  With $\bar{\partial}$-generalization of the Deift-Zhou steepest descent
method, we investigate the long-time asymptotics of the solution to the Cauchy
problem for the Hunter-Saxton (HS) equation \begin{eqnarray} &&u_{txx}-2\omega
u_x+2u_xu_{xx}+uu_{xxx}=0,\quad x\in \mathbb{R},\ t>0,\nonumber\\
&&u(x,0)=u_0(x), \nonumber \end{eqnarray} where $u_0\in H^{3,4}(\mathbb{R})$
and $\omega>0$ is a constant. Using the new scale $(y,t)$ and a series of
deformations to a Riemann-Hilbert problem associated with the Cauchy problem,
we obtain the long-time asymptotic approximations of the solution $u(x,t)$ in
two space-time regions: The solution of the HS equation decays as the speed of
$\mathcal{O}(t^{-1/2})$ in the region $y/t >0$; While in the region $y/t<0$,
the solution of the HS equation is depicted by a parabolic cylinder model with
an residual error order $\mathcal{O}(t^{-1+\frac{1}{2p}})$ with $ p>2$.
