Local tracking and stability for degenerate viscous shock waves (Q1867231)

The author of this interesting paper studies a scalar viscous conservation law with nonconstant diffusion, \(u_t+f(u)_x=(b(u)u_x)_x\), where \(u,x,f\in \mathbb{R}\), \(t\in \mathbb{R}_+\). The initial condition is \(u(0,x)=u_0(x)\), \(u_0(\pm \infty)=u_{\pm }\), \(b,f\in C^2(\mathbb{R})\). The stability of degenerate, or sonic, shock solutions \(\bar u(x-st)\) which satisfy the Rankine-Hugoniot condition, \(s(u_+-u_-)=f(u_+)-f(u_-)\) is discussed. It is considered a case of degenerate condition, that is, \(f^{\prime }(u_+)=s<f^{\prime }(u_-)\) (or symmetrically) \(f^{\prime }(u_+)<s=f^{\prime }(u_-)\). Without loss of generality it is taken \(s=0\) and thus \(f^{\prime }(u_+)=0<f^{\prime }(u_-)\). It is assumed that \(b(\bar u(x))\geq b_0>0\). Making use of the pointwise Green's function approach the author extends the linear analysis to an equation with nonintegrable coefficients. Some interesting estimates are obtained for a degenerate standing wave solution to the considered equation with first-order degeneracy (\(f^{\prime\prime }(u_{+})\neq 0\)). Then the orbital stability follows in every \(L^p\) norm.