QUESTION:If initial data $u_0 \in L^{\infty}$ In which sense weak solution of scalar conservation laws $u_t+f(u)_x=0$ $u(x,0)=u_0(x)$ takes initial data?
Explanation:
HYPERBOLIC SYSTEMS OF CONSERVATION LAWS, chapter 2 by Godlewski and Raviart theorem 3.1 states
"if $u_0 \in L^{\infty}(\mathbb{R}^d) \cap L^{1}(\mathbb{R}^d) \cap BV(\mathbb{R}^d)$ then the initial value problem has an entropy solution $u \in L^{\infty}(\mathbb{R}^d \times (0,+\infty)) \cap B(0,T;L^{\mathbb{R}^d})$, which satisfies
$\int_{\mathbb{R}^d} |u(x,t_1)-u(x,t_2)| \,dx \leq CTV(u_0)|t_1-t_2|$ for all $t_1,t_2 \geq 0$"
i.e $u(\cdot,t) \rightarrow u_0$ in $L^1$
Now if we assume $u_0$ to be in just $L^\infty(\mathbb{R^d})$ can we say
$u(\cdot,t) \rightarrow u_0$ in $L^1_{loc} (\mathbb{R^d})$?
if not in which sense $u(\cdot,t) \rightarrow u_0$
Best Answer
As far as I understand for $L^{\infty}$ solutions, the initial data is taken in $L^1_{loc}$ sense. Please correct me if I am wrong.