Abstract
In this chapter we consider the Cauchy problem for a scalar conservation law. Our goal is to show that subject to certain conditions, there exists a unique solution to the general initial value problem. Our method will be completely constructive, and we shall exhibit a procedure by which this solution can be constructed. This procedure is, of course, front tracking. The basic ingredient in the front-tracking algorithm is the solution of the Riemann problem.
Already in the example on traffic flow, we observed that conservation laws may have several weak solutions, and that some principle is needed to pick out the correct ones. The problem of lack of uniqueness for weak solutions is intrinsic in the theory of conservation laws. There are by now several different approaches to this problem, and they are commonly referred to as ‘‘entropy conditions.’’
Thus the solution of Riemann problems requires some mechanism to choose one of possibly several weak solutions. Therefore, before we turn to front tracking, we will discuss entropy conditions.
It is a capital mistake to theorise before one has data. Insensibly one begins to twist facts to suit theories,instead of theories to suit facts. — Sherlock Holmes, A Scandal in Bohemia (1891)
It is a capital mistake to theorise before one has data. Insensibly one begins to twist facts to suit theories,instead of theories to suit facts.
— Sherlock Holmes, A Scandal in Bohemia (1891)
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- 1.
The analysis up to and including (2.7) could have been carried out for systems on the line as well.
- 2.
So many layers I’ve peeled and peeled! Will the kernel never be revealed?
- 3.
\(\|f\|_{\mathrm{Lip}}\) is not a norm; after all, constants k have \(\|k\|_{\mathrm{Lip}}=0\).
- 4.
Beware! Here \(\frac{\partial\psi}{\partial t}\left(\frac{x+y}{2},\frac{t+s}{2}\right)\) means the partial derivative of ψ with respect to the second variable, and this derivative is evaluated at \(\left(\frac{x+y}{2},\frac{t+s}{2}\right)\).
- 5.
As in the previous equation, \(\frac{\partial\psi}{\partial x}\left(\frac{x+y}{2},\frac{t+s}{2}\right)\) means the partial derivative of ψ with respect to the first variable, and this derivative is evaluated at \(\left(\frac{x+y}{2},\frac{t+s}{2}\right)\).
- 6.
Recall that we had the same property in the linear case; see (1.54).
- 7.
Otherwise, we would have to replace the regular discretization by an irregular one containing the points where the second derivative does not exist.
- 8.
It is said that to tell if a cloth is new, you should examine the seams; yet what you thought you knew, may not be what it seems, for it might be the seams that are all that is new.
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Holden, H., Risebro, N.H. (2015). Scalar Conservation Laws. 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_2
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DOI: https://doi.org/10.1007/978-3-662-47507-2_2
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