Convergence rate of the vanishing viscosity limit for the Hunter-Saxton equation in the half space (Q2139621)

This paper deals with the following viscous problem \[ \left\{\!\!\!\!\! \begin{array}{lll} &\partial_t v_\varepsilon+u_\varepsilon\partial_xv_\varepsilon=-\frac{1}{2}(v_\varepsilon)^2+\varepsilon\partial_x^2v_\varepsilon, &x>0,t>0,\\ &\partial_x u_\varepsilon = v_\varepsilon, &x>0,t>0,\\ &u_\varepsilon(0,t) = 0, &t>0,\\ &\partial_xv_\varepsilon(0,t)=0, &t>0,\\ &v_\varepsilon(x,0)=v_0(x), &x>0. \end{array} \right. \] The authors consider the asymptotic behavior of the solutions to the previous problem in the half space as \(\varepsilon\) tends to zero.