2. Construction of the Asymptotic Solution for the First Interaction

Volume 2009, Article ID 101647,16pages doi:10.1155/2009/101647

Research Article

Uniform in Time Description for Weak Solutions of the Hopf Equation with Nonconvex Nonlinearity

Antonio Olivas Martinez and Georgy A. Omel’yanov

Departamento de Matematicas, Universidad de Sonora, Calle Rosales y Blvd Luis Encinas, s/n, 83000, Hermosillo, Sonora, Mexico

Correspondence should be addressed to Georgy A. Omel’yanov,[email protected] Received 7 July 2009; Revised 11 November 2009; Accepted 9 December 2009

Recommended by Ingo Witt

We consider the Riemann problem for the Hopf equation with concave-convex flux functions.

Applying the weak asymptotics method we construct a uniform in time description for the Cauchy data evolution and show that the use of this method implies automatically the appearance of the Oleinik E-condition.

Copyrightq2009 A. Olivas Martinez and G. A. Omel’yanov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

It is well known that the uniqueness problem for weak solutions of hyperbolic quasilinear systems remains unsolved up to now in the case of arbitrary jump amplitudes. Moreover, the approach which has been used successfully for shocks with sufficiently small amplitudes 1, 2 cannot be extended to the general case. On the other hand, there is a possibility to construct the unique stable solution passing to parabolic regularization. However, the vanishing viscosity method cannot be used effectively for nontrivial vector problems. Indeed, in the essentially nonintegrable case we, obviously, do not have the exact solution. Moreover, any traditional asymptotic method does not serve for the problem of nonlinear wave interaction since it leads to the appearance of a chain of partial differential equations, the first of them is nonlinear and, in fact, coincides with the original equation.

We are of opinion that a progress in this problem can be achieved in the framework of the weak asymptotics method; see, for example,3–5. In this method the approximated solutions are sought in the same form as in the Whitham method modified for nonlinear waves with localized fast variation 6, 7 for the original Whitham method for rapidly oscillating waves see 8. At the same time, the discrepancy in the weak asymptotics method is assumed to be small in the sense of the space of functionals D_x over test functions depending only on the “space” variable x. This somehow trivial modification

allows us to reduce the problem of describing interaction of nonlinear waves to solving some systems of ordinary differential equationsinstead of solving partial differential equations.

Respectively, the main characteristics of the solutionthe trajectory of the limiting singularity motion, etc.can be found by this method, whereas the shape of the real solution cannot be found.

Applications of the weak asymptotics method allowed among other to investigate the interaction of solitons for nonintegrable versions of the KdV and sine-Gordon equations9–

11, to describe uniformly in time the confluence of the shock waves for the Hopf equation with convex nonlinearities4, as well as to construct uniform in time asymptotics for the Riemann problem for isothermal gas dynamics 12–14 and delta-shock solutions for the so-called pressureless gas dynamics15,16. However, it should be necessary to verify the method application to each new type of problems.

As for the uniqueness problem, we are not ready now to consider the vector case; so we are going to simulate it and to investigate the Riemann problem for the scalar conservation law with nonconvex nonlinearity:

∂u

∂t ∂fu

∂x 0, t >0, x∈R¹, 1.1

u|_{t 0}

⎧⎨

⎩

u₋, x <0,

u, x >0. 1.2

Furthermore, the structure of the uniform in time asymptotics for a regularization of the problem1.1,1.2with an arbitraryfucan be very complicated. On the other hand, it is clear that we can define a sequence of time intervals and consider the asymptoticsu_ε for each time interval as a combination of local interacting solutions. Almost without loss of generality we can suppose that the local solutions correspond to convex or concave-convex parts of the nonlinearityfuε. That is why, in view of the result4, we restrict ourselves to the concave-convex case; that is, we will suppose that

ufu>0 u /0, f0 0, f0/0, lim

|u| → ∞fu ∞. 1.3 For definiteness we assume also that

u₋>0> u. 1.4

Let us recall that the solution of the initial-value problem is called stable if it depends continuously on the initial datasee, e.g.,2. Obviously, the stable solution to the problem 1.1–1.4is well knownsee, e.g.,17and it can be constructed using the characteristics method for1.1with regularized initial data. In particular, the stable solution will be the shock wave with amplitudeu₋−uif and only if the Oleinik E-condition

fu−fu₋

u−u₋ ≥ fu−fu₋

u−u₋ ≥ fu−fu

u−u 1.5

is satisfied for anyu∈u, u₋.

The same shock wave presents an example of nonstable weak solutions if the condition 1.5is violated. Let us note that this nonadmissible shock wave looks as if it is stable if fu−> fu.

Technically, our result consists of obtaining uniform in time asymptotic solutions for a regularization of the problem 1.1, 1.2. However, we consider as the main result the fact that the weak asymptotics method allows to construct the admissible limiting solution without any additional conditions. In particular, we obtain automatically the Oleinik E- condition for the shock wave solution.

The structure of the asymptotics construction is the following. Firstly we pass from the initial step function to a sequence of step functions such that each jump corresponds to a stable solutionin fact, to a shock wave or a centered rarefaction. Here we take into account the fact that weak asymptotics similarly to exact weak solution is not unique in the unstable case. At the same time, describing the collision of stable waves, we obtain automatically the stable scenario of interaction. Therefore, this passage from the Riemann problem to the problem of interaction of stable waves can be treated as a “regularization.” For our model example it means the transformation of the problem1.1,1.2to the following

“regularization”:

∂u_Δ

∂t ∂fu_Δ

∂x 0, t >0, x∈R¹, u_Δ|_{t 0} u u₋−uH

x₁⁰−x

u−uH x−x⁰₂

,

1.6

whereΔ x⁰₂−x⁰₁ > 0 is the “regularization” parameter,Hx is the Heaviside function, andu∈u, u₋. We choose the intermediate stateu <0 such that the left jumpat the point x x⁰₁ corresponds fort Δ to the stable shock wave, whereas the right jumpat the pointx x⁰₂corresponds to the centered rarefaction. Let us note that the problem1.6with Δ const is of interest by itself.

Next, we pass from1.6to the parabolic regularization:

∂u_Δε

∂t ∂fu_Δε

∂x ε∂²u_Δε

∂x² , t >0, x∈R¹,

u_Δε|_{t 0} u u₋−uω

x⁰₁−x ε

u−uω

x−x⁰₂ ε

,

1.7

whereωx/εis a regularization of the Heaviside function with the parameterε Δ. The contents of Sections 2 and 3 are the construction of the weak asymptotic solution to the problem1.7.

Finally, in conclusion, we consider the limiting solution both forε → 0 and forΔ → 0.

Completing this section let us formalize the concept of the weak asymptotics.

Definition 1.1. Letu_Δε u_Δεt, xbe a function that belongs toC^∞0, T×R¹_xfor eachε const >0 and toC0, T;DR¹_xuniformly inε∈0,const. One says thatu_Δεt, xis a weak

asymptotic modODεsolution of1.7if the relation

d dt

∞

−∞u_Δεψdx− ^∞

−∞fu_Δε∂ψ

∂xdx Oε 1.8

holds uniformly int∈0, Tfor any test functionψ ψx∈ DR¹_x.

Here and below the estimateOε^kis understood in theC0, Tsense:|Oε^k| ≤C_Tε^k fort∈0, T.

Definition 1.2. A functiongt, x, εis said to be of the valueO_Dε^kif the relation g, ψ ^∞

−∞gt, x, εψxdx O ε^k

1.9

holds for any test functionψ ψx∈ DR¹_x.

It is very important to note that the viscosity term in 1.7 has the value ODε and disappears in 1.8. The same is true for any parabolic regularization of the form εbu_xx. Thus, we see that the weak asymptotic modO_Dεsolution does not depend on the dissipative terms. In what follows we will omit the subindexΔforu.

2. Construction of the Asymptotic Solution for the First Interaction

To present the asymptotic ansatz for the problem1.7let us consider the possible scenario of the initial data1.6evolution. Our choice ofuin1.6implies that

fu< fu fu₋−fu

u₋−u u−u ∀u∈u, u₋, 2.1

fu> fu fu−fu

u−u u−u ∀u∈u, u. 2.2

Thus, the problem1.6solution should be the superposition of noninteracting shock wave and centered rarefaction during a sufficiently small time interval, namely,

u u u₋−uH

ϕ10t−x

r

x−x⁰₂ t

−u

H

x−ϕ20t

u−r

x−x⁰₂ t

H

x−ϕ30t ,

2.3

wheretΔ,ϕ10tis the shock wave phase:

ϕ₁₀t x⁰₁s₁₀t, s₁₀ ^deff fu₋−fu

u₋−u , 2.4

ϕk0 ϕk0tfork 2,3 are the characteristics:

ϕ₂₀ x⁰₂fut, ϕ₃₀ x⁰₂fut, 2.5

andr rx−x₂⁰/tis the centered rarefaction with the support betweenx ϕ₂₀andx ϕ₃₀:

r ∈ C^∞is such thatfrz z. 2.6

Assumption2.1 implies the intersection of the shock wave trajectoryϕ10 with the characteristicϕ₂₀ at some time instantt^∗₁ OΔ. Accordingly, the interaction between the shock and the singularity of the type x−ϕ₂₀^λ, 0 < λ < 1 i.e., with the left border of the rarefactionhas to occur, which will result in the appearance of a shock wave with a variable amplitude. Furthermore, this shock wave can interact with the right border of the rarefaction wave. So, generally speaking, the asymptotic ansatz needs to contain two fast variables. However, the distance between the characteristicsx ϕ₂₀tandx ϕ₃₀tat the first critical timet^∗₁ is greater than a constant forΔ const. Thus, the shock wave trajectory can intersect the characteristicx ϕ₃₀tonly at a second critical time instantt^∗₂ such that t^∗₂−t^∗₁≥const >0. Therefore, we can investigate the interaction process by stages.

Let us consider the first evolution stage for the solution of the problem 1.7. We present the asymptotic ansatz as a natural regularization of2.3:

uε u u₋−uω1 R−uω2 u−Rω3, 2.7 whereR Rx, t, ε∈ C^∞R¹×R¹×0,1is a function such that

Rx, t, ε

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

u ifx < ϕ₂₀−cε, r

x−x₂⁰ t

ifϕ₂₀ < x < ϕ₃₀, u ifx > ϕ30cε

2.8

with a constantc >0, ω₁ ω

−xϕ₁ ε

, ω₂ ω

x−ϕ₂ ε

, ω₃ ω

x−ϕ₃₀ ε

, 2.9

andωz/εis the Heaviside function regularization.

Furthermore, the phasesϕk ϕkτ, t,k 1,2,are assumed to be smooth functions such that

ϕkτ, t−→ϕk0t asτ −→∞, ϕkτ, t−→ϕk1t asτ−→ −∞ 2.10

exponentially fast, whereτdenotes the “fast time”:

τ ψ₀t

ε , ψ₀t ϕ₂₀t−ϕ₁₀t. 2.11

To simplify the formulas we also suppose that

ϕ11t ϕ21t. 2.12

We assume thatωtends to its limiting values

0 lim

η→ −∞ω η

, 1 lim

η→ ∞ω η

2.13

at an exponential rate. Moreover, since the limiting asε → 0 solution does not depend on the choice ofω, let

ω_η>0, ω η

ω

−η

1. 2.14

The first assumption2.10implies that the ansatz2.7describes the two noninteract- ing waves2.3fort≤t^∗₁−cε^α,α∈0,1. The second assumptions2.10and2.12imply that the ansatz2.7describes the union of the shock and the rarefaction waves fort≥t^∗₁cε^α.

2.2. Preliminary Calculations

To determine the asymptotics2.7we should calculate weak expansions of uε andfuε. Almost trivial calculations show that

uε u₋−u−−uH1 R−uH2 u−RH3ODε, 2.15

where

H_k H x−ϕ_k

fork 1,2, H₃ H

x−ϕ₃₀

. 2.16

Next, we have to calculate the weak expansion for the nonlinear term.

Lemma 2.1. Under the assumptions mentioned above the following relation holds:

fuε fu₋−u₋−uB1H₁

R2−uB2−fR2 fR H₂

fu−fR

H3ODε, 2.17

whereBiare the following convolutions:

B₁

∞

−∞ω η

f

u u₋−uω η

R₁−uω

−η−σ dη,

B₂

∞

−∞ω η

f

u u₋−uω

−η−σ

R₂−uω η

dη

2.18

with the properties

σlim→∞B1 fu−−fu

u₋−u , lim

σ→ −∞B1 fR1u₋−u−fu u₋−u ,

σlim→∞B2 fu, lim

σ→ −∞B2 fu−R2−u−fu− R2−u ,

2.19

σ στ, t, εcharacterizes the distance between the trajectoriesϕ1andϕ2, namely, σ ϕ₂−ϕ₁

ε , 2.20

andRk Rϕk, t, εfork 1,2, R3 Rϕ30, t, ε.

Sketch of the Proof

For eachψx∈ DR¹we have fuε, ψ

− ^∞

−∞fuεdφx dx dx fu− ^∞

−∞ψxdx ^∞

−∞

∂uε

∂xfuεφxdx,

2.21

whereφx _∞

xψxdx.

Next, the derivative∂u_ε/∂xcontains terms of valueO1/ε, sayωϕ1−x/ε/εand the termω2−ω3R_x. To calculate the first term we change the variable, sayη ϕ1−x/ε, and apply the Taylor expansion. Therefore,

− ^∞

−∞

1 εω

ϕ₁−x ε

fuεφxdx ^∞

−∞ω η

fuεφx_{x ϕ}

1−εηdη B1φ

ϕ1

Oε.

2.22

Finally, we note that

ω₂−ω₃ H x−ϕ₂

−H x−ϕ₃

ODε,

uε|_x∈ϕ2,ϕ30 u R−uω2 RO_Dε. 2.23

Thus,

∞

−∞R_xω2−ω3fuεφxdx ^ϕ30

ϕ2

R_xfRφxdxOε

φxfR^{x ϕ30}

x ϕ2 ^ϕ30

ϕ2

fRψxdxOε.

2.24

This implies the formula2.17.

To calculate the limiting values2.19of the convolutionsBi it is enough to use the stabilization properties2.13of the functionωη.

Remark 2.2. The convolutions Bi are the functions of σ,τ, andt. At the same time we can treatBi as functions ofσ,τ, andε. Indeed, let us denote byx^∗₁ the intersection point of the trajectoriesx ϕ₁₀tandx ϕ₂₀t, that is,x^∗₁ ϕ₁₀t^∗₁ ϕ₂₀t^∗₁. Then, by virtue of2.4 and2.5

ϕ₁₀t x^∗₁s₁₀ t−t^∗₁

, ϕ₂₀ x^∗₁fu t−t^∗₁

. 2.25

Consequently,

τ ψ₀ ε

t−t^∗₁

, ψ₀ ^defffu−s₁₀, 2.26

B_iσ, τ, t|_{t t}^∗

1ετ/ψ₀ deffB_iσ, τ, ε. 2.27 Substituting the expressions2.15and2.17into the left-hand side of1.8, we derive our main relation for obtaining the parameters of the asymptotic solution2.7:

u−−u dϕ1

dt −B1

δ

x−ϕ1

−R2−u dϕ2

dt −B2

δ

x−ϕ2

∂R

∂t ∂fR

∂x

H x−ϕ₂

−H

x−ϕ₃₀

ODε.

2.28

2.3. Analysis of the Singularity Dynamics

Let us consider the system that is obtained by setting equal to zero the coefficients of theδ functions in relation2.28, namely,

dϕ_k

dt B_k, k 1,2. 2.29

Before the interaction τ → ∞ the first assumption 2.10 for k 1,2 implies σ → τ → ∞. Therefore, the limiting relations2.19verify the concordance of2.29with our definition2.4and2.5ofϕ₁₀andϕ₂₀.

To find the limiting behavior ofϕk after the interactionτ → −∞let us reduce the system2.29to a scalar equation. In view of2.20and2.26

d

ϕ₂−ϕ₁

dt ψ₀dσ

dτ. 2.30

Hence, by subtracting one equation in2.29from the other we obtain

ψ₀dσ

dτ B2−B1deffFσ, τ, ε, 2.31

where we take into accountRemark 2.2. Using the first assumption2.10again we complete 2.31with the condition

τlim→∞

σ

τ 1. 2.32

To study this problem let us analyze the functionFσ, τ, ε.

Lemma 2.3. The valueσ 0 is the unique critical point for the problems2.31and2.32and is achieved forτ → −∞.

Proof. First we calculate

F|_{σ 0} ^∞

−∞

f

u₋ R2−u₋ω η

−f

R1 u₋−R1ω

η

×ωηdη_{σ 0} fR2−fu₋

R₂−u₋ − fu₋−fR1 u₋−R₁

σ 0 0

2.33

sinceσ 0 impliesϕ1 ϕ2. Next we note that the assumption2.1implies the inequality:

F|σ→∞ ψ₀ <0.

Let us consider now the functionF for|σ|bounded by a constant. Sinceϕ₂−ϕ₁ σε Oεfor such values ofσ, we can conclude thatRk−u Oε,k 1,2. Therefore, with accuracyOε

Fσ, τ, ε ^∞

−∞ω η−σ

f

u₋−u−−uω η

dη−fu−−fu

u₋−u . 2.34 In fact, the integral in the right-hand side of 2.34 is the average of f with the kernelω. For concave-convex functions f the derivative fu₋ −u₋ −uωη decreases monotonically fromfu− > 0 to its minimal valuef0 < 0 whenηgoes form−∞to the valueη η0whereη0is such thatu₋−u−−uωη0 0. Next, whenηgoes formη0to∞, the derivative increases monotonically fromf0to the limiting valuefu<0. At the same time,ωη−σ > 0 is a soliton-type exponentially vanishing function concentrated around the pointη σ. This implies that the behavior of the integral as a function ofσis the same as the behavior offu₋−u₋−uωηas the function ofη. Therefore, the integral diagram

has the unique solution of the equationFσ,·,· const for any nonnegative const< fu−. Thus, the equationFσ,·,· 0 has the unique solutionσ 0; moreoverF_σ|_{σ 0}<0.

Furthermore,

∂Fσ, τ, ε

∂τ

σ 0 R_2τ

∞

−∞ω η

ω η

f

u u₋−uω

−η

R2−uω η

dη

−R_1τ

∞

−∞ω

−η ω

η f

u u₋−uω η

R1−uω

−η dη

σ 0 0 2.35 sinceR₁ R₂andR₁

τ R₂

τ forσ 0.

By induction we obtain the equality d^mFστ, τ, ε

dτ^m

σ 0 0 ∀m∈N 2.36

which implies the statement ofLemma 2.3.

Consequently,ϕ₁ andϕ₂converge after the first interaction that confirms the a priori supposition 2.12. To obtain the limiting trajectoryx ϕ₁₁ ϕ₂₁ of the shock wave, it is enough to pass to the limitτ → −∞in one of the equalities2.29. Obviously, we obtain the following equation:

dϕ11

dt

fu−−fr u₋−r

x ϕ11

. 2.37

Let us come back to the relation 2.28. Definingϕ_k in accordance with 2.29, we transform2.28to the following form:

∂R

∂t ∂fR

∂x

H x−ϕ2

−H x−ϕ30

ODε. 2.38

For each test functionψwe have ∂R

∂t ∂fR

∂x

H x−ϕ2

−H x−ϕ30

, ψ

± Ω±

∂R

∂t ∂fR

∂x

ψxdx ^ϕ30

ϕ20

∂R

∂t ∂fR

∂x

ψxdx,

2.39

where

Ω−

x:ϕ2< x < ϕ20

, Ω

x:ϕ20< x < ϕ2

. 2.40

Forϕ20 < x < ϕ30the functionRcoincides with the centered rarefactionr, thus

∂r

∂t ∂fr

∂x 0, 2.41

and the last integral in2.39is equal to zero. Forx∈Ω±we note that, according to definition 2.8, either R const or |ϕ2τ, t−ϕ₂₀t| ≤ cε,c const. SinceR_tand R_x are bounded uniformly int >0, we conclude that the first integrals in2.39have the valueOε.

This completes the construction of the asymptotic solution2.7.

Obviously, fort∈0, t^∗₁−c₁ε^α,c₁>0,α∈0,1, the formula2.7is transformed to the form

u_ε u u₋−uω

ϕ₁₀t−x ε

R−uω

x−ϕ₂₀t ε

u−Rω

x−ϕ30t ε

,

2.42

which is the limit of2.7asτ → ∞,σ → ∞.

Fort ∈t^∗₁c₂ε^α, t^∗₁c₃ε^α,c₃ > c₂ >0,α∈0,1, the formula2.7is transformed to the form

uε u₋ R−u₋ω

x−ϕ11t ε

u−Rω

x−ϕ30t ε

, 2.43

which is the limit of2.7asτ → −∞,σ → 0. This implies the following.

Lemma 2.4. The weak asymptotic modODε solution 2.7 describes uniformly in time the evolution of the problem1.7solution from the state2.42to the state2.43whentincreases from 0 tot^∗₁cε^α.

Clearly, passing to the limit asε → 0 we obtain the well-known result for the stable scenario of the collision of the shock wave and the centered rarefaction, when the shock wave enters into the rarefaction domain and propagates with variables velocity and amplitudesee 2.37and2.41.

3. The Shock Wave Propagation over the Centered Rarefaction

Let us consider the evolution of the problem1.7solution fort > t^∗₁. The behavior of2.37 solution is well knownsee, e.g.,18: the trajectoryx ϕ₁₁ crosses all the characteristics X futx⁰₂if

fu−−fu

u₋−u > fu foru∈u, u 3.1

and tends to the characteristicX futx₂⁰withusuch that fu−−fu

u₋−u fu. 3.2

Ifu < u, the resulting solution for the problem 1.7 will be a combination of the smoothed shock wave with amplitudeu₋ −u and the front trajectory ϕ₁₁ fut x⁰₂ and the regularization for the centered rarefactiondefined near the domain bounded by the characteristicsX futx⁰₂ andX futx₂⁰. Obviously,u ≡u₋ forx < ϕ₁₁tand u≡uforx≥Xt. Therefore, we obtain the following.

Theorem 3.1. Letu<u. Then the weak asymptotic mod O_Dεsolution2.7describes uniformly in time the evolution of the initial data1.7into the described above regularization for the combination of the shock wave and the centered rarefaction.

Ifu >u, there occurs the collision of the shock wave and the weak singularity of the x−ϕ30^λ₋ type, 0 < λ < 1in the limit asε → 0. To describe this collision let us construct again a weak asymptotic modO_Dεsolution. In a similar way to2.7we write

u_ε u₋ R−u₋ω1 u−Rω3, 3.3

whereR Rx, t, εis defined in2.8and

ω_k ω

x−ϕ_k ε

, k 1,3. 3.4

We suppose that the phasesϕ_k ϕ_kτ1, tare smooth functions such that

ϕ₁τ1, t−→ϕ₁₁t, ϕ₃τ1, t−→ϕ₃₀t asτ₁−→∞, 3.5 ϕ₁τ1, t−→ϕt, ϕ₃τ1, t−→ϕ₃₁t asτ₁ −→ −∞, 3.6

exponentially fast, where the “fast time”τ₁is defined as follows:

τ₁ ψ₁t

ε , ψ₁t ϕ₃₀t−ϕ₁₁t. 3.7

To simplify the formulas we also suppose that

ϕt ϕ₃₁t. 3.8

The assumptions3.5,3.6, and3.8imply that the ansatz3.3coincides with the solution described inSection 2asτ1 → ∞and tends to the shock wave asτ1 → −∞.

Repeating the analysis ofSection 2we obtain the following statement.

Lemma 3.2. Under the assumptions mentioned above the following relations hold:

u_ε u₋ R−u₋H x−ϕ₁

u−RH x−ϕ₃

O_Dε, fuε fu−

R1−u₋C1−fR1 fR H₁

u−R3C3fR3−fR

H3ODε,

3.9

whereC_iare the convolutions

C₁

∞

−∞ω η

f

u₋ R₁−u₋ω η

u−R₁ω

η−σ₁ dη,

C₃

∞

−∞ω η

f

u₋ R₃−u₋ω ησ₁

u−R₃ω η

dη

3.10

with the properties

σlim→∞C₁ fu₋−fR1

u₋−R₁ , lim

σ→ −∞C₁ fuu₋−R₁−fu u₋−R₁ ,

σlim→∞C3 fu, lim

σ→ −∞C3 fu−u−R3−fu− u−R3 ,

3.11

σ₁ σ₁τ1, t, εcharacterizes the distance between the trajectoriesϕ₁andϕ₃, namely,

σ₁

ϕ₃−ϕ₁

ε , 3.12

andRk Rϕk, t, εfork 1,3.

Substituting the expressions 3.9 into the left-hand side of 1.8 we derive the following relation for obtaining the asymptotic parameters:

−R1−u₋ dϕ1

dt −C₁

δ x−ϕ₁

−u−R₃ dϕ3

dt −C₃

δ x−ϕ₃

∂R

∂t ∂fR

∂x

H1−H₃ ODε.

3.13

To calculate the trajectories ϕ₁ and ϕ₃ we set the coefficients of the δ-functions in relation3.13equal to zero, namely,

dϕ_k

dt C_k, k 1,3. 3.14

Lemma 3.3. Under the assumptionu>u, system 3.14describes the confluence of the trajectories ϕ₁andϕ₃.

Proof. Before the interactionτ1 → ∞ σ1 → ∞, so that we obtain again the Rankine- Hugoniot condition2.37forϕ₁₁. Moreover, we obtain the second formula in2.5for the characteristicϕ₃₀.

Subtracting the above relations we pass to the equation

d ϕ3−ϕ1

dt ψ₁dσ1

dτ₁ C₃−C₁^deffF₁σ1, τ₁, ε, 3.15

where we puttin terms ofτ₁andε.

Suppositions3.5complete equation3.15with the condition

τ1lim→∞

σ₁

τ1 1. 3.16

The last step of the proof is similar to Lemma 2.3 verification of the following statement.

Lemma 3.4. The valueσ₁ 0 is the unique critical point for the problems3.15and3.16and is achieved forτ1 → −∞.

Consequently,ϕ₁ and ϕ₃ converge after the second interaction that confirms the a priori supposition3.8. Passing in3.14to the limitτ₁ → −∞we find the Rankine-Hugoniot condition

dϕ dt

fu₋−fu

u₋−u 3.17

for the limiting trajectoryx ϕ ϕ₃₁ of the shock wave with the amplitudeu₋−u. Thus, the suppositionu∈u, u is explicitly the stability condition for the limiting shock wave.

Finally we note that the relation

∂u_ε

∂t ∂fuε

∂x O_Dε, forϕ1< x < ϕ3 3.18

can be proved in a similar way as inSection 2.

Summarizing the above arguments we obtain the following assertion.

Theorem 3.5. Letu>u. Then the weak asymptotic mod ODεsolutions2.7and3.3describes uniformly in time the evolution of the initial data1.6to the smoothed shock wave with amplitude u₋−u.

4. Conclusion

Concluding all the result we obtain the following uniform in time description of the problem 1.7solution: the frontϕ₁of the smoothed shock wave and the left frontϕ₂of the smoothed centered rarefaction merge during the time intervalt^∗₁−cε^α, t^∗₁cε^α, 0< α <1, in accordance with2.29. Ifu <u, then the further evolution of the front ϕ11 ≡ϕ21is described by2.37 whereas the right front of the rarefaction wave remains the characteristic ϕ₃₀. In the case u uthe trajectoryϕ11 tends toϕ30 ast → ∞. Ifu > u, then the trajectories ϕ11 andϕ3

merge during the time intervalt^∗₂−cε^α, t^∗₂cε^αin accordance with3.14and the resulting trajectory fort≥t^∗₂cε^αcoincides with the shock wave front3.17.

The conditionu >u, in view of 2.37and the assumption1.3, is equivalent to the inequality

fu≤fu fu−fu−

u−u₋ u−u ∀u∈u, u₋, 4.1 which is explicitly the Oleinik E-condition.

In the limit asε → 0 butΔ const all the trajectories loose the smoothness remaining continuous. However, the condition 4.1 does not depend on ε; so it remains valid for limiting solution.

To calculate the limit asΔ → 0 it is enough to note thatt^∗₁ OΔand|ϕ30−ϕ₂₀||_{t t}^∗₁ OΔ. Therefore, the problems 1.1 and 1.2 solution will be, in accordance with the condition4.1, either the shock wave with amplitudeu₋−uor the union of the shock wave with amplitudeu₋−uand the centered rarefactionwith support between the characteristics futandfut.

Acknowledgment

The research was supported by CONACYT under Grant 55463Mexico.

References

1 S. Bianchini and A. Bressan, “Vanishing viscosity solutions of nonlinear hyperbolic systems,” Annals of Mathematics, vol. 161, no. 1, pp. 223–342, 2005.

2 C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 2000.

3 V. G. Danilov, “Generalized solutions describing singularity interaction,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 8, pp. 481–494, 2002.

4 V. G. Danilov and V. M. Shelkovich, “Propagation and interaction of shock waves of quasilinear equation,” Nonlinear Studies, vol. 8, no. 1, pp. 135–169, 2001.

5 V. G. Danilov, G. A. Omel’yanov, and V. M. Shelkovich, “Weak asymptotics method and interaction of nonlinear waves,” in Asymptotic Methods for Wave and Quantum Problems, M. Karasev, Ed., vol. 208 of American Mathematical Society Translations, Series 2, pp. 33–163, American Mathematical Society, Providence, RI, USA, 2003.

6 V. P. Maslov and G. A. Omel’janov, “Asymptotic soliton-like solutions of equations with small dispersion,” Akademiya Nauk SSSR i Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk, vol. 36, no. 3, pp. 63–126, 1981, English translation: Russian Mathematical Surveys, vol. 36, no.3, pp. 73-149, 1981.

7 V. P. Maslov and G. A. Omel’yanov, Geometric Asymptotics for Nonlinear PDE, I., American Mathematical Society, Providence, RI, USA, 2001.

8 G. B. Whitham, Linear and Nonlinear Waves, Pure and Applied Mathematic, John Wiley & Sons, New York, NY, USA, 1974.

9 V. G. Danilov and G. A. Omel’yanov, “Weak asymptotics method and the interaction of infinitely narrowδ-solitons,” Nonlinear Analysis: Theory, Methods & Applications, vol. 54, no. 4, pp. 773–799, 2003.

10 M. G. Garcia-Alvarado and G. A. Omel’yanov, “Kink-antikink interaction for semilinear wave equations with a small parameter,” Electronic Journal of Differential Equations, vol. 2009, no. 45, pp.

1–26, 2009.

11 D. A. Kulagin and G. A. Omel’yanov, “Interaction of kinks for semilinear wave equations with a small parameter,” Nonlinear Analysis: Theory, Methods & Applications, vol. 65, no. 2, pp. 347–378, 2006.

12 M. G. Garc´ıa-Alvarado, R. F. Espinoza, and G. A. Omel’yanov, “Interaction of shock waves in gas dynamics: uniform in time asymptotics,” International Journal of Mathematics and Mathematical Sciences, no. 19, pp. 3111–3126, 2005.

13 R. F. Espinoza and G. A. Omel’yanov, “Asymptotic behavior for the centered-rarefaction appearance problem,” Electronic Journal of Differential Equations, no. 148, pp. 1–25, 2005.

14 R. F. Espinoza and G. A. Omel’yanov, “Weak asymptotics for the problem of interaction of two shock waves,” Nonlinear Phenomena in Complex Systems, vol. 8, no. 4, pp. 331–341, 2005.

15 V. G. Danilov and V. M. Shelkovich, “Dynamics of propagation and interaction ofδ-shock waves in conservation law systems,” Journal of Differential Equations, vol. 211, no. 2, pp. 333–381, 2005.

16 V. G. Danilov and D. Mitrovic, “Delta shock wave formation in the case of triangular hyperbolic system of conservation laws,” Journal of Differential Equations, vol. 245, no. 12, pp. 3704–3734, 2008.

17 Ph. G. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory of Classical and Nonclassical Shock Wave, Lectures in Mathematics ETH Z ¨urich, Birkh¨auser, Basel, Switzerland, 2002.

18 B. L. Rozhdestvenskii and N. N. Yanenko, System of Quasilinear Equations and their Applications to Gas Dynamics, Nauka, Moscow, Russia, 1978, English translation: American Mathematical Society, Providence, RI, USA, 1983.