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.

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 [

We are of opinion that a progress in this problem can be achieved in the framework of the weak asymptotics method; see, for example, [

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 [

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:

Furthermore, the structure of the uniform in time asymptotics for a regularization of the problem (

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., [

The same shock wave presents an example of nonstable weak solutions if the condition (

Technically, our result consists of obtaining uniform in time asymptotic solutions for a regularization of the problem (

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 (

Next, we pass from (

Finally, in conclusion, we consider the limiting solution both for

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

Let

Here and below the estimate

A function

It is very important to note that the viscosity term in (

To present the asymptotic ansatz for the problem (

Assumption (

Let us consider the first evolution stage for the solution of the problem (

Furthermore, the phases

To simplify the formulas we also suppose that

We assume that

The first assumption (

To determine the asymptotics (

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

Under the assumptions mentioned above the following relation holds:

For each

Next, the derivative

To calculate the limiting values (

The convolutions

Substituting the expressions (

Let us consider the system that is obtained by setting equal to zero the coefficients of the

To find the limiting behavior of

To study this problem let us analyze the function

The value

First we calculate

Let us consider now the function

In fact, the integral in the right-hand side of (

Furthermore,

By induction we obtain the equality

Consequently,

Let us come back to the relation (

For each test function

For

This completes the construction of the asymptotic solution (

Obviously, for

For

The weak asymptotic

Clearly, passing to the limit as

Let us consider the evolution of the problem (

If

Let

If

To simplify the formulas we also suppose that

The assumptions (

Repeating the analysis of Section

Under the assumptions mentioned above the following relations hold:

Substituting the expressions (

To calculate the trajectories

Under the assumption

Before the interaction (

Subtracting the above relations we pass to the equation

Suppositions (

The last step of the proof is similar to Lemma

The value

Consequently,

Finally we note that the relation

Summarizing the above arguments we obtain the following assertion.

Let

Concluding all the result we obtain the following uniform in time description of the problem (

The condition

In the limit as

To calculate the limit as

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