Abstract
The goal of this chapter is to show that the limit found by front tracking, that is, the weak solution of the initial value problem
is stable in L 1 with respect to perturbations in the initial data. In other words, if \(v=v(x,t)\) is another solution found by front tracking, then
for some constant C. Furthermore, we shall show that under some mild extra entropy conditions, every weak solution coincides with the solution constructed by front tracking.
Ma per seguir virtute e conoscenza. — Dante Alighieri (1265–1321), La Divina Commedia Hard to comprehend? It means ‘‘[but to] pursue virtue and knowledge.’’
Ma per seguir virtute e conoscenza.
— Dante Alighieri (1265–1321), La Divina Commedia
Hard to comprehend? It means ‘‘[but to] pursue virtue and knowledge.’’
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Keywords
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The goal of this chapter is to show that the limit found by front tracking, that is, the weak solution of the initial value problem
is stable in L 1 with respect to perturbations in the initial data. In other words, if \(v=v(x,t)\) is another solution found by front tracking, then
for some constant C. Furthermore, we shall show that under some mild extra entropy conditions, every weak solution coincides with the solution constructed by front tracking.
FormalPara ⋄ Example 7.1 (A special system)As an example for this chapter we shall consider the special \(2\times 2\) system
For simplicity assume that u > 0 and v > 0. The Jacobian matrix reads
with eigenvalues and eigenvectors
The system is clearly strictly hyperbolic. Observe that
and hence the first family is linearly degenerate. The corresponding wave curve \(W_{1}(u_{l},v_{l})=C_{1}(u_{l},v_{l})\) is given by (cf. Theorem 5.7)
or (see Fig. 7.1)
The corresponding eigenvalue \(\lambda_{1}\) is constant along each hyperbola.
With the chosen normalization of r 2 we find that
and hence the second-wave family is genuinely nonlinear. The rarefaction curves of the second family are solutions of
and thus
We see that these are straight lines emanating from the origin, and \(\lambda_{2}\) increases as u increases. Consequently, R 2 consists of the ray
The rarefaction speed is given by
To find the shocks in the second family, we use the Rankine–Hugoniot relation
which implies
(Observe that the solution with \(u/u_{l}=v_{l}/v\) coincides with the wave curve of the linearly degenerate first family.) The shock part of this curve S 2 consists of the line
The shock speed is given by
Hence the Hugoniot locus and rarefaction curves coincide for this system. Systems with this property are called Temple class systems after Temple 177; . Furthermore, the system is linearly degenerate in the first family and genuinely nonlinear in the second. Summing up, the solution of the Riemann problem for (7.2) is as follows: First the middle state is given by
If \(u_{l}/v_{l}\leq u_{r}/v_{r}\), the second wave is a rarefaction wave, and the solution can be written as
In the shock case, that is, when \(u_{l}/v_{l}> u_{r}/v_{r}\), the solution reads
If we set
and thus
the solution of the Riemann problem will be especially simple in \((\eta,\xi)\) coordinates. See Fig. 7.1. Given left and right states \(\left(\eta_{l},\xi_{l}\right)\), \(\left(\eta_{r},\xi_{r}\right)\), the middle state is given by \(\left(\eta_{l},\xi_{r}\right)\). Consequently, measured in \((\eta,\xi)\) coordinates, the total variation of the solution of the Riemann problem equals the total variation of the initial data. This means that we do not need the Glimm functional to show that a front-tracking approximation to the solution of (7.2) has bounded total variation. With this in mind it is easy to show (using the methods of the previous chapters) that there exists a weak solution to the initial value problem for (7.2) whenever the total variation of the initial data is bounded.
We may use these variables to parameterize the wave curves as follows:
For future use we note that the rarefaction and shock speeds are as follows:
⋄
The curves W in \((u,v)\) coordinates (a) and \((\eta,\xi)\) coordinates (b)
As a reminder we now summarize some properties of the front-tracking approximation for a fixed δ.
-
1.
For all positive times t, \(u^{\delta}(x,t)\) has finitely many discontinuities, each having position \(x_{i}(t)\). These discontinuities can be of two types: shock fronts or approximate rarefaction fronts. Furthermore, only finitely many interactions between discontinuities occur for \(t\geq 0\).
-
2.
Along each shock front, the left and right states
$$\displaystyle u_{l,r}=u^{\delta}\left(x_{i}\mp,t\right)$$(7.7)are related by
$$\displaystyle u_{r}=S_{\hat{{\imath}}}\left(\epsilon_{i}\right)u_{l}+e_{i},$$where \(\epsilon_{i}\) is the strength of the shock and \(\hat{{\imath}}\) is the family of the shock. The ‘‘error’’ e i is a vector of small magnitude. Furthermore, the speed of the shock, \(\dot{x}\), satisfies
$$\displaystyle\left|\dot{x}-\mu_{\hat{{\imath}}}\left(u_{l},u_{r}\right)\right|\leq\mathcal{O}\left(1\right)\delta,$$(7.8)where \(\mu_{\hat{{\imath}}}\left(u_{l},u_{r}\right)\) is the \(\hat{{\imath}}\)th eigenvalue of the averaged matrix
$$\displaystyle M\left(u_{l},u_{r}\right)=\int_{0}^{1}df\left((1-\alpha)u_{l}+\alpha u_{r}\right)\,d\alpha;$$ -
3.
Along each rarefaction front, the values u l and u r are related by
$$\displaystyle u_{r}=R_{\hat{{\imath}}}\left(\epsilon_{i}\right)u_{l}+e_{i}.$$(7.9)Also,
$$\displaystyle\left|\dot{x}-\lambda_{\hat{{\imath}}}\left(u_{r}\right)\right|\leq\mathcal{O}\left(1\right)\delta\quad\text{and}\quad\left|\dot{x}-\lambda_{\hat{{\imath}}}\left(u_{l}\right)\right|\leq\mathcal{O}\left(1\right)\delta,$$(7.10)where \(\lambda_{\hat{{\imath}}}\left(u\right)\) is the \(\hat{{\imath}}\)th eigenvalue of \(df\left(u\right)\).
-
4.
The total magnitude of all errors is small:
$$\displaystyle\sum_{i}\left|e_{i}\right|\leq\delta.$$(7.11)
Also, recall that for a suitable constant C 0 the Glimm functional
is nonincreasing for each collision of fronts, where T and Q are defined by (6.23) and (6.22), respectively, and that the interaction potential
is strictly decreasing for each collision of fronts.
7.1 Stability
Details are always vulgar. — Oscar Wilde, The Picture of Dorian Gray (1891)
Now let \(v^{\delta}\) be another front-tracking solution with initial condition v 0. To compare \(u^{\delta}\) and \(v^{\delta}\) in the L 1-norm, i.e., to estimate \({\left\|u^{\delta}-v^{\delta}\right\|}_{1}\), we introduce the vector \(q=q(x,t)=(q_{1},\dots,q_{n})\) by
and the intermediate states \(\omega_{i}\),
with velocities
As in Chapt. 5, \(H_{k}(\epsilon)u\) denotes the kth Hugoniot curve through u, parameterized such that
Note that in the definition of q we use both parts of this curve, not only the part where \(\epsilon<0\). The vector q represents a ‘‘solution’’ of the Riemann problem with left state \(u^{\delta}\) and right state \(v^{\delta}\) using only shocks. (For \(\epsilon> 0\) these will be weak solutions; that is, they satisfy the Rankine–Hugoniot condition. However, they will not be Lax shocks.)
Later in this section we shall use the fact that genuine nonlinearity implies that \(\mu_{k}\left(u,H_{k}(\epsilon)u\right)\) will be increasing in ϵ, i.e.,
for some constant c depending only on f.
As our model problem showed, the L 1 distance is more difficult to control than the ‘‘q-distance.’’ However, it turns out that even the q-distance is not quite enough, and we need to introduce a weighted form. We let \(\mathcal{D}\left(u^{\delta}\right)\) and \(\mathcal{D}\left(v^{\delta}\right)\) denote the sets of all discontinuities in u and v, respectively, and define the functional \(\Phi\left(u^{\delta},v^{\delta}\right)\) as
Here the weights W k are defined as
where \(Q\left(u^{\delta}\right)\) and \(Q\left(v^{\delta}\right)\) are the interaction potentials of \(u^{\delta}\) and \(v^{\delta}\), respectively; cf. (6.22). The quantity A k is the total strength of all waves in \(u^{\delta}\) or \(v^{\delta}\) that approach the k -wave \(q_{k}(x)\). More precisely, if the kth field is linearly degenerate, then
The summation is over all discontinuities \(x_{i}\in\mathcal{D}\left(u^{\delta}\right)\cup\mathcal{D}(v^{\delta})\). If the kth field is genuinely nonlinear, we must also account for waves of the same family approaching each other, and define
In plain words, a q k shock is approached by k-waves in \(u^{\delta}\) from the left, and k-waves in \(v^{\delta}\) from the right. Similarly, a q k rarefaction wave is approached by k-waves in \(v^{\delta}\) from the left and k-waves in \(u^{\delta}\) from the right.
Once the values of the constants \(\kappa_{1}\) and \(\kappa_{2}\) are determined, we will assume that the total variations of \(u^{\delta}\) and \(v^{\delta}\) are so small that
In this case we see that Φ is equivalent to the L 1 norm; i.e., there exists a finite constant C 1 such that
We can also define, with obvious modifications, \(\Phi(u^{\delta_{1}}(t),v^{\delta_{2}}(t))\) with two different parameters \(\delta_{1}\) and \(\delta_{2}\). Our first goal will be to show that
for all \(0\leq t\leq s\). Once this inequality is in place, we can show that the sequence of front-tracking approximations is a Cauchy sequence in L 1 for
Letting \(\delta_{1}\) and \(\delta_{2}\) tend to zero, we have the convergence of the whole sequence, and not only a subsequence.
The first step in order to prove (7.21) is to choose \(\kappa_{2}\) so large that the weights W k do not increase when fronts in \(u^{\delta_{1}}\) or \(v^{\delta_{2}}\) collide. This is possible, since the total variations of both \(u^{\delta_{1}}\) and \(v^{\delta_{2}}\) are uniformly small; hence the terms \(\kappa_{1}A_{k}\) are uniformly bounded, and by the interaction estimate, Q decreases for all collisions. This ensures the inequalities (7.19).
Then we must examine how Φ changes between collisions. Observe that \(\Phi(t)\) is piecewise linear and continuous in t. Let
We differentiate Φ and find that
where
The second equality in (7.22) is obtained by adding terms
and observing that there is only a finite number of terms in the sum in (7.22).
⋄ Example 7.2 (Example 7.1 (cont’d.))
Let us check how this works for our special system. The two front-tracking approximations are denoted by u and v, and for simplicity we omit the superscript δ. These are made by approximating a rarefaction wave between \(\eta_{l}=n\delta\) and \(\eta_{r}=m\delta\), \(m> n\), by a series of discontinuities with speed \(3j\delta\), \(j=n,\dots,m-1\). In other words, we use the characteristic speed to the left of the discontinuity. The functions u and v are well defined by standard techniques.
Since we managed this far without the interaction potential, we define the weights also without these (they are needed only to bound the weights, anyway). Hence for the example we use
Now we shall estimate
To this end we consider a fixed discontinuity at x (to simplify the notation we do not use a subscript on this discontinuity) in one of the functions, say v. This discontinuity gives a contribution to the right-hand side of (7.24), denoted by \(E_{1}+E_{2}\), where
For this \(2\times 2\) system we have
To estimate \(E_{1}+E_{2}\) we study several cases. ⋄
Case 1
Assume first that the jump at x is a contact discontinuity, that is, of the first family, in which case
and consequently,
Furthermore,
Then
For the weights of the second family we find that
To estimate \(\mu_{2}^{-}-\dot{x}\) we exploit that \(\mu_{2}^{-}\) is a discontinuity of the second family, while \(\dot{x}\) is a contact discontinuity of the first family. Thus we can estimate from below their difference by the smallest difference in speeds between waves in the first- and second-wave families. We find that \(\mu_{2}^{-}-\dot{x}\geq c=\min_{u,v}\left\{{\eta}\right\}> 0\). Hence
Then
if \(\kappa c\geq\sup_{x}W_{1}(x)\). (Throughout this argument we will choose larger and larger κ.) This inequality (7.28) is the desired estimate when x is a contact discontinuity.
Case 2
The case that x is a genuinely nonlinear wave, that is, belongs to the second family, is more complicated. There are two distinct cases, that of an (approximate) rarefaction wave and that of a shock wave. First we treat the term E 1, which is common to the two cases. Here
Consequently,
We split the estimate for E 2 into several cases.
Case 2a (rarefaction wave)
First we consider the case that x is an approximate rarefaction wave. By the construction of v we have
The speeds appearing in E 2 are given by
We define the auxiliary speed
It is easily seen that
We have several subcases. First we assume that \(q_{2}^{-}> 0\), in which case \(q_{2}^{+}> 0\) as well; see Fig. 7.2.
In this case \(A_{2}^{+}=A_{2}^{-}+\left|\epsilon\right|\). Hence
We need to estimate the term \(\left\{\left(q_{2}^{-}+\epsilon\right)\left(\mu_{2}^{+}-\tilde{\mu}\right)-q_{2}^{-}\left(\mu_{2}^{-}-\tilde{\mu}\right)\right\}\). This estimate is contained in Lemma 7.4 in the general case, and it is verified directly for this model right after the proof of Lemma 7.4. We obtain
and thus
We estimate \(\mu_{2}^{-}-\dot{x}\leq-q_{2}^{-}\leq 0\), and hence
for some constant M if we choose κ big enough. We have used that \(W_{2}^{+}\) is bounded. Therefore,
Now for the case \(q_{2}^{-}<0\). Here we have two further subcases, \(q_{2}^{+}<0\) and \(q_{2}^{+}> 0\). First we assume that \(q_{2}^{+}<0\), and thus both \(q_{2}^{-}\) and \(q_{2}^{+}\) are negative. Note that
Thus
where we have used Lemma 7.4 (with \(\varepsilon=\epsilon\), \(\varepsilon^{\prime}=q_{2}^{+}\)) and chosen κ sufficiently large. Thus we conclude that \(E_{1}+E_{2}\leq M\left|\epsilon\right|\delta\) in this case as well.
Now for the last case in which \(\epsilon> 0\), namely \(q_{2}^{-}<0<q_{2}^{+}\). Since \(q_{2}^{+}=q_{2}^{-}+\epsilon\), we have
Furthermore, \(A_{2}^{+}=A_{2}^{-}\), and thus \(W_{2}^{+}=W_{2}^{-}\). We see that
and hence
for some constant M.
\(q_{2}^{-}> 0\)
Case 2b (shock wave)
When x is a shock front, we have \(\epsilon<0\). In this case,
We first consider the case \(q_{2}^{-}<0\). Then
and we obtain
Lemma 7.4 (with \(\varepsilon^{\prime}=\epsilon\), \(\varepsilon=q_{2}^{-}\)) implies
Furthermore,
If \(q_{2}^{-}> 0\), then there are two further cases to be considered, depending on the sign of \(q_{2}^{+}\). We first consider the case \(q_{2}^{+}<0\), and thus \(q_{2}^{+}<0<q_{2}^{-}\). Now \(A_{2}^{+}=A_{2}^{-}\). Furthermore,
Thus
and we easily obtain
This leaves the final case \(q_{2}^{\pm}> 0\). In this case we have that \(A_{2}^{+}=A_{2}^{-}+\left|\epsilon\right|\). We still have
and thus
Furthermore, by Lemma 7.4, we have that
Then we calculate
if κ is sufficiently large. This is the last case.
Now we have shown that in all cases,
Summing over all discontinuities in u and v we conclude that
for some finite constant \(C^{\prime}\) independent of δ.
We shall now show that
and this estimate is easily seen to imply (7.21). To prove (7.30) we shall need some preliminary results:
Lemma 7.3
Assume that the vectors \(\epsilon=\left(\epsilon_{1},\dots,\epsilon_{n}\right)\), \(\epsilon^{\prime}=\left(\epsilon^{\prime}_{1},\dots,\epsilon^{\prime}_{n}\right)\), and \(\epsilon^{\prime\prime}=\left(\epsilon^{\prime\prime}_{1},\dots,\epsilon^{\prime\prime}_{n}\right)\) satisfy
for some vector u, where
Then
If the scalar ϵ and the vector \(\epsilon^{\prime}=\left(\epsilon^{\prime}_{1},\dots,\epsilon^{\prime}_{n}\right)\) satisfy
where R l denotes the lth rarefaction curve, then
Proof
The proof of this lemma is a straightforward modification of the proof of the interaction estimate (6.18). □
Lemma 7.4
Let \(\bar{\omega}\in\Omega\) be sufficiently small, and let \(\varepsilon\) and \(\varepsilon^{\prime}\) be real numbers. Define
Then one has
Proof
The proof of this is again in the spirit of the proof of the interaction estimate, equation (6.13). Let the function Ψ be defined as
Then Ψ is at least twice differentiable, and satisfies
Consequently,
From this the lemma follows. □
⋄ Example 7.5 (Lemma 7.4 for Example 7.1)
If k = 2, let \(\bar{\omega}\), \(\omega^{\prime}\), and \(\omega^{\prime\prime}\) denote the η-coordinate, since only this will influence the speeds. Then a straightforward calculation yields
verifying the lemma in this case. ⋄
If the kth characteristic field is genuinely nonlinear, then the characteristic speed \(\lambda_{k}\left(H_{k}(\epsilon)\omega\right)\) is increasing in ϵ, and we can even choose the parameterization such that
for all sufficiently small ϵ and ω. This also implies that \(\mu_{k}(\omega,H_{k}(\epsilon)\omega)\) is strictly increasing in ϵ. However, the Hugoniot locus through the point ω does not in general coincide with the Hugoniot locus through the point \(H_{k}(q)\omega\). Therefore, it is not so straightforward comparing speeds defined on different Hugoniot loci. When proving (7.30) we shall need to do this, and we repeatedly use the following lemma:
Lemma 7.6
For some state ω define
Then Ψ is at least twice differentiable for all \(k=1,\dots,n\). Furthermore, if the kth characteristic field is genuinely nonlinear, then for sufficiently small \(\left|q\right|\) and \(\left|\epsilon\right|\),
where c depends only on f for all sufficiently small \(\left|\omega\right|\).
Proof
Let the vector \(\epsilon^{\prime}\) be defined by \(\mathcal{H}(\epsilon^{\prime})\omega=H_{k}(\epsilon)H_{k}(q)\omega\). Then by Lemma 7.3,
Consequently,
Using this we find that
Therefore,
Hence, we compute
for sufficiently small \(\left|\epsilon\right|\). The value of the constant \(c^{\prime}\) (and its existence) depends on the genuine nonlinearity of the system and hence on f. Since \(\Psi^{\prime}\) is continuous for small \(\left|q\right|\), the lemma follows. □
We shall prove (7.30) in the case that the front at x i is a front in \(v^{\delta_{2}}\); the case in which it is a front in \(u^{\delta_{1}}\) is completely analogous. We therefore fix i, and study the relation between \(q_{k}^{-}\) and \(q_{k}^{+}\). Since the front is going to be fixed from now on, we drop the subscript i. For simplicity we write \(\delta=\delta_{2}\). Assume the the family of the front x is l and the front has strength ϵ. The situation is as in Fig. 7.3.
A key observation is that we can regard the waves \(q_{k}^{+}\) as the result of an interaction between the waves \(q_{k}^{-}\) and ϵ; similarly, the waves \(-q_{k}^{-}\) are the result of an interaction between ϵ and \(-q_{k}^{+}\).
The setting in the proof of (7.30)
Regarding the weights, from (7.16) and (7.18) we find that
while for \(k=l\) we obtain
The proof of (7.30) is a study of cases. We split the estimate into two subgroups, depending on whether the front at x is an approximate rarefaction wave or a shock. Within each subgroup we discuss three subcases depending on the signs of \(q_{l}^{\pm}\). In all cases we discuss the terms E k (\(k\neq l\)) and E l separately. For \(k\neq l\) we write E k (recall that we dropped the subscript i) as
By the strict hyperbolicity of the system, we have that
where c is some fixed constant depending on the system. Thus we always have that
We begin with the case that the front at x is an approximate rarefaction wave (\(\epsilon> 0\)). In this case,
for some vector \(\tilde{q}\). Hence
From (7.31) and (7.40) we obtain
and from (7.32) and (7.41) we obtain
This implies that
Furthermore, since ϵ is an approximate rarefaction, \(0\leq\epsilon\leq\delta\). Therefore, we can replace \(\tilde{q}_{l}\) with ϵ and \(\tilde{q}_{k}\) (\(k\neq l\)) with zero on the right-hand side of (7.42), making an error of \(\mathcal{O}\left(1\right)\delta\). Indeed,
Using (7.43) and the fact that \(\epsilon\leq\delta\), we conclude that
Similarly,
Since in this case \(0\leq\epsilon\leq\delta\), and the total variation is small, we can assume that the right-hand sides of (7.44)–(7.45) are smaller than \(\epsilon+\mathcal{O}\left(1\right)\left|e\right|\). Also, the error e is small; cf. (7.11). Then
We can also use the estimates (7.44) and (7.45) to make a simplifying assumption throughout the rest of our calculations. Since the total variation of \(u-v\) is uniformly bounded, we can assume that the right-hand sides of (7.44) and (7.45) are bounded by
In particular, we then find that
Hence if \(\epsilon> 0\), from the left inequality we find that
or
and if \(\epsilon<0\), from the right inequality above,
or
If \(\epsilon> 0\) and \(q_{\ell}^{-}\geq q_{\ell}^{+}\) or \(\epsilon<0\) and \(q_{\ell}^{+}\geq q_{\ell}^{-}\), then \(\left|\epsilon\right|\leq\mathcal{O}\left(1\right)\left|e\right|\). In this case we find for \(k\neq l\), or \(k=l\) and \(q_{\ell}^{-}q_{\ell}^{+}> 0\), that
If \(k=l\) and \(q_{\ell}^{-}q_{\ell}^{+}<0\), then for \(\epsilon> 0\) we have that \(q_{\ell}^{+}-q_{\ell}^{-}\geq\mathcal{O}\left(1\right)\left|e\right|\), so if \(q_{\ell}^{+}<q_{\ell}^{-}\), we must have that
Similarly, if \(\epsilon<0\) and \(q_{\ell}^{+}> q_{\ell}^{-}\), we obtain
Then we find that
These observations imply that if \(\left|\epsilon\right|=\mathcal{O}\left(1\right)\left|e\right|\), we have that
which is what we want to show. Thus in the following we can assume that either
or
Now follows a discussion of several different cases, depending on whether the front is an approximate rarefaction wave or a shock wave, and on the signs of \(q_{\ell}^{-}\) and \(q_{\ell}^{+}\).
Case R1
\(0<q_{l}^{-}<q_{l}^{+}\), \(\epsilon> 0\).
For \(k\neq l\) we recall (7.38) that
The second term in (7.50) is less than or equal to (cf. (7.39))
Furthermore, by (7.45),
By the continuity of \(\mu_{k}\),
Hence from (7.38), we find that
For \(k=l\) the situation is more complicated. We define states and speeds
see Fig. 7.4.
Recall that
Now by Lemma 7.4, with \(\omega=\omega_{l-1}^{-}\), \(\varepsilon=q_{\ell}^{-}\), and \(\varepsilon^{\prime\prime}=q_{\ell}^{-}+\epsilon\),
We also find that (cf. (7.10) and the fact that \(\mu_{l}(u,u)=\lambda_{l}(u)\))
Furthermore,
Since the lth field is genuinely nonlinear, then by Lemma 7.6,
for some constant c > 0 depending only on the system. Recall that in this case,
Moreover, ϵ, \(q_{\ell}^{+}\), and \(q_{\ell}^{-}\) are positive. Using the above inequalities, we compute
Adding (7.57) and (7.51), we obtain
which holds for sufficiently large \(\kappa_{1}\). This implies (7.30) in Case R1.
The situation for \(0<q_{l}^{-}<q_{l}^{+}\), \(\epsilon> 0\), and \(k=l\)
Case R2
\(q_{l}^{-}<q_{l}^{+}<0\), \(\epsilon> 0\).
Writing E k as in (7.38), and using (7.44) (instead of (7.45) as in the previous case), we find for \(k\neq l\) that
For \(k=l\) the situation is similar to the previous case. We define auxiliary states and speeds
see Fig. 7.5.
Recall that
In this case we use (7.33) with \(\bar{\omega}=\omega_{\ell}^{+}\), \(\varepsilon=q_{\ell}^{+}\), and \(\varepsilon^{\prime}=-\epsilon\). This gives
As in (7.54), we find that
We also obtain the analogue of (7.55), namely,
By genuine nonlinearity, using Lemma 7.6, we find that
for some constant c. Now
Using the above estimates (7.61)–(7.64), we compute
Finally,
by choosing \(\kappa_{1}\) larger if necessary. Hence (7.30) holds in this case as well.
The situation for \(q_{l}^{-}<q_{l}^{+}<0\), \(\epsilon> 0\), and \(k=l\)
Case R3
\(q_{l}^{-}<0<q_{l}^{+}\), \(\epsilon> 0\).
Since the front at x is a rarefaction front, both estimates (7.51) and (7.59) hold. Moreover, we have that
Then from \(AD+BC\leq(A+B)(D+C)\) for positive A, B, C, and D, we obtain
Using (7.51) for \(k<l\) and (7.59) for \(k> l\), and choosing \(\kappa_{1}\) sufficiently large, we obtain (7.30).
Now we shall study the cases in which the front at x is a shock front. Also, here we prove (7.30) in three cases depending on \(q_{\ell}^{-}\) and \(q_{\ell}^{+}\). If the front at x is a shock front, then by the construction of the front-tracking approximation, we have
or
where \(q^{\pm}=\left(q_{1}^{\pm},\dots,q_{n}^{\pm}\right)\), and e is the error of the front at x. Then we can use (7.31) and continuity of the mapping H to find that
We also have that
or
by the continuity of H. From this we obtain
Case S1
\(0<q_{\ell}^{+}<q_{\ell}^{-}\), \(\epsilon<0\).
If \(k\neq l\), then we can write E k as (7.38) and use the arguments leading to (7.51) and the estimate (7.69) to obtain
For \(k=l\) we define the auxiliary states and speeds as in (7.60); see Fig. 7.6.
Then the estimate (7.61) holds. Also, using (7.69) we find that
Moreover,
By Lemma 7.6, we have
In this case
and
We estimate
As before, setting \(\kappa_{1}\) sufficiently large, (7.75) and (7.70) imply
which is (7.30).
The situation for \(0<q_{\ell}^{+}<q_{\ell}^{-}\), \(\epsilon<0\), and \(k=l\)
Case S2
\(q_{\ell}^{+}<q_{\ell}^{-}<0\), \(\epsilon<0\).
In this case we proceed as in Case S1, but using (7.68) instead of (7.69). For \(k\neq l\) this gives the estimate
We now define the intermediate states \(\tilde{\omega}_{\ell}\), \(\omega_{\ell}^{\star}\) and the speeds \(\tilde{\mu}_{\ell}\) and \(\mu_{\ell}^{\star}\) as in (7.52); see Fig. 7.7.
Then the estimate (7.53) holds. As in Case R1, we compute
and
In this case, genuine nonlinearity and Lemma 7.6 imply that
with c > 0. Moreover, now
Now we can use the (by now) familiar technique of estimating E l :
Combining (7.81) and (7.77), we obtain
which is (7.30).
The situation for \(q_{\ell}^{+}<q_{\ell}^{-}<0\), \(\epsilon<0\), and \(k=l\)
Case S3
\(q_{\ell}^{+}<0<q_{\ell}^{-}\), \(\epsilon<0\).
For \(k\neq l\), the estimate (7.77) remains valid.
Next we consider the case \(k=l\). The \(\mathcal{O}\left(1\right)\) that multiplies \(\left|\epsilon\right|\) in (7.69) (or (7.69)) is proportional to the total variation of the initial data. Hence we can assume that this is arbitrarily small by choosing \(\mathrm{T.V.}\left(u_{0}\right)\) sufficiently small. Since all terms \(q_{j}^{\pm}\) are bounded, we can and will assume that
Without loss of generality we may assume that \(\left|q_{l}^{+}\right|\geq\left|q_{l}^{-}\right|\). This implies that
Thus
or
which can be rewritten as
From this we conclude that
We define the auxiliary states \(\tilde{\omega}_{\ell}\), \(\omega_{\ell}^{\star}\) and the speeds \(\tilde{\mu}_{\ell}\) and \(\mu_{\ell}^{\star}\) as in (7.52); see Fig. 7.8.
Then estimates (7.78) and (7.79) hold.
By Lemma 7.6 we have that
for a positive constant c. Recalling that \(W_{\ell}^{-}\geq 1\), and using (7.89), (7.90), and the estimates (7.78) and (7.79) (which remain valid in this case), we compute
Now (7.77) and (7.91) are used to balance the terms containing the factor \(\sum_{k\neq l}\left|q_{k}^{-}\right|\). The remaining term,
can be made negative by choosing \(\mathrm{T.V.}\left(u_{0}\right)\) (and hence \(\mathcal{O}\left(1\right)\)) sufficiently small.
Hence also in this case (7.30) holds.
The situation for \(q_{\ell}^{+}<0<q_{\ell}^{-}\), \(\epsilon<0\), and \(k=l\)
Finally, if \(q_{\ell}^{-}\) or \(q_{\ell}^{+}\) is zero, (7.30) can easily be shown to be a limit of one of the previous cases.
Summing up, we have proved the following theorem:
Theorem 7.7
Let \(u^{\delta_{1}}\) and \(v^{\delta_{2}}\) be front-tracking approximations, with accuracies defined by \(\delta_{1}\), \(\delta_{2}\),
For sufficiently small M there exist constants \(\kappa_{1}\), \(\kappa_{2}\), and C 2 such that the functional Φ defined by (7.15) and (7.16) satisfies (7.21). Furthermore, there exists a constant C (independent of \(\delta_{1}\) and \(\delta_{2}\)) such that
To state the next theorem we need the following definition. Let the domain \(\mathcal{D}\) be defined as the L 1 closure of the set
that is, \(\mathcal{D}=\overline{\mathcal{D}}_{0}\). Since the total variation is small, we will assume that all possible values of u are in a (small) neighborhood \(\Omega\subset\mathbb{R}^{n}\).
Theorem 7.8
Let \(f_{j}\in C^{2}(\mathbb{R}^{n})\), \(j=1,\dots,n\). Consider the strictly hyperbolic equation \(u_{t}+f(u)_{x}=0\). Assume that each wave family is either genuinely nonlinear or linearly degenerate. For all initial data u 0 in \(\mathcal{D}\), defined by (7.94), every sequence of front-tracking approximations \(u^{\delta}\) converges to a unique limit u as \(\delta\to 0\). Furthermore, let u and v denote solutions
with initial data u 0 and v 0, respectively, obtained as a limit of a front-tracking approximation. Then
Proof
First we use (7.93) to conclude that every front-tracking approximation \(u^{\delta}\) has a unique limit u as \(\delta\to 0\). Then we take the limit \(\delta\to 0\) in (7.93) to conclude that (7.95) holds. □
Note that this also gives an error estimate for front tracking for systems. If we denote the limit of the sequence \(\left\{{u^{\delta}}\right\}\) by u and \(v^{\delta_{2}}=u^{\delta}\), then by letting \(\delta_{2}\to 0\) in (7.93)
for some finite constant C. Hence front tracking for systems is a first-order method.
7.2 Uniqueness
Let \({\mathcal{S}}_{t}\) denote the map that maps initial data u 0 into the solution u of
at time t, that is, \(u={\mathcal{S}}_{t}u_{0}\). In Chapt. 6 we showed the existence of the semigroup \({\mathcal{S}}_{t}\), and in the previous section its stability for initial data in the class \(\mathcal{D}\) as limits of approximate solutions obtained by front tracking. Thus we know that it satisfies
for all \(t,s\geq 0\) and u, v in \(\mathcal{D}\).
In this section we prove uniqueness of solutions that have initial data in \(\mathcal{D}\).
We want to demonstrate that every other solution u coincides with this semigroup. To do this we will basically need three assumptions. The first is that u is a weak solution, the second is that it satisfies Lax’s entropy conditions across discontinuities, and the third is that it has locally bounded variation on a certain family of curves. Concretely, we define an entropy solution of
to be a bounded measurable function \(u=u(x,t)\) of bounded total variation satisfying the following conditions:
- A :
-
The function \(u=u(x,t)\) is a weak solution of the Cauchy problem (7.1) taking values in \(\mathcal{D}\), i.e.,
$$\displaystyle\int_{0}^{T}\int_{\mathbb{R}}(u\varphi_{t}+f(u)\varphi_{x})\,dx\,dt+\int_{\mathbb{R}}\varphi(x,0)u_{0}(x)\,dx=0$$(7.96)for all test functions \(\varphi\) whose support is contained in the strip \([0,T\rangle\).
- B :
-
Assume that u has a jump discontinuity at some point \((x,t)\), i.e., there exist states \(u_{l,r}\in\Omega\) and speed σ such that if we let
$$\displaystyle U(y,s)=\begin{cases}u_{l}&\text{for $y<x+\sigma(s-t)$,}\\ u_{r}&\text{for $y\geq x+\sigma(s-t)$,}\end{cases}$$(7.97)then
$$\displaystyle\lim_{\rho\to 0}\frac{1}{\rho^{2}}\int_{t-\rho}^{t+\rho}\int_{x-\rho}^{x+\rho}\left|u(y,s)-U(y,s)\right|\,dy\,ds=0.$$(7.98)Furthermore, there exists k such that
$$\displaystyle\lambda_{k}\left(u_{l}\right)\geq\sigma\geq\lambda_{k}\left(u_{r}\right).$$(7.99) - C :
-
There exists a \(\theta> 0\) such that for all Lipschitz functions γ with Lipschitz constant not exceeding θ, the total variation of \(u(x,\gamma(x))\) is locally bounded.
Remark 7.9
One can prove, see Exercise 7.1, that the front-tracking solution constructed in the previous chapter is an entropy solution of the conservation law.
There is a direct argument showing that any weak solution, whether it is a limit of a front-tracking approximation or not, satisfies a Lipshitz continuity in time of the spatial L 1-norm, as long as the solution has a uniform bound on the total variation. We present that argument here.
Theorem 7.10
Let \(u_{0}\in\mathcal{D}\), and let u denote any weak solution of (7.1) such that \(\mathrm{T.V.}\left(u(t)\right)\leq C\). Then
Proof
Let \(0<s<t<T\), and let \(\alpha_{h}\) be a smooth approximation to the characteristic function of the interval \([s,t]\), so that
Furthermore, define
where ϕ is any smooth function with compact support. If we insert this into the weak formulation
and let \(h\to 0\), we obtain
From this we obtain
which proves the claim. Here we first used Exercise A.1, Theorem A.4, subsequently the definition (A.1) for \(\mathrm{T.V.}\left(f\right)\), and finally the Lipschitz continuity of f and the bound on the total variation on u. □
Remark 7.11
This argument provides an alternative to the proof of the Lipschitz continuity in Theorem 2.15 in the scalar case.
Before we can compare an arbitrary entropy solution to the semigroup solution, we need some preliminary results. Firstly, Theorem 7.10 says that every function \(u(\,\cdot\,,t)\) taking values in \(\mathcal{D}\) and satisfying A is L 1 Lipschitz continuous:
for \(t\geq s\).
Furthermore, by the structure theorem for functions of bounded variation 193; , Theorem 5.9.6, u is continuous almost everywhere. For the sake of definiteness, we shall assume that all functions in \(\mathcal{D}\) are right continuous. Also, there exists a set \(\mathcal{N}\) of zero Lebesgue measure in the interval \([0,T]\) such that for \(t\in[0,T]\setminus\mathcal{N}\), the function \(u(\,\cdot\,,t)\) either is continuous at x or has a jump discontinuity there. Intuitively, the set \(\mathcal{N}\) can be thought of as the set of times when collisions of discontinuities occur.
Lemma 7.12
Proof
Let \(P_{\lambda}\) denote the parallelogram
Integrating the conservation law over \(P_{\lambda}\), we obtain
If we furthermore integrate this with respect to λ from \(\lambda=0\) to \(\lambda=\rho\), and divide by \(\rho^{2}\), we obtain
Now let \(\rho\to 0\). Then
Hence
This concludes the proof of the lemma. □
The next lemma states that if u satisfies C, then the discontinuities cannot cluster too tightly together.
Lemma 7.13
Assume that \(u\colon[0,T]\to\mathcal{D}\) satisfies C. Let \(t\in[0,T]\) and \(\varepsilon> 0\). Then the set
has no limit points.
Proof
Assume that \(B_{t,\varepsilon}\) has a limit point, denoted by x 0. Then there is a monotone sequence \(\left\{x_{i}\right\}_{i=1}^{\infty}\) in \(B_{t,\varepsilon}\) converging to x 0. Without loss of generality we assume that the sequence is decreasing. Since \(u(x,t)\) is right continuous, we can find a point z i in \((x_{i},x_{i-1})\) such that
Now choose \(s_{i}> t\) and \(y_{i}\in(z_{i+1},z_{i})\) such that
We define a curve \(\gamma(x)\) for \(x\in\left[x_{0},x_{1}\right]\) passing through all the points \(\left(z_{i},t\right)\) and \(\left(y_{i},s_{i}\right)\) by
Then γ is Lipschitz continuous with Lipschitz constant θ, and we have that
for all \(i\in\mathbb{N}\). This means that the total variation of \(u(x,\gamma(x))\) is infinite, violating C, concluding the proof of the lemma. □
In the following, we let \(\sigma^{\star}\) be a number strictly larger than the absolute value of every characteristic speed, and we also demand that \(\sigma^{\star}\geq 1/\theta\), where θ is the constant in C. The next lemma says that if u satisfies C, then discontinuities cannot propagate faster than \(\sigma^{\star}\). Precisely, we have the following result.
Lemma 7.14
Assume that \(u\colon[0,T]\to\mathcal{D}\) satisfies C. Then for \((x,t)\in(0,T)\times\mathbb{R}\),
Proof
We assume that the lemma does not hold. Then, for some \(\left(x_{0},t\right)\) there exist decreasing sequences \(s_{j}\to t\) and \(y_{j}\to x_{0}\) such that
for some \(\varepsilon> 0\) and \(j\in\mathbb{N}\). Now let
where as before θ is defined by C. Now we define a subsequence of \(\left\{(y_{j},s_{j})\right\}\) as follows. Set \(j_{1}=1\) and for \(i\geq 1\) define
Then
for all i. Let γ be the curve defined in (7.103). Since we have that \(z_{i}\to x_{0}\), we have that
for sufficiently large i. Consequently,
and the total variation of \(u(x,\gamma(x))\) is infinite, contradicting C. □
The next lemma concerns properties of the semigroup \({\mathcal{S}}_{t}\). We assume that u is a continuous function \(u\colon[0,T]\to\mathcal{D}\), and wish to estimate \({\mathcal{S}}_{T}u(0)-u(T)\). Let h be a small number such that \(Nh=T\). Then we can calculate
Letting h tend to zero, we obtain the following lemma:
Lemma 7.15
Assume that \(u\colon[0,T]\to\mathcal{D}\) is Lipschitz continuous in the L 1-norm. Then for every interval \([a,b]\), we have
Proof
For ease of notation we set
Observe that by finite speed of propagation, we can define \(u(x,0)\) to be zero outside of \([a,b]\), and the Lipschitz continuity of the semigroup will look identical written in the norm \(\left\|\,\cdot\,\right\|\) to how it looked before. Let
Note that ϕ is measurable, and for all h > 0, the function
is continuous. Hence we have that
and therefore ϕ is Borel measurable. Define functions
The function ψ is a Lipschitz function, and hence
Furthermore, Rademacher’s theoremFootnote 1 implies that there exists a null set \(\mathcal{N}_{1}\subseteq[0,T]\) such that Ψ and ψ are differentiable outside \(\mathcal{N}_{1}\). Furthermore, using that Lebesgue measurable functions are approximately continuous almost everywhere (see 64; , p. 47), we conclude that there exists another null set \(\mathcal{N}_{2}\) such that ϕ is continuous outside \(\mathcal{N}_{2}\). Let \(\mathcal{N}=\mathcal{N}_{1}\cup\mathcal{N}_{2}\). Outside \(\mathcal{N}\) we have
Using properties of the semigroup we infer
which implies
Thus \(\psi^{\prime}\leq 0\) almost everywhere, and we conclude that
which proves the lemma. □
The next two lemmas are technical results valid for functions satisfying (7.97) and (7.98).
Lemma 7.16
Assume that \(u\colon[0,T]\to\mathcal{D}\) is Lipschitz continuous, and that for some \((x,t)\) equations (7.97) and (7.98) hold. Then for all positive α we have
Proof
We assume that the limit in (7.109) is not zero. Then there exist sequences \(\rho_{i}\to 0\) and \(\left|h_{i}\right|<\rho_{i}\) and a \(\delta> 0\) such that
for all i. Without loss of generality we assume that \(h_{i}> 0\), and let
Then the map \(h\mapsto v(\,\cdot\,,h)\) is Lipschitz continuous with respect to the L 1 norm, since
From (7.111) we obtain
We can (safely) assume that \(\delta/\widetilde{M}<1\) (if this is not so, then (7.111) will hold for smaller δ as well). We integrate the last inequality with respect to h, for h in \(\left[-\rho_{i},\rho_{i}\right]\). Since \(\big[h_{i}-\rho_{i}\delta/\widetilde{M},h_{i}\big]\subset\left[-\rho_{i},\rho_{i}\right]\), we obtain
Comparing this with (7.97) and (7.98) yields a contradiction. The limit (7.111) is proved similarly. □
For the next lemma, recall that a (signed) Radon measure is a (signed) regular Borel measureFootnote 2 that is finite on compact sets.
Lemma 7.17
Assume that w is in \(L^{1}\left((a,b)\,;\,\mathbb{R}^{n}\right)\) such that for some Radon measure μ, we have that
Then
Proof
First observe that the assumptions of the lemma also hold if the closed interval on the right-hand side of (7.112) is replaced by an open interval. We have that
Secondly, since w is in L 1, it can be approximated by piecewise constant functions. Let v be a piecewise constant function with discontinuities located at \(a=x_{0}<x_{1}<\cdots<x_{N}=b\), and
Then we have
Since \(\varepsilon\) can be made arbitrarily small, this proves the lemma. □
Next we need two results that state how well the semigroup is approximated firstly by the solution of a Riemann problem with states that are close to the initial state for the semigroup, and secondly by the solution of the linearized equation. To define this precisely, let \(\omega_{0}\) be a function in \(\mathcal{D}\), fix a point x on the real line (which will remain fixed throughout the next lemma and its proof), and let \(\omega(y,t)\) be the solution of the Riemann problem
(If \(\omega_{0}\) is continuous at x, then \(\omega(y,t)=\omega_{0}(x)\) is constant.) Define \(\tilde{A}=df(\omega_{0}(x+))\), and let \(\tilde{u}\) be the solution of the linearized equation
Furthermore, define \(\hat{u}(y,t)\) by
Then we can state the following lemma.
Lemma 7.18
Let \(\omega_{0}\in\mathcal{D}\). Then we have
for all x and all positive h and ρ such that \(x-\rho+h\sigma^{\star}<x+\rho-h\sigma^{\star}\).
Proof
We first prove (7.117). In the proof of this we shall need the following general result:
Let \(\bar{v}\) be the solution of \(\bar{v}_{t}+f(\bar{v})_{y}=0\) with Riemann initial data
for some states \(u_{l,r}\in\Omega\). We have that this Riemann problem is solved by waves separating constant states \(u_{l}=v_{0},v_{1},\dots,v_{n}=u_{r}\). Let u c be a constant in Ω and set \(A^{c}=df(u^{c})\). Assume that u l and u r satisfy
i.e., \(\lambda^{c}_{k}\) is the kth eigenvalue and \(u_{l}-u_{r}\) is the kth eigenvector of A c. Let \(\tilde{v}\) be defined by
(\(\tilde{v}\) solves \(u_{t}+A^{c}u_{y}=0\) with a single jump at y = 0 from u l to u r as initial data). We wish to estimate
Note that since \(\bar{v}\) and \(\tilde{v}\) are equal outside the range of integration, the limits in the integral can be replaced by \(\mp\infty\).
Due to the hyperbolicity of the system, the vectors \(\{r_{j}(u)\}_{j=1}^{n}\) form a basis in \(\mathbb{R}^{n}\), and hence we can find unique numbers \(\bar{\varepsilon}^{l,r}_{j}\) such that
From \(u_{r}-u_{l}=\varepsilon^{c}r_{k}(u^{c})\) for some \(\varepsilon^{c}\) it follows that
Thus we conclude (using an identical argument for the right state) that
Let \(\varepsilon_{i}\) denote the strength of the ith wave in \(\bar{v}\). Then, by construction of the solution of the Riemann problem, for \(i<k\) we have that
while for \(i> k\) we find that
for some constant C. Assume that the k-wave in \(\bar{v}\) moves with speed in the interval \([\underline{\lambda}_{k},\bar{\lambda}^{k}]\); i.e., if the k-wave is a shock, then \(\underline{\lambda}_{k}=\bar{\lambda}^{k}=\mu_{k}\left(v_{k-1},v_{k}\right)\), and if the wave is a rarefaction wave, then \(\underline{\lambda}_{k}=\lambda_{k}(v_{k-1})\) and \(\bar{\lambda}_{k}=\lambda_{k}(v_{k})\). Set \(\underline{s}=\min(\underline{\lambda}_{k},\tilde{\lambda}_{k})\) and \(\bar{s}=\max(\bar{\lambda}_{k},\tilde{\lambda}_{k})\). Then we can write I as
Next we note that the first integral above can be estimated as
and similarly,
Using (7.119), we obtain
for some constant C. It remains to estimate I 2. We first assume that the k-wave in \(\bar{v}\) is a shock wave and that \(\lambda_{k}^{c}> \mu_{k}(v_{k-1},v_{k})\). Then
by the above estimates for \(\left|v_{k}-u_{r}\right|\) and \(\left|v_{k-1}-u_{l}\right|\). If \(\lambda_{k}^{c}\leq\mu_{k}\left(v_{k-1},v_{k}\right)\) or the k-wave is a rarefaction wave, we similarly establish (7.121). Combining this with (7.120), we find that
Having established this preliminary estimate, we turn to the proof of (7.117). Let \(\bar{\omega}_{0}\) be a piecewise constant approximation to \(\omega_{0}\) such that
Furthermore, let v be the solution of the linear hyperbolic problem
where again \(\tilde{A}=df(\omega_{0}(x+))\). Let the eigenvalues and the right and left eigenvectors of \(\tilde{A}\) be denoted by \(\tilde{\lambda}_{k}\), \(\tilde{r}_{k}\), and \(\tilde{l}_{k}\), respectively, for \(k=1,\dots,n\), normalized so that
Then it is not too difficult to verify (see Sect. 1.1.6) that \(v(y,t)\) is given by
We can also construct v by front tracking. Since the eigenvalues are constant and the initial data piecewise constant, front tracking will give the exact solution. Hence v will be piecewise constant with a finite number of jumps occurring at \(x_{i}(t)\), where we have that
for all t where we do not have a collision of fronts, that is, for all but a finite number of t’s. Now we apply the estimate (7.122) to each individual front x i . Then we obtain, introducing \(v_{i}^{\pm}=v(x_{i}(t)\pm,t)\),
Recall that \(\tilde{A}=df(\omega_{0}(x+))\) and that \(\tilde{u}\) was defined by (7.114), that is,
In analogy to formula (7.125) we have that \(\tilde{u}\) satisfies
Regarding the difference between \(\tilde{u}\) and v, we find that
By the Lipschitz continuity of the semigroup we have that
Furthermore, by Lemma 7.15 with \(a=x-\rho\), \(b=x+\rho\), \(T=h\), and t = 0, and using (7.126), we obtain
Consequently, using (7.132), (7.131), and (7.130), we find that
Since ϵ is arbitrary, this proves (7.117).
Now we turn to the proof of (7.116). First we define z to be the function
where \(\left|\lambda\right|\leq\sigma^{\star}\). Recall that \(\bar{v}(y,t)\) denotes the solution of \(\bar{v}_{t}+f(\bar{v})_{y}=0\) with Riemann initial data
Then trivially we have that
Let \(\bar{\omega}_{0}\) be as (7.123) but replacing the TV bound by
Recall that \(\hat{u}(y,t)\) was defined in (7.115) by
Let J h be the set
and let \(\hat{v}\) be the function defined by
Then we have that
Note that the bound (7.131) remains valid. We need a replacement for (7.126). In this case we wish to estimate
For \(\left|x-y\right|> \sigma^{\star}t\), the function \(\bar{v}(y,t)\) is discontinuous across lines located at x i . In addition, it may be discontinuous across the lines \(\left|x-y\right|=\sigma^{\star}t\). Inside the region \(\left|x-y\right|\leq\sigma^{\star}t\), v is an exact entropy solution, coinciding with the semigroup solution. Using this and (7.133), we find that
Now using Lemma 7.15, we find that
As before, since ϵ is arbitrary, (7.131), (7.134), and (7.136) imply (7.116). □
Remark 7.19
Note that if \(\omega_{0}\) is continuous at x, then Lemma 7.18 and (7.117) say that the linearized equation gives a good local approximation of the action of the semigroup. If \(\omega_{0}\) has a discontinuity at x, then
does not become small as ρ tends to zero; hence we must resort to (7.116) in this case. Since the total variation of every function in \(\mathcal{D}\) is small, (7.117) is a much stronger estimate than (7.116).
Now that the preliminary technicalities are out of the way, we can set about proving that an entropy solution coincides with the semigroup.
Let u be an entropy solution. To prove that \(u(\,\cdot\,,t)={\mathcal{S}}_{t}u_{0}\), it suffices to show, applying Lemma 7.15, that
for all \(a<b\), and for all \(t\in[0,T]\setminus\mathcal{N}\).
Assume therefore that \(t\notin\mathcal{N}\). Then by the structure theorem, see 193; , Theorem 5.9.6, there exists a null set \(\mathcal{N}\subset[0,T]\) such that outside that set, u either is continuous or has a jump discontinuity (as a function of x). Therefore, we split the argument into two cases, one in which u has a jump discontinuity, and one in which u is continuous or has a small jump in the sense that it is not in the set \(B_{t,\varepsilon}\).
Consider first a point \((x,t)\) where u has jump discontinuity.Footnote 3 By condition B there exist \(u_{l,r}\in\Omega\) and σ such that the limit (7.98) holds when U is defined by (7.97). Using a change of variables, we find that
by Lemma 7.16. Hence for small positive h, we have that
for some small \(\tilde{\varepsilon}\) to be determined later. By Lemma 7.14 we have \(U(y,s)=\hat{u}(y,s-t)\), where \(\hat{u}\) is defined by (7.115) with \(\omega_{0}(y)=u(y,t)\), and U is defined by (7.97), in some forward neighborhood of \((x,t)\). Then using (7.138) and (7.116), we obtain
for all h sufficiently small, since we compute the total variation on a shrinking interval excluding the jump in u at x.
Now we consider points \((x,t)\) where u either is continuous or has a small jump discontinuity. Hence we can choose an interval \(\langle c,d\rangle\) centered at x such that \(B_{t,\varepsilon}\cap(c,d)=\emptyset\). Recall that \(B_{t,\varepsilon}\), defined in (7.102), is the set of points where \(u(\,\cdot\,,t)\) has a jump larger than \(\varepsilon\). Let the family of trapezoids \(\Gamma_{h}\) be defined by
Now we claim that for h sufficiently small, we have that for all \((y,s)\in\Gamma_{h}\),
To prove this, we argue as follows: By Lemma 7.14, discontinuities in u cannot propagate faster than \(\sigma^{\star}\); hence \(u(\,\cdot\,,t)\) is continuous in the lower corners of \(\Gamma_{h}\), and the estimate surely holds for \((y,s)\) located there. We must prove (7.140) for \((y,s)\) in a region \([c+h^{\prime},d-h^{\prime}]\times[t,t+h]\), where \(h^{\prime}\) is given and we can be free to choose h small. Now also \([c+h^{\prime},d-h^{\prime}]\cap B_{\varepsilon,t}=\emptyset\); hence for each \(y\in[c+h^{\prime},d-h^{\prime}]\) we can find \(\xi_{y}\), h y such that the estimate (7.140) is valid for
Now we can cover the compact interval \([c+h^{\prime},d-h^{\prime}]\) with a finite number of intervals of the form \((y_{i}-\xi_{y_{i}},y_{i}+\xi_{y_{i}})\), and choose
Then we obtain (7.140) for \((y,s)\) in \([c+h^{\prime},d-h^{\prime}]\times[t,t+h]\).
Now we must compare u and \(\tilde{u}\) near \((x,t)\). The eigenvectors of \(\tilde{A}=df(u(x,t))\) are normalized according to (7.124). Observe that trivially
Then
To aid us here we use Lemma 7.17. Let \(x_{1}<x_{2}\) be in the interval \((c+\sigma^{\star}h,d-\sigma^{\star}h)\). Then we shall estimate
If we integrate the conservation law over the region
we find that
Taking the inner product with \(\tilde{l}_{k}\), we obtain
where we have defined
Let \(A^{\star}\) denote the matrix
Then
Since
Therefore,
Returning to (7.141) and using Lemma 7.17, we find that
Now we use (7.117), (7.145), and (7.140) to obtain
By Lemma 7.13, the set \(B_{t,\varepsilon}\) contains only a finite number of points; \(x_{1}<x_{2}<\cdots<x_{N}\), where \(u(\,\cdot\,,t)\) has a discontinuity larger than \(\varepsilon\). We can cover the set \([a,b]\setminus\cup_{i}\left\{x_{i}\right\}\) by a finite number of open intervals \((c_{j},d_{j})\), \(j=1,\dots,M\), such that:
- (a):
-
\(x_{i}\notin\cup_{j}(c_{j},d_{j})=\emptyset\) for \(i=1,\dots,N\).
- (b):
-
\(\mathrm{T.V.}_{(c_{j},d_{j})}\left(u\left(\,\cdot\,,t\right)\right)\leq 2\varepsilon\) for \(j=1,\dots,M\).
- (c):
-
Every \(x\in[a,b]\) is contained in at most two distinct intervals \((c_{i},d_{i})\).
We have established that for sufficiently small h,
by (7.139) choosing \(\tilde{\varepsilon}=\varepsilon/(2N)\). Also,
for all i, j, and \(\varepsilon> 0\). Combining this, we find that
Since \(\varepsilon\) can be arbitrarily small, (7.137) holds, and we have proved the following theorem:
Theorem 7.20
Let \(f_{j}\in C^{2}(\mathbb{R}^{n})\), \(j=1,\dots,n\). Consider the strictly hyperbolic equation \(u_{t}+f(u)_{x}=0\). Assume that each wave family is either genuinely nonlinear or linearly degenerate. For every \(u_{0}\in\mathcal{D}\), defined by (7.94), the initial value problem
has a unique weak entropy solution satisfying conditions A–C, see Sect. 7.2. Furthermore, this solution can be found by the front-tracking construction.
7.3 Notes
The material in Sect. 7.1 is taken almost entirely from the fundamental result of Bressan, Liu, and Yang 33; ; there is really only an \(\mathcal{O}\left(\left|e\right|\right)\) difference.
Stability of front-tracking approximations to systems of conservation laws was first proved by Bressan and Colombo in 28; , in which they used a pseudopolygon technique to ‘‘differentiate’’ the front-tracking approximation with respect to the initial location of the fronts. This approach was later used to prove stability for many special systems; see 47; , 8; , 3; , 4; .
The same results as those in Sect. 7.1 of this chapter have also been obtained by Bressan, Crasta, and Piccoli, using a variant of the pseudopolygon approach 29; . This leads to many technicalities, and 29; is heavy reading indeed!
The material in Sect. 7.2 is taken from the works of Bressan 23; ; 24; ; 25; ; 26; and coworkers, notably Lewicka 32; , Goatin 30; , and LeFloch 31; .
There are few earlier results on uniqueness of solutions to systems of conservation laws; most notable are those by Bressan 20; , where uniqueness and stability are obtained for Temple class systems where every characteristic field is linearly degenerate, and in 22; for more general Temple class systems.
Continuity in L 1 with respect to the initial data was also proved by Hu and LeFloch 100; using a variant of Holmgren’s technique. See also 77; .
Stability for some non-strictly hyperbolic systems of conservation laws (these are really only ‘‘quasisystems’’) has been proved by Winther and Tveito 185; and Klingenberg and Risebro 114; .
We end this chapter with a suitable quotation:
This is really easy:
— Rinaldo Colombo, private communication
7.4 Exercises
-
7.1
Show that the solution of the Cauchy problem obtained by the front-tracking construction of Chapt. 6 is an entropy solution in the sense of conditions A–C in Sect. 7.2.
-
7.2
The proof of Theorem 7.8 was carried out in detail only in the genuinely nonlinear case. Do the necessary estimates in the case of a linearly degenerate wave family.
Notes
- 1.
Rademacher’s theorem states that a Lipschitz function is differentiable almost everywhere; see 64; , p. 81.
- 2.
A Borel measure μ is regular if it is outer regular on all Borel sets (i.e., \(\mu(B)=\inf\{\mu(A)\mid A\supseteq B, \text{$A$ open}\}\) for all Borel sets B) and inner regular on all open sets (i.e., \(\mu(U)=\sup\{\mu(K)\mid K\subset U, \text{$K$ compact}\}\) for all open sets U).
- 3.
The following argument is valid for every jump discontinuity, but will be applied only to jumps in \(B_{t,\varepsilon}\).
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Holden, H., Risebro, N.H. (2015). Well-Posedness of the Cauchy Problem. In: Front Tracking for Hyperbolic Conservation Laws. Applied Mathematical Sciences, vol 152. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47507-2_7
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