Research Article | Open Access
Smooth Approximations of Global in Time Solutions to Scalar Conservation Laws
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.
In the current paper we present an approach for constructing uniform and global in time approximate solutions to a Cauchy problem for one dimensional scalar conservation law with arbitrary smooth nonlinear convex flux. This approximating solution (see Definition 2.1) is continuous, piecewise smooth for ( is a regularization parameter) and tends to an admissible weak solution  of the corresponding Cauchy problem. More precisely, we consider the problem where we also assume that is piecewise monotonic. It is well known that, since , in the interval where the initial data decreases, the shock wave will arise sooner or later (see Figure 3). In order to be more precise in our considerations, we will assume that the initial data are decreasing everywhere. The same procedure can be applied if the initial data are continuous piecewise monotonic functions.
Also, for the initial data we will assume that they are such that for every fixed 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  (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  constructs a global approximating solution via a set of functional series which are defined in appropriate domains in . 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  and the wave front tracking .
Here, we use different technique, so-called the weak asymptotic method [6–14]. In the framework of our approach, the process of shock wave formation is considered as interaction of weak discontinuities, that is, nonlinear waves whose derivatives are the Heaviside type functions.
We stress that the formulas which we will give here are much simpler than the ones obtained by using the matching method. More precisely, our approximate solution is found almost in the same way as the classical solution of a Cauchy problem, by using the method of characteristics.
However, unlike the standard characteristics, we use so-called “new characteristics” which do not intersect in the moment of bifurcation, but they remain on the distance , where is a parameter of approximation.
The difference between standard and new characteristics is plotted on Figure 2 (one of possible scenarios). As one can see from there, the new characteristics do not intersect each other except ones emanating from the intervals and . 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 and .
Therefore, we can go back along the trajectory given by a “new characteristics” and thus obtain value of the approximate solution at every point . 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 and right neighborhood of 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 , and this issue is far from being trivial.
Next important moment of the approach presented here is the following. In one-dimensional case, the solution 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 -axis. In the linear hyperbolic case this situation is described by using Maslov's canonical operator (integral Fourier operator) [15–18] and entire Lagrange manifold.
In the linear parabolic case (corresponding to the flux ), it is also possible to apply the method of tunnel canonical Maslov's operator [15–17] and for the construction of the solution only “essential” part of the Lagrange manifold (not the entire one) is used (see [15–18]). This essential part exactly coincides with the shock wave profile in one-dimensional case. Approximative analytic description of that “essential” part of the Lagrange manifold is given in this paper.
In the sequel we will often use the notion of smooth description in of a process. Therefore, we give formal definition of the notion. Definition 1.1. By smooth in description of a process one implies function which is smooth in and approximately (in the weak sense) solves an equation that governs the process.
The paper is organized as follows.
In Section 2, we give basic notions and notations of the weak asymptotic method and describe two problematic situations arising when solving the problem. In Section 3, we resolve the first problematic situation that arises in the construction of global approximating solution to the considered problem—we describe smoothly in 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  (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 axis, we need to describe smoothly in 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. By , , one denotes the family of distributions depending on and such that for any test function , the estimateholds, where the estimate on the right-hand side is understood in the usual Landau sense and locally uniformly in , that is, for .Definition 2.2. The family of functions is called a weak asymptotic solution of (1.1) and (1.2) if there exists such that in the space , one hasThe following theorem is the basic theorem of the method. We also call it the nonlinear superposition law.Theorem 2.3 (see ). Suppose that the functions , , satisfy , and where is the Schwartz space of rapidly decreasing functions. For the bounded functions defined on , one has where is the Heaviside function and , , and for one has
At the beginning we make some remarks. Consider the point such thatAssume for the simplicity that such is unique. In that case the shock wave will firstly arise on the characteristics along which the point moves. More precisely, blow up of the classical solution happens on the height in the moment and in the point (i.e., initial data changes as plotted on Figure 3). The point we call the point of the gradient catastrophe.
In order to describe phenomenon of the gradient catastrophe smoothly in , on the first step we define the function such thatwhere and are constants determined from the conditionsAfter that, we replace the piece of original initial data in a small neighborhood of the point by the function , and by constants in the intervals and so that the continuity is preserved (see Figure 4). This change of initial condition provides the shock wave to arise from the interval in the moment . The amplitude of the shock wave is going to be . On the other hand, for , the solution to our problem (with transformed initial data) will be continuous function (see Figure 4 again).
As we will see, with such new initial data it is much easier to find global smooth approximating solution. Namely, it appears that the most difficult issue here is to describe evolution of the initial data in the moment of blowing up of the classical solution, and that it is much easier to do it if we have “line” of gradient catastrophe (in this case it is the line ; see Figure 4) than the point of gradient catastrophe (in the case plotted in Figure 3 it is the point ). We address the reader to [2, 3] in order to understand difficulties which are caused if we have only a point of gradient catastrophe.
The second problematic situation which we meet is shock waves interaction. This is simpler case and we can describe this process smoothly in 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 , . Next,where is the Heaviside function and , , are from Theorem 2.3. The remaining functions will be defined in what follows.
On the first step, we assume that the function from (1.2) has the form (see Figure 4 with )where is a positive constant. Furthermore, the functions , and are nonincreasing and smooth, and they satisfyThis assumption is here in order to obtain the situation such that on characteristics emanating from the intervals and we can have gradient catastrophe only for (i.e., after the straightening the curve connecting the points and ).
In order to clarify as much as possible the presentation, in this section we will assume more then this. Namely, it is well known that for the solution of (1.1) and (3.2) will admit the shock wave moving along the line given by the Cauchy problem defining the Rankine-Hugoniot conditionsHere, and are classical solutions to the following Cauchy problems, respectively (in a subdomain of where they exist)
Before we globally define the “new characteristics”
(denoted ), we need to define extremal “new
characteristics,” that is, the ones emanating from and .
The proof is relatively simple and it relies on the basic ODE
theory.Lemma 3.1. The curves and for and are given by the following Cauchy
problems:Introduce the functiondescribing the relation between
standard characteristics emanating from and ,
The function given bydescribing the distance between two nonintersecting curves and , satisfies the following Cauchy problemwhere and it tends to a constant as .Proof. The proof follows from the definition of the curves and , and standard ODE theory (see Figure 5 and put ). For details one can consult .
After inessential changes caused by replacing the
constant by the function ,
the proof of the following lemma can be found in the frame of [11, Theorem 5].Lemma 3.2. Consider the
set of solutions , ,
to the following Cauchy problemwhere is a constant large enough (which provides
global solvability of the implicit equation , , ;
see [11, Theorem 5]).
The curves , one calls the “new characteristics” emanating
from the interval .
Then, for arbitrary two, the curves and are nonintersecting.Note that we extended a little bit the interval . This is necessary in order to prove that the “new characteristics” do not mutually intersect (see ). Also note that this does not affect the weak asymptotic solution of the problem since we perturbed initial function for .
The previous lemma gives a formula for determining the “new characteristics” emanating from the interval , and states that they do not intersect each other along entire time axis. Remark that the “new characteristics” emanate from the interval and that we state that only them are non-intersecting. The characteristics emanating out of the interval are standard and they can intersect each other. Still, it will happen “late” enough (see (3.4)–(3.6)).
In order to define the new characteristics along entire -axis, we introduce the following notationsNote that and are standard characteristics emanating from the intervals and , respectively. Now, we can define the functions representing the “new characteristics” emanating out of .
For we putSimilarly, for ,For better understanding of the previous formulas see Figure 2. Actually, as well as are equal to the standard characteristics before they come “close” to the shock curve. After that, they are parallel to the shock curve.
Finally, we can write formula for the “new characteristics” which holds for (and this is set for which we need the “new characteristics”):where is defined in Lemma 3.2. Denote by the inverse function to the function , , , of the “new characteristics” defined by (3.15). Clearly, the function is not defined in the regionsTherefore, we introduce the following extension of :
To continue, assume that the increase of strength of the shock wave starts in the moment for a constant (actually, this is the moment when standard characteristics emanating out of the interval start to intersect with the shock curve). Then, we introduce the functions and so that they are equal to zero for and defined for as follows:
(i) for such that ,(ii) for such that .
Note that the functions and have the same regularity as the functions and , respectively. Therefore,
Theorem 3.3. The weak asymptotic solution of
problem (1.1) and (1.2) has the formwhere , ,
satisfy the conditions from Theorem 2.3 and the functions , , ,
are given in Lemma 3.1.
The function is given bywhere is defined by (3.17).
The functions and are classical solutions to Cauchy problems (3.5) and (3.6), respectively.Proof. The proof completely follows the construction from :
(a)we substitute (3.19) in (1.1);(b)we use formula (2.3);(c)we divide the real line on three intervals , , and . In that way we get three equations in each of the intervals, and we solve them separately in each of them.
We substitute the function from (3.19) in (1.1). Using Theorem 2.3 we getFor more detailed computation see .
Considering the previous expression for and we get the following equations:which is true by definition of the functions and (see (3.5) and (3.6)). Thus, (3.21) reduces to
We pass to the interval .
Notice that for we have . Therefore, the situation is the same as in  (see proof of Theorem 5 from ).
Consider the interval for a . In that interval we have (see again [10, 11])and from here, since , we also haveHaving this in mind, from (3.23) we getMultiplying the last relation by a test function and integrating over we get (recall that is fixed)Since for we haveand since the functions and are continuous, from the definition of distribution we see that (3.27) is fulfilled.
Details of the construction for can be found in .
Since given by (3.2) is the weak asymptotic solution in the intervals and and common part of the intervals is large enough (more precisely, it is enough to be for an ), we see that 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:where , , are continuous decreasing functions. Furthermore, we assume that , satisfyClearly, the function has two admissible jumps at the points and . Those jumps start to move according to the Rankine-Hugoniot conditions until they merge at the moment . Furthermore, since we want to investigate interaction of shock waves appearing in the initial data, we assume thatSuch conditions provide that the gradient catastrophe will not happen before the two shock waves collide.
By , , , respectively, we denote classical solutions of the following Cauchy problems (in the regions of where they exist)where the flux is the same as the one from (1.1). Since the initial functions , are decreasing, the solutions to (4.4) will be also decreasing.
By , , , and , , we denote the solutions to the following Cauchy problemsNote that continuous solutions to those Cauchy problems always exist since those are actually Rankine-Hugoniot conditions for the admissible shock placed at , , and corresponding to Cauchy problem (1.1) with initial conditions, respectively (below we imply )
Furthermore, since the function decreases and is convex it is clear that .
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 formwhere , ,
and the functions , ,
are the ones from Theorem 2.3.
Here,The function is given by the relationand the functions , , are given by the formulas whereand , , are given by (4.26).
Proof. In the sequel we use the following
By and the Heaviside and Dirac distributions, respectively, andWe start by substituting (4.7) in (1.1) and applying Theorem 2.3. We obtainwhere, as usual, , . If we equalize by zero the sum of the coefficients multiplying and , respectively, we get Now, we subtract (4.15) and (4.16). Using (4.10) and (4.11), and passing from the variable to the variable , we obtain the following Cauchy problem (see also [19, pages 108–110])whereWe can rewrite the function in the following manner:
Now, we return to (4.17). From the standard ODE theory we see that the solution of (4.17) tends to the stationary solution of (4.17). Clearly, is the minimal root to the equation in (Cauchy theorem for existence and uniqueness of the solution to an ODE with an initial condition; see Figure 5). Since after the interaction we have .
It remained to determine the functions , . In the sequel, we let
Substituting expressions (4.10) and (4.11) in (4.15) and (4.16), respectively, we obtain the equations
Furthermore, notice that if , we haveTherefore, we will determine , , so that they satisfy the following differential equations in (compare with (4.21))and then we will prove thatNotice that differential equation (4.23) is defined only on the interval . Therefore, it is necessary to extend continuously the functions after the moment . Put for ,Since for we have , it follows that the extension is well defined. It is not difficult to find the solution to (4.23)Using the l'Hospital theorem for limits we get from hereproving (4.24) and thus (4.22). This, in turn, proves that , , are approximate solutions to (4.21), and thus, (4.10) and (4.11) are fulfilled.
In that way, we have eliminated the coefficients multiplying distribution in (4.14). Now, we have to annulate the coefficients multiplying distribution in (4.14), more precisely, we need to get
Since the functions , satisfy (4.4), we can rewrite the previous expression in the formIf we multiply this by , we havesince (it is clear from the definition but one can check in ). In that way, we see that (4.14) is correct which proves the theorem.
Finally, notice that since represents the weak asymptotic solution to the considered problem we have where is the weak solution to the considered problem. Therefore, the functions , , up to , satisfy the Rankine-Hugoniot conditions. In turn, from there it follows that for we haveand for we have
5. Scalar Conservation Law with Decreasing Initial Data
In this section, we consider (1.1) with the following initial condition:We assume that the function decreases and that it has finite number of points in which the functionreaches maximum. Denote this set of points by . Assume also that .
Now, we continue in the following way. Around every , , we allocate -neighborhoods of the form , . Then, we transform the initial data , replacing it by the function , , such that for every , we have the following (compare Figures 3 and 4)where the constants and are determined by the conditionsSo, we have replaced the original initial data by the functionObviously, we have the following estimate fulfilled . Beside that, since the function reaches its maximum at the points , then there exist neighborhoods , , such that for every we haveThe moment of “straightening” of the curves connecting the points and , according to Section 2 and the Lagrange mean value theorem is given by
From here we see that for every we have