$u$ weak solution $\Rightarrow$ $|u|$ weak solution

Question

Assume that $u(t,x)$ is a weak solution to equation $$ u_t + \sum_{i=1}^n (a_iu)_x = 0, \ \ \ u(0,x)=u_0(x)$$ where $a_i \in W_{loc}^{1,\infty}([0,T)\times\mathbb{R}^N))$ and $u_0\in L^1_{loc} (\mathbb{R}^n)$. I have to show that $|u(t,x)|$ is also a weak solution, namely $v=|u|$ verifies $$ \int_{\Omega} L^* (\phi) v \ dxdt = \int_{\mathbb{R}^n} u_0(x)\phi (0,x)\ dx $$ where $\Omega = [0,T]\times \mathbb{R}^n$ and $L^* (\phi) = -\phi_t -a\nabla\phi $, $a=(a_1,...,a_n),\phi$ test function. I am stucked, any help is welcome! thank you.