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

Abstract

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.

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 equations (instead of solving partial differential equations). Respectively, the main characteristics of the solution (the 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 equations [9–11], to describe uniformly in time the confluence of the shock waves for the Hopf equation with convex nonlinearities [4], 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 dynamics [15, 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𝜕𝑓(𝑢)𝜕𝑥=0,𝑡>0,𝑥∈ℝ¹,𝑢|_𝑡=0=𝑢₋𝑢,𝑥<0,₊,𝑥>0. (1.1)(1.2)

Furthermore, the structure of the uniform in time asymptotics for a regularization of the problem (1.1), (1.2) with an arbitrary 𝑓(𝑢) can be very complicated. On the other hand, it is clear that we can define a sequence of time intervals and consider the asymptotics 𝑢_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 nonlinearity 𝑓(𝑢_𝜀). That is why, in view of the result [4], we restrict ourselves to the concave-convex case; that is, we will suppose that 𝑢𝑓^′′(𝑢)>0(𝑢≠0),𝑓^′′(0)=0,𝑓^′′′(0)≠0,lim_|𝑢|→∞𝑓^′(𝑢)=∞. (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 data (see, e.g., [2]). Obviously, the stable solution to the problem (1.1)–(1.4) is well known (see, e.g., [17]) and it can be constructed using the characteristics method for (1.1) with regularized initial data. In particular, the stable solution will be the shock wave with amplitude 𝑢₋−𝑢₊ if and only if the Oleinik E-condition

𝑓(𝑢)−𝑓(𝑢₋)𝑢−𝑢₋≥𝑓(𝑢₊)−𝑓(𝑢₋)𝑢₊−𝑢₋≥𝑓(𝑢₊)−𝑓(𝑢)𝑢₊−𝑢 (1.5) is satisfied for any 𝑢∈[𝑢₊,𝑢₋].

The same shock wave presents an example of nonstable weak solutions if the condition (1.5) is violated. Let us note that this nonadmissible shock wave looks as if it is stable if 𝑓^′(𝑢₋)>𝑓^′(𝑢₊).

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 solution (in 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 problem (1.1), (1.2) to the following “regularization”: 𝜕𝑢_Δ+𝜕𝑡𝜕𝑓(𝑢_Δ)𝜕𝑥=0,𝑡>0,𝑥∈ℝ¹,𝑢_Δ|_𝑡=0=𝑢+(𝑢₋−𝑢)𝐻(𝑥⁰₁−𝑥)+(𝑢₊−𝑢)𝐻(𝑥−𝑥⁰₂), (1.6) where Δ=𝑥⁰₂−𝑥⁰₁>0 is the “regularization” parameter, 𝐻(𝑥) is the Heaviside function, and 𝑢∈(𝑢₊,𝑢₋). We choose the intermediate state 𝑢<0 such that the left jump (at the point 𝑥=𝑥⁰₁) corresponds for 𝑡≪Δ to the stable shock wave, whereas the right jump (at the point 𝑥=𝑥⁰₂) corresponds to the centered rarefaction. Let us note that the problem (1.6) with Δ=const is of interest by itself.

Next, we pass from (1.6) to the parabolic regularization:

𝜕𝑢_Δ𝜀+𝜕𝑡𝜕𝑓(𝑢_Δ𝜀)𝜕𝜕𝑥=𝜀²𝑢_Δ𝜀𝜕𝑥²,𝑡>0,𝑥∈ℝ¹,𝑢_Δ𝜀|_𝑡=0=𝑢+(𝑢₋−𝑢)𝜔((𝑥⁰₁−𝑥)𝜀)+(𝑢₊−𝑢)𝜔((𝑥−𝑥⁰₂)𝜀), (1.7) where 𝜔(𝑥/𝜀) is a regularization of the Heaviside function with the parameter 𝜀≪Δ. The contents of Sections 2 and 3 are the construction of the weak asymptotic solution to the problem (1.7).

Finally, in conclusion, we consider the limiting solution both for 𝜀→0 and for Δ→0.

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

Definition 1.1. Let 𝑢_Δ𝜀=𝑢_Δ𝜀(𝑡,𝑥) be a function that belongs to 𝒞^∞([0,𝑇]×ℝ¹_𝑥) for each 𝜀=const>0 and to 𝒞([0,𝑇];𝒟^′(ℝ¹_𝑥)) uniformly in 𝜀∈[0,const]. One says that 𝑢_Δ𝜀(𝑡,𝑥) is a weak asymptotic mod𝒪_𝒟′(𝜀) solution of (1.7) if the relation 𝑑𝑑𝑡^∞_−∞𝑢_Δ𝜀𝜓𝑑𝑥−^∞_−∞𝑓(𝑢_Δ𝜀)𝜕𝜓𝜕𝑥𝑑𝑥=𝒪(𝜀) (1.8) holds uniformly in 𝑡∈(0,𝑇] for any test function 𝜓=𝜓(𝑥)∈𝒟(ℝ¹_𝑥).

Here and below the estimate 𝒪(𝜀^𝑘) is understood in the 𝒞([0,𝑇]) sense: |𝒪(𝜀^𝑘)|≤𝐶_𝑇𝜀^𝑘 for 𝑡∈[0,𝑇].

Definition 1.2. A function 𝑔(𝑡,𝑥,𝜀) is said to be of the value 𝒪_𝒟′(𝜀^𝑘) if the relation (𝑔,𝜓)=^∞_−∞𝑔(𝑡,𝑥,𝜀)𝜓(𝑥)𝑑𝑥=𝒪(𝜀^𝑘) (1.9) holds for any test function 𝜓=𝜓(𝑥)∈𝒟(ℝ¹_𝑥).

It is very important to note that the viscosity term in (1.7) has the value 𝒪_𝒟′(𝜀) and disappears in (1.8). The same is true for any parabolic regularization of the form 𝜀(𝑏(𝑢))_𝑥𝑥. Thus, we see that the weak asymptotic mod𝒪_𝒟′(𝜀) solution does not depend on the dissipative terms. In what follows we will omit the subindex Δ for 𝑢.

2. Construction of the Asymptotic Solution for the First Interaction

2.1. Asymptotic Ansatz

To present the asymptotic ansatz for the problem (1.7) let us consider the possible scenario of the initial data (1.6) evolution. Our choice of 𝑢 in (1.6) implies that

𝑓(𝑢)<𝑓(𝑢)+𝑓(𝑢₋)−𝑓(𝑢)𝑢₋−𝑢(𝑢−𝑢)∀𝑢∈(𝑢,𝑢₋),𝑓(𝑢)>𝑓(𝑢)+𝑓(𝑢)−𝑓(𝑢₊)𝑢−𝑢₊(𝑢−𝑢)∀𝑢∈(𝑢₊,𝑢). (2.1)(2.2) Thus, the problem (1.6) solution should be the superposition of noninteracting shock wave and centered rarefaction during a sufficiently small time interval, namely, 𝑢=𝑢+(𝑢₋−𝑢)𝐻(𝜑₁₀(𝑡)−𝑥)+{𝑟(𝑥−𝑥⁰₂𝑡)−𝑢}𝐻(𝑥−𝜑₂₀(𝑡))+{𝑢₊−𝑟(𝑥−𝑥⁰₂𝑡)}𝐻(𝑥−𝜑₃₀(𝑡)), (2.3) where 𝑡≪Δ, 𝜑₁₀(𝑡) is the shock wave phase:

𝜑₁₀(𝑡)=𝑥⁰₁+𝑠₁₀𝑡,𝑠=₁₀^def𝑓(𝑢₋)−𝑓(𝑢)𝑢₋−𝑢, (2.4)𝜑_𝑘0=𝜑_𝑘0(𝑡) for 𝑘=2,3 are the characteristics:

𝜑₂₀=𝑥⁰₂+𝑓^′(𝑢)𝑡,𝜑₃₀=𝑥⁰₂+𝑓^′(𝑢₊)𝑡, (2.5) and 𝑟=𝑟((𝑥−𝑥⁰₂)/𝑡) is the centered rarefaction with the support between 𝑥=𝜑₂₀ and 𝑥=𝜑₃₀:

𝑟∈𝒞^∞issuchthat𝑓^′(𝑟(𝑧))=𝑧. (2.6)

Assumption (2.1) implies the intersection of the shock wave trajectory 𝜑₁₀ with the characteristic 𝜑₂₀ at some time instant 𝑡^∗₁=𝒪(Δ). 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 rarefaction) has 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 characteristics 𝑥=𝜑₂₀(𝑡) and 𝑥=𝜑₃₀(𝑡) at the first critical time 𝑡^∗₁ is greater than a constant for Δ=const. Thus, the shock wave trajectory can intersect the characteristic 𝑥=𝜑₃₀(𝑡) only at a second critical time instant 𝑡^∗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 of (2.3):

𝑢_𝜀=𝑢+(𝑢₋−𝑢)𝜔₁+(𝑅−𝑢)𝜔₂+(𝑢₊−𝑅)𝜔₃, (2.7) where 𝑅=𝑅(𝑥,𝑡,𝜀)∈𝒞^∞(ℝ¹×ℝ¹₊×[0,1]) is a function such that

𝑅(𝑥,𝑡,𝜀)=𝑢if𝑥<𝜑₂₀−𝑐𝜀,𝑟(𝑥−𝑥⁰₂𝑡)if𝜑₂₀<𝑥<𝜑₃₀,𝑢₊if𝑥>𝜑₃₀+𝑐𝜀 (2.8) with a constant 𝑐>0,

𝜔₁=𝜔(−𝑥+𝜑₁𝜀),𝜔₂=𝜔(𝑥−𝜑₂𝜀),𝜔₃=𝜔(𝑥−𝜑₃₀𝜀), (2.9) and 𝜔(𝑧/𝜀) is the Heaviside function regularization.

Furthermore, the phases 𝜑_𝑘=𝜑_𝑘(𝜏,𝑡), 𝑘=1,2, are assumed to be smooth functions such that

𝜑_𝑘(𝜏,𝑡)→𝜑_𝑘0(𝑡)as𝜏→+∞,𝜑_𝑘(𝜏,𝑡)→𝜑_𝑘1(𝑡)as𝜏→−∞ (2.10) exponentially fast, where 𝜏 denotes the “fast time”:

𝜓𝜏=⁰(𝑡)𝜀,𝜓₀(𝑡)=𝜑₂₀(𝑡)−𝜑₁₀(𝑡). (2.11)

To simplify the formulas we also suppose that

𝜑₁₁(𝑡)=𝜑₂₁(𝑡). (2.12)

We assume that 𝜔 tends to its limiting values

0=lim_{𝜂→−∞}𝜔(𝜂),1=lim_𝜂→∞𝜔(𝜂) (2.13) at an exponential rate. Moreover, since the limiting as 𝜀→0 solution does not depend on the choice of 𝜔, let

𝜔^′_𝜂>0,𝜔(𝜂)+𝜔(−𝜂)=1. (2.14)

The first assumption (2.10) implies that the ansatz (2.7) describes the two noninteracting waves (2.3) for 𝑡≤𝑡^∗₁−𝑐𝜀^𝛼, 𝛼∈(0,1). The second assumptions (2.10) and (2.12) imply that the ansatz (2.7) describes the union of the shock and the rarefaction waves for 𝑡≥𝑡^∗₁+𝑐𝜀^𝛼.

2.2. Preliminary Calculations

To determine the asymptotics (2.7) we should calculate weak expansions of 𝑢_𝜀 and 𝑓(𝑢_𝜀). Almost trivial calculations show that

𝑢_𝜀=𝑢₋−(𝑢₋−𝑢)𝐻₁+(𝑅−𝑢)𝐻₂+(𝑢₊−𝑅)𝐻₃+𝒪_𝒟′(𝜀), (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: 𝑓(𝑢_𝜀)=𝑓(𝑢₋)−(𝑢₋−𝑢)𝐵₁𝐻₁+{(𝑅₂−𝑢)𝐵₂−𝑓(𝑅₂)+𝑓(𝑅)}𝐻₂+{𝑓(𝑢₊)−𝑓(𝑅)}𝐻₃+𝒪_𝒟′(𝜀), (2.17) where 𝐵_𝑖 are the following convolutions: 𝐵₁=^∞_−∞𝜔^′(𝜂)𝑓^′(𝑢+(𝑢₋−𝑢)𝜔(𝜂)+(𝑅₁−𝐵𝑢)𝜔(−𝜂−𝜎))𝑑𝜂,₂=^∞_−∞𝜔^′(𝜂)𝑓^′(𝑢+(𝑢₋−𝑢)𝜔(−𝜂−𝜎)+(𝑅₂−𝑢)𝜔(𝜂))𝑑𝜂 (2.18) with the properties lim_𝜎→+∞𝐵₁=𝑓(𝑢₋)−𝑓(𝑢)𝑢₋−𝑢,lim_{𝜎→−∞}𝐵₁=𝑓(𝑅₁+𝑢₋−𝑢)−𝑓(𝑢)𝑢₋−𝑢,lim_𝜎→+∞𝐵₂=𝑓^′(𝑢),lim_{𝜎→−∞}𝐵₂=𝑓(𝑢₋+𝑅₂−𝑢)−𝑓(𝑢₋)𝑅₂−𝑢, (2.19)𝜎=𝜎(𝜏,𝑡,𝜀) characterizes the distance between the trajectories 𝜑₁ and 𝜑₂, namely, 𝜑𝜎=²−𝜑₁𝜀, (2.20) and 𝑅_𝑘=𝑅(𝜑_𝑘,𝑡,𝜀) for 𝑘=1,2,𝑅₃=𝑅(𝜑₃₀,𝑡,𝜀).

Sketch of the Proof
For each 𝜓(𝑥)∈𝒟(ℝ¹) we have (𝑓(𝑢_𝜀),𝜓)=−^∞_−∞𝑓(𝑢_𝜀)𝑑𝜙(𝑥)𝑑𝑥𝑑𝑥=𝑓(𝑢₋)^∞_−∞𝜓(𝑥)𝑑𝑥+^∞_−∞𝜕𝑢_𝜀𝑓𝜕𝑥^′(𝑢_𝜀)𝜙(𝑥)𝑑𝑥, (2.21) where 𝜙(𝑥)=∫^∞_𝑥𝜓(𝑥^′)𝑑𝑥^′.
Next, the derivative 𝜕𝑢_𝜀/𝜕𝑥 contains terms of value 𝒪(1/𝜀), say 𝜔^′((𝜑₁−𝑥)/𝜀)/𝜀 and the term (𝜔₂−𝜔₃)𝑅^′_𝑥. To calculate the first term we change the variable, say 𝜂=(𝜑₁−𝑥)/𝜀, and apply the Taylor expansion. Therefore,
−^∞_−∞1𝜀𝜔^′(𝜑₁−𝑥𝜀)𝑓^′(𝑢_𝜀)𝜙(𝑥)𝑑𝑥=^∞_−∞𝜔^′(𝜂)𝑓^′(𝑢_𝜀)𝜙(𝑥)|_{𝑥=𝜑1−𝜀𝜂}𝑑𝜂=𝐵₁𝜙(𝜑₁)+𝒪(𝜀). (2.22) Finally, we note that 𝜔₂−𝜔₃=𝐻(𝑥−𝜑₂)−𝐻(𝑥−𝜑₃)+𝒪_𝒟′𝑢(𝜀),_𝜀|_{𝑥∈[𝜑2,𝜑30]}=𝑢+(𝑅−𝑢)𝜔₂=𝑅+𝒪_𝒟′(𝜀). (2.23) Thus, ^∞_−∞𝑅^′_𝑥(𝜔₂−𝜔₃)𝑓^′(𝑢_𝜀)𝜙(𝑥)𝑑𝑥=^𝜑30_𝜑2𝑅^′_𝑥𝑓^′(𝑅)𝜙(𝑥)𝑑𝑥+𝒪(𝜀)=𝜙(𝑥)𝑓(𝑅)|^𝑥=𝜑30_𝑥=𝜑2+^𝜑30_𝜑2𝑓(𝑅)𝜓(𝑥)𝑑𝑥+𝒪(𝜀). (2.24) This implies the formula (2.17).
To calculate the limiting values (2.19) of the convolutions 𝐵_𝑖 it is enough to use the stabilization properties (2.13) of the function 𝜔(𝜂).

Remark 2.2. The convolutions 𝐵_𝑖 are the functions of 𝜎, 𝜏, and 𝑡. At the same time we can treat 𝐵_𝑖 as functions of 𝜎, 𝜏, and 𝜀. Indeed, let us denote by 𝑥^∗₁ the intersection point of the trajectories 𝑥=𝜑₁₀(𝑡) and 𝑥=𝜑₂₀(𝑡), that is, 𝑥^∗₁=𝜑₁₀(𝑡^∗₁)=𝜑₂₀(𝑡^∗₁). Then, by virtue of (2.4) and (2.5) 𝜑₁₀(𝑡)=𝑥^∗₁+𝑠₁₀(𝑡−𝑡^∗₁),𝜑₂₀=𝑥^∗₁+𝑓^′(𝑢)(𝑡−𝑡^∗₁). (2.25) Consequently, 𝜓𝜏=^′0𝜀(𝑡−𝑡^∗₁),𝜓=^′₀^def𝑓^′(𝑢)−𝑠₁₀,𝐵_𝑖(𝜎,𝜏,𝑡)|=_{𝑡=𝑡∗1+𝜀𝜏/𝜓′0}^def𝐵_𝑖(𝜎,𝜏,𝜀). (2.26)(2.27)

Substituting the expressions (2.15) and (2.17) into the left-hand side of (1.8), we derive our main relation for obtaining the parameters of the asymptotic solution (2.7):

(𝑢₋−𝑢){𝑑𝜑₁𝑑𝑡−𝐵₁}𝛿(𝑥−𝜑₁)−(𝑅₂−𝑢){𝑑𝜑₂𝑑𝑡−𝐵₂}𝛿(𝑥−𝜑₂)+{𝜕𝑅+𝜕𝑡𝜕𝑓(𝑅)𝜕𝑥}(𝐻(𝑥−𝜑₂)−𝐻(𝑥−𝜑₃₀))=𝒪_𝒟′(𝜀). (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 the 𝛿 functions in relation (2.28), namely, 𝑑𝜑_𝑘𝑑𝑡=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 relations (2.19) verify the concordance of (2.29) with our definition (2.4) and (2.5) of 𝜑₁₀ and 𝜑₂₀.

To find the limiting behavior of 𝜑_𝑘 after the interaction (𝜏→−∞) let us reduce the system (2.29) to a scalar equation. In view of (2.20) and (2.26)

𝑑(𝜑₂−𝜑₁)𝑑𝑡=𝜓^′₀𝑑𝜎.𝑑𝜏 (2.30) Hence, by subtracting one equation in (2.29) from the other we obtain

𝜓^′₀𝑑𝜎=𝐵𝑑𝜏²−𝐵=₁^def𝐹(𝜎,𝜏,𝜀), (2.31) where we take into account Remark 2.2. Using the first assumption (2.10) again we complete (2.31) with the condition

lim_𝜏→+∞𝜎𝜏=1. (2.32)

To study this problem let us analyze the function 𝐹(𝜎,𝜏,𝜀).

Lemma 2.3. The value 𝜎=0 is the unique critical point for the problems (2.31) and (2.32) and is achieved for 𝜏→−∞.

Proof. First we calculate 𝐹|_𝜎=0=^∞_−∞{𝑓^′(𝑢₋+(𝑅₂−𝑢₋)𝜔(𝜂))−𝑓^′(𝑅₁+(𝑢₋−𝑅₁)𝜔(𝜂))}×𝜔^′(𝜂)𝑑𝜂|_𝜎=0=𝑓(𝑅₂)−𝑓(𝑢₋)𝑅₂−𝑢₋−𝑓(𝑢₋)−𝑓(𝑅₁)𝑢₋−𝑅₁|_𝜎=0=0 (2.33) since 𝜎=0 implies 𝜑₁=𝜑₂. Next we note that the assumption (2.1) implies the inequality: 𝐹|_𝜎→+∞=𝜓^′₀<0.
Let us consider now the function 𝐹 for |𝜎| bounded by a constant. Since 𝜑₂−𝜑₁=𝜎𝜀=𝒪(𝜀) for such values of 𝜎, we can conclude that 𝑅_𝑘−𝑢=𝒪(𝜀), 𝑘=1,2. Therefore, with accuracy 𝒪(𝜀)
𝐹(𝜎,𝜏,𝜀)=^∞_−∞𝜔^′(𝜂−𝜎)𝑓^′(𝑢₋−(𝑢₋−𝑢)𝜔(𝜂))𝑑𝜂−𝑓(𝑢₋)−𝑓(𝑢)𝑢₋−𝑢. (2.34)
In fact, the integral in the right-hand side of (2.34) is the average of 𝑓^′ with the kernel 𝜔^′. For concave-convex functions 𝑓 the derivative 𝑓^′(𝑢₋−(𝑢₋−𝑢)𝜔(𝜂)) decreases monotonically from 𝑓^′(𝑢₋)>0 to its minimal value 𝑓^′(0)<0 when 𝜂 goes form −∞ to the value 𝜂=𝜂₀ where 𝜂₀ is such that 𝑢₋−(𝑢₋−𝑢)𝜔(𝜂₀)=0. Next, when 𝜂 goes form 𝜂₀ to +∞, the derivative increases monotonically from 𝑓^′(0) to the limiting value 𝑓^′(𝑢)<0. At the same time, 𝜔^′(𝜂−𝜎)>0 is a soliton-type exponentially vanishing function concentrated around the point 𝜂=𝜎. This implies that the behavior of the integral as a function of 𝜎 is the same as the behavior of 𝑓^′(𝑢₋−(𝑢₋−𝑢)𝜔(𝜂)) as the function of 𝜂. Therefore, the integral diagram has the unique solution of the equation 𝐹(𝜎,⋅,⋅)=const for any nonnegative const<𝑓^′(𝑢₋). Thus, the equation 𝐹(𝜎,⋅,⋅)=0 has the unique solution 𝜎=0; moreover 𝐹^′_𝜎|_𝜎=0<0.
Furthermore, 𝜕𝐹(𝜎,𝜏,𝜀)|𝜕𝜏_𝜎=0=𝑅^′_2𝜏^∞_−∞𝜔(𝜂)𝜔^′(𝜂)𝑓^′′(𝑢+(𝑢₋−𝑢)𝜔(−𝜂)+(𝑅₂−𝑢)𝜔(𝜂))𝑑𝜂−𝑅^′_1𝜏^∞_−∞𝜔(−𝜂)𝜔^′(𝜂)𝑓^′′(𝑢+(𝑢₋−𝑢)𝜔(𝜂)+(𝑅₁−𝑢)𝜔(−𝜂))𝑑𝜂|_𝜎=0=0 (2.35) since 𝑅₁=𝑅₂ and 𝑅^′_1𝜏=𝑅^′_2𝜏 for 𝜎=0.
By induction we obtain the equality
𝑑^𝑚𝐹(𝜎(𝜏),𝜏,𝜀)𝑑𝜏^𝑚|_𝜎=0=0∀𝑚∈ℕ (2.36) which implies the statement of Lemma 2.3.

Consequently, 𝜑₁ and 𝜑₂ converge after the first interaction that confirms the a priori supposition (2.12). To obtain the limiting trajectory 𝑥=𝜑₁₁=𝜑₂₁ of the shock wave, it is enough to pass to the limit 𝜏→−∞ in one of the equalities (2.29). Obviously, we obtain the following equation:

𝑑𝜑₁₁=𝑑𝑡𝑓(𝑢₋)−𝑓(𝑟)𝑢₋|−𝑟_𝑥=𝜑11. (2.37)

Let us come back to the relation (2.28). Defining 𝜑_𝑘 in accordance with (2.29), we transform (2.28) to the following form: {𝜕𝑅+𝜕𝑡𝜕𝑓(𝑅)𝜕𝑥}(𝐻(𝑥−𝜑₂)−𝐻(𝑥−𝜑₃₀))=𝒪_𝒟′(𝜀). (2.38)

For each test function 𝜓 we have

({𝜕𝑅+𝜕𝑡𝜕𝑓(𝑅)𝜕𝑥}(𝐻(𝑥−𝜑₂)−𝐻(𝑥−𝜑₃₀)),𝜓)=_±Ω±{𝜕𝑅+𝜕𝑡𝜕𝑓(𝑅)𝜕𝑥}𝜓(𝑥)𝑑𝑥+^𝜑30_𝜑20{𝜕𝑅+𝜕𝑡𝜕𝑓(𝑅)𝜕𝑥}𝜓(𝑥)𝑑𝑥, (2.39) where

Ω₋={𝑥∶𝜑₂<𝑥<𝜑₂₀},Ω₊={𝑥∶𝜑₂₀<𝑥<𝜑₂}. (2.40)

For 𝜑₂₀<𝑥<𝜑₃₀ the function 𝑅 coincides with the centered rarefaction 𝑟, thus

𝜕𝑟+𝜕𝑡𝜕𝑓(𝑟)𝜕𝑥=0, (2.41) and the last integral in (2.39) is equal to zero. For 𝑥∈Ω_± we note that, according to definition (2.8), either 𝑅=const or |𝜑₂(𝜏,𝑡)−𝜑₂₀(𝑡)|≤𝑐𝜀, 𝑐=const. Since 𝑅^′_𝑡 and 𝑅^′_𝑥 are bounded uniformly in 𝑡>0, we conclude that the first integrals in (2.39) have the value 𝒪(𝜀).

This completes the construction of the asymptotic solution (2.7).

Obviously, for 𝑡∈(0,𝑡^∗₁−𝑐₁𝜀^𝛼], 𝑐₁>0, 𝛼∈(0,1), the formula (2.7) is transformed to the form

𝑢_𝜀=𝑢+(𝑢₋−𝜑𝑢)𝜔(¹⁰(𝑡)−𝑥𝜀)+(𝑅−𝑢)𝜔(𝑥−𝜑₂₀(𝑡)𝜀)+(𝑢₊−𝑅)𝜔(𝑥−𝜑₃₀(𝑡)𝜀), (2.42) which is the limit of (2.7) as 𝜏→+∞, 𝜎→+∞.

For 𝑡∈[𝑡^∗₁+𝑐₂𝜀^𝛼,𝑡^∗₁+𝑐₃𝜀^𝛼], 𝑐₃>𝑐₂>0, 𝛼∈(0,1), the formula (2.7) is transformed to the form

𝑢_𝜀=𝑢₋+(𝑅−𝑢₋)𝜔(𝑥−𝜑₁₁(𝑡)𝜀)+(𝑢₊−𝑅)𝜔(𝑥−𝜑₃₀(𝑡)𝜀), (2.43) which is the limit of (2.7) as 𝜏→−∞, 𝜎→0. This implies the following.

Lemma 2.4. The weak asymptotic mod𝒪_𝒟′(𝜀) solution (2.7) describes uniformly in time the evolution of the problem (1.7) solution from the state (2.42) to the state (2.43) when 𝑡 increases from 0 to 𝑡^∗₁+𝑐𝜀^𝛼.

Clearly, passing to the limit as 𝜀→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 amplitude (see (2.37) and (2.41)).

3. The Shock Wave Propagation over the Centered Rarefaction

Let us consider the evolution of the problem (1.7) solution for 𝑡>𝑡^∗₁. The behavior of (2.37) solution is well known (see, e.g., [18]): the trajectory 𝑥=𝜑₁₁ crosses all the characteristics 𝑋=𝑓^′(𝑢)𝑡+𝑥⁰₂ if

𝑓(𝑢₋)−𝑓(𝑢)𝑢₋−𝑢>𝑓^′(𝑢)for𝑢∈(𝑢,𝑢] (3.1) and tends to the characteristic 𝑋=𝑓^′(𝑢)𝑡+𝑥⁰₂ with 𝑢 such that

𝑓(𝑢₋)−𝑓(𝑢)𝑢₋−𝑢=𝑓^′(𝑢). (3.2)

If 𝑢₊<𝑢, the resulting solution for the problem (1.7) will be a combination of the smoothed shock wave (with amplitude 𝑢₋−𝑢 and the front trajectory 𝜑₁₁=𝑓^′(𝑢)𝑡+𝑥⁰₂) and the regularization for the centered rarefaction (defined near the domain bounded by the characteristics 𝑋=𝑓^′(𝑢)𝑡+𝑥⁰₂ and 𝑋₊=𝑓^′(𝑢₊)𝑡+𝑥⁰₂). Obviously, 𝑢≡𝑢₋ for 𝑥<𝜑₁₁(𝑡) and 𝑢≡𝑢₊ for 𝑥≥𝑋₊(𝑡). Therefore, we obtain the following.

Theorem 3.1. Let 𝑢₊<𝑢. Then the weak asymptotic mod𝒪_𝒟′(𝜀) solution (2.7) describes uniformly in time the evolution of the initial data (1.7) into the described above regularization for the combination of the shock wave and the centered rarefaction.

If 𝑢₊>𝑢, there occurs the collision of the shock wave and the weak singularity of the (𝑥−𝜑₃₀)^𝜆₋ type, 0<𝜆<1 (in the limit as 𝜀→0). To describe this collision let us construct again a weak asymptotic mod𝒪_𝒟′(𝜀) solution. In a similar way to (2.7) we write

𝑢_𝜀=𝑢₋+(𝑅−𝑢₋)𝜔₁+(𝑢₊−𝑅)𝜔₃, (3.3) where 𝑅=𝑅(𝑥,𝑡,𝜀) is defined in (2.8) and

𝜔_𝑘=𝜔(𝑥−𝜑_𝑘𝜀),𝑘=1,3. (3.4) We suppose that the phases 𝜑_𝑘=𝜑_𝑘(𝜏₁,𝑡) are smooth functions such that

𝜑₁(𝜏₁,𝑡)→𝜑₁₁(𝑡),𝜑₃(𝜏₁,𝑡)→𝜑₃₀(𝑡)as𝜏₁𝜑→+∞,₁(𝜏₁,𝑡)→𝜑(𝑡),𝜑₃(𝜏₁,𝑡)→𝜑₃₁(𝑡)as𝜏₁→−∞, (3.5)(3.6) exponentially fast, where the “fast time” 𝜏₁ is defined as follows:

𝜏₁=𝜓₁(𝑡)𝜀,𝜓₁(𝑡)=𝜑₃₀(𝑡)−𝜑₁₁(𝑡). (3.7)

To simplify the formulas we also suppose that

𝜑(𝑡)=𝜑₃₁(𝑡). (3.8)

The assumptions (3.5), (3.6), and (3.8) imply that the ansatz (3.3) coincides with the solution described in Section 2 as 𝜏₁→+∞ and tends to the shock wave as 𝜏₁→−∞.

Repeating the analysis of Section 2 we 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₊−𝑅)𝐻(𝑥−𝜑₃)+𝒪_𝒟′(𝜀),𝑓(𝑢_𝜀)=𝑓(𝑢₋)+{(𝑅₁−𝑢₋)𝐶₁−𝑓(𝑅₁)+𝑓(𝑅)}𝐻₁+{(𝑢₊−𝑅₃)𝐶₃+𝑓(𝑅₃)−𝑓(𝑅)}𝐻₃+𝒪_𝒟′(𝜀), (3.9) where 𝐶_𝑖 are the convolutions 𝐶₁=^∞_−∞𝜔^′(𝜂)𝑓^′(𝑢₋+(𝑅₁−𝑢₋)𝜔(𝜂)+(𝑢₊−𝑅₁)𝜔(𝜂−𝜎₁𝐶))𝑑𝜂,₃=^∞_−∞𝜔^′(𝜂)𝑓^′(𝑢₋+(𝑅₃−𝑢₋)𝜔(𝜂+𝜎₁)+(𝑢₊−𝑅₃)𝜔(𝜂))𝑑𝜂 (3.10) with the properties lim_𝜎→+∞𝐶₁=𝑓(𝑢₋)−𝑓(𝑅₁)𝑢₋−𝑅₁,lim_{𝜎→−∞}𝐶₁=𝑓(𝑢₊+𝑢₋−𝑅₁)−𝑓(𝑢₊)𝑢₋−𝑅₁,lim_𝜎→+∞𝐶₃=𝑓^′(𝑢₊),lim_{𝜎→−∞}𝐶₃=𝑓(𝑢₋+𝑢₊−𝑅₃)−𝑓(𝑢₋)𝑢₊−𝑅₃, (3.11)𝜎₁=𝜎₁(𝜏₁,𝑡,𝜀) characterizes the distance between the trajectories 𝜑₁ and 𝜑₃, namely, 𝜎₁=(𝜑₃−𝜑₁)𝜀, (3.12) and 𝑅_𝑘=𝑅(𝜑_𝑘,𝑡,𝜀) for 𝑘=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₋){𝑑𝜑₁𝑑𝑡−𝐶₁}𝛿(𝑥−𝜑₁)−(𝑢₊−𝑅₃){𝑑𝜑₃𝑑𝑡−𝐶₃}𝛿(𝑥−𝜑₃)+{𝜕𝑅+𝜕𝑡𝜕𝑓(𝑅)𝜕𝑥}(𝐻₁−𝐻₃)=𝒪_𝒟′(𝜀). (3.13)

To calculate the trajectories 𝜑₁ and 𝜑₃ we set the coefficients of the 𝛿-functions in relation (3.13) equal to zero, namely,

𝑑𝜑_𝑘𝑑𝑡=𝐶_𝑘,𝑘=1,3. (3.14)

Lemma 3.3. Under the assumption 𝑢₊>𝑢, system (3.14) describes the confluence of the trajectories 𝜑₁ and 𝜑₃.

Proof. Before the interaction (𝜏₁→+∞) 𝜎₁→+∞, so that we obtain again the Rankine-Hugoniot condition (2.37) for 𝜑₁₁. Moreover, we obtain the second formula in (2.5) for the characteristic 𝜑₃₀.
Subtracting the above relations we pass to the equation 𝑑(𝜑₃−𝜑₁)𝑑𝑡=𝜓^′₁𝑑𝜎₁𝑑𝜏₁=𝐶₃−𝐶=₁^def𝐹₁(𝜎₁,𝜏₁,𝜀), (3.15) where we put 𝑡 in terms of 𝜏₁ and 𝜀.
Suppositions (3.5) complete equation (3.15) with the condition lim_𝜏1→+∞𝜎₁𝜏₁=1. (3.16)
The last step of the proof is similar to Lemma 2.3 verification of the following statement.
Lemma 3.4. The value 𝜎₁=0 is the unique critical point for the problems (3.15) and (3.16) and is achieved for 𝜏₁→−∞.

Consequently, 𝜑₁ and 𝜑₃ converge after the second interaction that confirms the a priori supposition (3.8). Passing in (3.14) to the limit 𝜏₁→−∞ we find the Rankine-Hugoniot condition

𝑑𝜑=𝑑𝑡𝑓(𝑢₋)−𝑓(𝑢₊)𝑢₋−𝑢₊ (3.17) for the limiting trajectory 𝑥=𝜑=𝜑₃₁ of the shock wave with the amplitude 𝑢₋−𝑢₊. Thus, the supposition 𝑢₊∈(𝑢,𝑢) is explicitly the stability condition for the limiting shock wave.

Finally we note that the relation

𝜕𝑢_𝜀+𝜕𝑡𝜕𝑓(𝑢_𝜀)𝜕𝑥=𝒪_𝒟′(𝜀),for𝜑₁<𝑥<𝜑₃ (3.18) can be proved in a similar way as in Section 2.

Summarizing the above arguments we obtain the following assertion.

Theorem 3.5. Let 𝑢₊>𝑢. Then the weak asymptotic mod𝒪_𝒟′(𝜀) solutions (2.7) and (3.3) describes uniformly in time the evolution of the initial data (1.6) to 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 (1.7) solution: the front 𝜑₁ of the smoothed shock wave and the left front 𝜑₂ of the smoothed centered rarefaction merge during the time interval (𝑡^∗₁−𝑐𝜀^𝛼,𝑡^∗₁+𝑐𝜀^𝛼), 0<𝛼<1, in accordance with (2.29). If 𝑢₊<𝑢, then the further evolution of the front 𝜑₁₁≡𝜑₂₁ is described by (2.37) whereas the right front of the rarefaction wave remains the characteristic 𝜑₃₀. In the case 𝑢₊=𝑢 the trajectory 𝜑₁₁ tends to 𝜑₃₀ as 𝑡→∞. If 𝑢₊>𝑢, then the trajectories 𝜑₁₁ and 𝜑₃ merge during the time interval (𝑡^∗₂−𝑐𝜀^𝛼,𝑡^∗₂+𝑐𝜀^𝛼) in accordance with (3.14) and the resulting trajectory for 𝑡≥𝑡^∗₂+𝑐𝜀^𝛼 coincides with the shock wave front (3.17).

The condition 𝑢₊>𝑢, in view of (2.37) and the assumption (1.3), is equivalent to the inequality

𝑓(𝑢)≤𝑓(𝑢₊)+𝑓(𝑢₊)−𝑓(𝑢₋)𝑢₊−𝑢₋(𝑢−𝑢₊)∀𝑢∈[𝑢₊,𝑢₋], (4.1) which is explicitly the Oleinik E-condition.

In the limit as 𝜀→0 but Δ=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 as Δ→0 it is enough to note that 𝑡^∗₁=𝒪(Δ) and |𝜑₃₀−𝜑₂₀||_𝑡=𝑡^∗₁=𝒪(Δ). Therefore, the problems (1.1) and (1.2) solution will be, in accordance with the condition (4.1), either the shock wave with amplitude 𝑢₋−𝑢₊ or the union of the shock wave (with amplitude 𝑢₋−𝑢) and the centered rarefaction (with support between the characteristics 𝑓^′(𝑢)𝑡 and 𝑓^′(𝑢₊)𝑡).

Acknowledgment

The research was supported by CONACYT under Grant 55463 (Mexico).