Consider the initital value problem given by \begin{eqnarray} u_t+f(u)_x=0\\ u(x,0)=u_0(x) \end{eqnarray} where $f$ is Lipschitz continuous and $u_0 \in L^{\infty}(\mathbb{R}).$
$u\in L^{\infty}(\mathbb{R} \times \mathbb{R}^+)$ is called weak solution to the above IVP if it satisfies the following, \begin{eqnarray} \int_{\mathbb{R}^\times\mathbb{R}^+} \left(u\phi_t + f(u)\phi_x\right) dxdt + \int_{\mathbb{R}} u_0(x)\phi(0,x) dx = 0, \quad \forall \phi \in C_c^1(\mathbb{R} \times [0,\infty)). \end{eqnarray} $u\in L^{\infty}(\mathbb{R} \times \mathbb{R}^+)$ is called entropic weak solution to the above IVP if it satisfies the following, \begin{eqnarray} \int_{\mathbb{R}^\times\mathbb{R}^+} \left(|u-k|\phi_t + \hbox{sgn}(u-k)(f(u)-f(k))\phi_x \right) dxdt + \int_{\mathbb{R}} u_0(x)\phi(0,x) dx \geq 0, \quad \forall k\in \mathbb{R} \text{ and } \phi \in C_c^1(\mathbb{R} \times [0,\infty)). \end{eqnarray} It is pretty natural to expect the following,
- If $u$ is a smooth solution to the IVP, then $u$ is also entropic weak solution
- For $f(u)=au,$ $a \in \mathbb{R},$ if $u$ is a weak solution then it is entropic weak solution, cf. this post.
I am unable to prove both of them.
P.S: I could prove if $u$ is smooth solution then it is weak solution. I feel some modification of this proof should give (1), which I could not.
Any comments or references would be greatly appreciated.