t1*).
For the part of the functionv[2](x,t1*) disposed between ϕ11(t1*,0) and ϕ21(t1*,0),
we repeat the procedure from the beginning of the section. Namely, we approximate
the function v1(x,t1*) in the interval (x[2]−2εμ,x[2]+2εμ) by the function u[2](x) such that(5.16)f′(u[2](x))=−Kx+b,x∈(x[2]−εμ,x[2]+εμ),u[2](x)=v1(x[2]−2εμ,t1*),x∈(x[2]−2εμ,x[2]−εμ),u[2](x)=v1(x[2]+2εμ,t1*),x∈(x[2]+εμ,x[2]+2εμ),where K and b are appropriate constants.
Denote this new function byv˜[2](x).
Now, we solve (1.1) with the function v˜[2] as initial data but starting from t=t1*.
According to Sections 3 and 4, the weak asymptotic solution to (1.1) with initial condition u(x,t1*)=v˜[2](x) in the interval [t1*,t2*+(t3*−t2*)/2),
where t3* 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 form(5.17)uε[2](x,t)=v0(x,t)+(v0[2](x,t)−v0(x,t))ω(ϕ1(t,τ1)−xε)+(u1[2](x01(x,t,ε))−v0[2](x,t))ω(ϕ[2]1(t,τ[2])−xε)+(v1[2](x,t)−u1[2](x01(x,t,ε)))ω(ϕ[2]2(t,τ[2])−xε)+(v2(x,t)−v1[2](x,t))ω(ϕ2(t,τ2)−xε)+(v3(x,t)−v2(x,t,ε))ω(ϕ3(t,τ2)−xε)+⋯+(vn−1(x,t)−vn−2(x,t))ω(ϕn−1(t,τn−1)−xε)+(vn(x,t)−vn−1(x,t))ω(ϕn(t,τn)−xε).Here, the functions ϕi, i=1,4,…,n,
are independent on ε and they are given by the following Cauchy
problems(5.18)ϕ1t=f(v0[2](ϕ1,t))−f(v0(ϕ1,t))v0[2](ϕ1,t)−v0(ϕ1,t),ϕ1(t1*)=ϕ11(t1*,0),ϕit=f(vi−1(ϕi,t))−f(vi(ϕi,t))vi−1(ϕi,t)−vi(ϕi,t),ϕi(t1*)=ϕi1(t1*,0),where i=4,…,n, t∈[t1*,t2*+(t3*−t2*)/2).
If byϕ20(t) and ϕ30(t) we denote global solutions of the following
Cauchy problems (see corresponding equations (4.7)):(5.19)ϕ20t=f(v2(ϕ20,t))−f(v1[2](ϕ20,t))v2(ϕ20,t)−v1[2](ϕ20,t),ϕ20(t1*)=ϕ21(t1*,0),ϕ30t=f(v3(ϕ20,t))−f(v2(ϕ20,t))v3(ϕ30,t)−v2(ϕ30,t),ϕ30(t1*)=ϕ31(t1*,0),t∈[t1*,t2*+(t3*−t2*)2),we have the quantity τ2 defined by(5.20)τ2=ϕ20(t)−ϕ30(t)ε.If we also put(5.21)ϕ[2]10(t)=f′(v(x[2]+εμ))t+x[2]+εμ,ϕ[2]20(t)=f′(v(x[2]−εμ))t+x[2]−εμ,t∈[t1*,t2*+(t3*−t2*)2),then we have the quantity τ[2] given by(5.22)τ[2]=ϕ[2]10(t)−ϕ[2]20(t)ε.Note that ϕ[2]i0, i=1,2,
are standard characteristics emanating from x[2]+εμ and x[2]−εμ.
The functionsvi(x,t), i=0,2,3,…,n are classical solutions to Cauchy problems
(5.14) in the regions t∈[t1*,t2*+(t3*−t2*)/2), x∈R,
while v0[2](x,t) and v1[2](x,t) satisfy, respectively, the following
equations:(5.23)v0[2]t+f′(v0[2])v0[2]x=0,v0[2](x,0)=v(x),t∈[t1*,t2*+(t3*−t2*)2),x∈R,v1[2]t+f′(v1[2])v1[2]x=0,v1[2](x,0)=v(x).The function u1[2](x01(x,t,ε)) is analogical to the function u1(x,t,ε) from (5.11) with an obvious difference in
indexing.
The functionsϕi, i=1,4,5,…,n,
are independent on ε and they are given by Rankine-Hugoniot
conditions(5.24)ϕit=f(vi(ϕi,t))−f(vi−1(ϕi,t))vi(ϕi,t)−vi−1(ϕi,t),ϕ1t=f(v0[1](ϕ1,t))−f(v0(ϕ1))v0[1](ϕ1,t)−v0(ϕ1,t),ϕi(t1*)=ϕi1(t1*,0),i=1,4,5,…,n.
Thus, we have described the weak asymptotic solution
to (1.1), (5.1) in the intervals [0,t1*+(t2*−t1*)/2) and [t1*,t2*+(t3*−t2*)/2) through the functions uε[1] and uε[2],
respectively. Notice, then, the partition of unity η(t)∈C01(R+) of the time axis satisfying(5.25)η(t)≡1,t≤t1*,η(t)≡0,t≥t1*+(t3*−t2*)2.If we put(5.26)uε(x,t)=η(t)uε[1](x,t)+(1−η(t))uε[2](x,t),x∈R,t∈[0,t3*),and use the fact that in the
interval (t1*,t1*+(t2*−t1*)/2) we have(5.27)uε[1]=uε[2]+𝒪𝒟'(ε),we see that the function uε gives the weak asymptotic solution to (1.1) and (5.1) in the interval [0,t3*).
Repeating described procedure for t>t3*,
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(5.28)ut+f(ux)=0,x∈R,t∈R+.By introducing the following
change of the unknown function u=∫x0xw(x′)dx′ and differentiating with respect to x∈R we get the equation(5.29)wt+(f(w))x=0,x∈R,t∈R+.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).



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*Source: 350762-2009-03-10.xml*