I am reading here and there (see for instance Denis Serre systems of conservation laws 1- p.36), the following for which I can't spot the mistake I am making that prevents me from arriving to the same conclusion:
A weak solution to a scalar conservation law is $u \in L^1_{loc}(\mathbb{R}\times[0;s])$ such that:
$$\iint_{\mathbb{R}\times[0;s]}u(x,t)\phi_t(x,t) + f(u(x,t))\phi_x(x,t) dxdt + \int_{\mathbb{R}}u_0(x)\phi(0,x)dx =0,\\ \forall \phi \in D(\mathbb{R}\times[0;s]).$$
It is then infered that if $u$ is a weak solution, then $v(x,t):= u(-x,s-t)$ is a weak solution with initial datum $v_0(x)=u(-x,s)$ beyond the fact that one should give a meaning to $u(-x,s)$ a change of variable in the weak formulation does not give to me what I should get, making it I obtain:
$$\begin{aligned}&\iint_{\mathbb{R}\times[0;s]}v(x,t)\phi_t(x,t) + f(v(x,t))\phi_x(x,t) dxdt + \int_{\mathbb{R}}v_0(x)\phi(0,x)dx \\ &\quad = \iint_{\mathbb{R}\times[0;s]}u(x,t)\phi_t(-x,s-t) + f(u(x,t))\phi_x(-x,s-t) dxdt\\ &\phantom{\quad =} + \int_{\mathbb{R}}u(x,0)\phi(-x,s)dx\\ &\quad = 2\int_{\mathbb{R}}u(x,0)\phi(-x,s)dx\\ &\quad \neq 0\\ \end{aligned}$$