Suppose we have for $T\in\mathbb{R}_{>0}$ a conservation law of the following general form \begin{align} \dot{u}(t,x)+(a(t,x,u(t,x)))_{x}&=0 && (t,x)\in(0,T)\times\mathbb{R}\\ u(0,x)&=u_{0}(x) &&x\in\mathbb{R} \end{align} with $u_{0}$ and $a$ given and $a$ sufficiently smooth. We call it initial value problem (IVP).
If one is interested in defining non-classical solutions, there are several different definitions of "weak" solutions:
$u\in C([0,T];L^{1}_{\text{loc}}(\mathbb{R}))$ is called weak solution of the IVP, iff \begin{equation} \int_{0}^{T}\int_{\mathbb{R}}u(t,x)\phi_{t}(t,x)+a(t,x,u(t,x))\phi_{x}(t,x)\ d x\ dt+\int_{\mathbb{R}}u_{0}(x)\phi(0,x)\ dx=0\quad \forall \phi\in C^{1}_{\text{c}}((-1,T)\times\mathbb{R}). \end{equation} In my understanding this is quite a nice function space, since we can indeed evaluate $u$ at every time $t$ (reasonable for a kind of time evolution and the semi group property) and in particular for $t=0$ as a $L^{1}_{\text{loc}}$ function.
$u$ is called distributional solution if $u$ is a measureable function on $(0,T)\times\mathbb{R}$ and satisfies the integral equality in item 1.
$u$ is called integral solution, if $u\in L^{\infty}((0,T)\times\mathbb{R})$ is satisfying the integral equality in item 1.
Now we are interested in existence, uniqueness, and the corresponding entropy conditions. As I have seen, in the literature there is not a very precise statement of which kind of weak solution (sometimes every of these 3 types of solutions are called weak solutions) is necessary to apply entropy conditions and to get kind of uniqueness and existence. Even more, there seems to be some mixing between them.
So, if anybody has a good and mathematical rigorous reference concerning the different types of weak solutions, it would be great.
In addition, when will these three types of "weak" solutions be identical? Could it make sense to consider solutions which do not change continuously in time if one measures the space in the $L^{p}$ topology?
I am thankful for every comment or help and thank everyone for reading in advance!
Alex
Any weak solution is a distributional solution.. However when we are dealing with initial value problem, it makes more sense to talk about weak solution as one would expect in some sense solution takes the initial data ($u(\cdot,t) \rightarrow u_0(\cdot)$ as $t \rightarrow 0$ in $L^1_{loc}$). Since it is an evolution equation it is natural to seek for a solution which is continuous in time in some norm...