I am trying to do a numerical approximation of the viscosity solution for the Hamilton-Jacobi problem
$\hspace{5cm}\displaystyle \frac{du}{dt}+\frac{|u_x|^2}{2}=0 \hspace{0.5cm} x \in \mathbb{R},\hspace{0.3cm}t \in [0,+\infty[$
$\hspace{5cm}\displaystyle u(x,0)=\sin(x)\hspace{0.5cm}x \in \mathbb{R}$
So I need to calculate the explicit solution to compare them. I read about Lax-Hopf formula applied to the general problem and the solution should be
$$u(x,t)=\inf_{y \in \mathbb{R}} \left \{ \sin(y)+\frac{|x-y|^2}{2t}\right \}$$
Can someone give me some help/ hint to find a explicit form for this infimus? Greetings