Firstly, I would like to introduce two problems.
A Riemann problem for a system of conservation laws given in divergent form:
$$(1) \hspace{1cm} \begin{cases}
u_t+f(u)_x=0 \\[2ex]
u(x,0)= \begin{cases}
u_l, x<0 \\[2ex]
u_r, x>0,
\end{cases}
\end{cases}
$$
and a Riemann problem for a systems of conservation laws given in nondivergent form:
$$(2) \hspace{1cm} \begin{cases}
u_t+A(u)u_x=0 \\[2ex]
u(x,0)= \begin{cases}
u_l, x<0 \\[2ex]
u_r, x>0.
\end{cases}
\end{cases}
$$
In the problem $(2)$, $A(u)\neq Dg$ (so we can't transform problem $(2)$ into the problem $(1)$). In both problems $u_l , u_r$ are constants in $\mathbb{R}^n$, $n\geq 1$ and $u \in \mathbb{R}^n$, $x \in \mathbb{R}$, $t \in [0,T]$.
We said that a problem $(1)$ has a weak solution if the following identity is valid:
$$\int_{0}^T \int_{\mathbb{R}} [u \psi_{t} + f(u) \psi_{x}] \; dx dt + \int_{\mathbb{R}} u_{0}(x) \psi (x,0) \; dx = 0$$
for every test function $\psi \in C_0^\infty(\mathbb{R} \times [0,T]) $. More informations could be found, for example, in
[Dafermos]. In most books this is more precisely called weak solution in the distributional sense.
My questions are:
- How does the weak solutions of a problem $(2)$ could look?
- Do we use the test functions or not, or we define it completely different than for a problem $(1)$?
The only place where I have found some kind of weak solution of the system $(2)$ are some papers of Philippe Le Floch, so I assume that some type of weak solution for a problem $(2)$ exists. Also it would be nice if this weak solution would be given in some weak measure sense (using maybe Radon or Borel measures).
By "Cauret, J.J.,Colombeau, J.F., Le Roux, A.J., Discontinuous generalized solutions of nonlinear nonconservative hyperbolic equations, 1989," we can't use distribution theory and test functions in the solutions of a problem $(2)$.
In the problem $(1)$, in the weak solutions, we move all the derivatives to the test functions. We could do that because the system is given in the divergent form. So test functions here are very useful. On the other hand, in the problem $(2)$ we couldn't do that because system is not given in the divergent form.
Any help with this would be great whether it is some reference in the literature or good old fashioned way (by writting the answer).
Best Answer
Discontinuous solutions $u$ require that we introduce the concept of weak solutions, in the sense of distributions. However, in the nondivergent (a.k.a. nonconservative or quasilinear) form of the PDE, the standard notion of weak solution does not apply. In facts, if $u$ is discontinuous, then the nonconservative product $A(u) u_x$ is the product between a discontinuity and a Dirac delta, which isn't a well-defined distribution. For functions of bounded variation, Dal Maso-Le Floch-Murat [1] proposed a notion of weak solution to quasilinear equations:
The notation $[\cdot]_\phi$ denotes a specific piecewise-defined Borel measure, and $\phi$ is a fixed Lipschitz continuous family of paths (see dedicated literature for details).
[1] G Dal Maso, P Le Floch, F Murat: Definition and weak stability of nonconservative products. J. Math. Pures Appl. 74, 483-548 (1995)