\(L^p\)-decay rates for perturbations of degenerate scalar viscous shock waves
The author of this interesting paper considers a scalar conservation law with second-order viscosity in one dimension of form \[ u_t+f(u)_x=\big(b(u)u_x\big)_x, \] where \(f,b\in C^2\), \(b>0\), and \((x,t)\in \mathbb R\times [0,+\infty )\). It is supposed that the triple \((u_+,u_-,s)\in\mathbb R^3\), with \(u_-\neq u_+\). Without loss of generality, it is taken \(u_+<u_-\). Suppose also that the above stated triple \((u_+,u_-,s)\) constitutes a generalized shock front that is a weak solution of the form \[ u(x,t)=u_-, \quad x-st<0, \] \[ u(x,t)=u_+, \quad x-st>0 \] to the underlying inviscid conservation law \(u_t+f(u)_x=0\), which satisfies both Rankine-Hugoniot jump condition \(-s(u_+-u_-)+f(u_+)-f(u_-)=0\), and the generalized entropy condition \(-s(u-u_-)+ f(u)-f(u_-)<0\), for all \(u\in (u_+,u_-)\). A special interest here is the existence of traveling waves of form \(u(x,t)=\overline{u}(x-st)\) (shock profiles) for the initial problem stated above. Here \(\overline{u}\) satisfies the relations \[ (b(\overline{u})\overline{u}')'=f(\overline{u})'-s\overline{u}', \qquad \overline{u}(\xi )\to u_{\pm }, \] as \(\xi\to \pm \infty \). Here, \((\cdot )'=d/d\xi \) denotes differentiation w.r.t. the moving (Galilean) variable \(\xi =x-st\), and \(s\) is the shock speed. Assume that the flux function \(f(u)\) changes its convexity in \(u\in (u_{+},u_{-})\). The author considers sonic (or degenerate) shocks, namely, waves whose speed matches one of the characteristic speeds with \(s=f'(u_{+})\) or \(s=f'(u_{-})\). The main result is that sharper decay rates for zero-mass perturbations of viscous degenerate shocks can be obtained using energy methods and interpolation inequalities. The \(L^p\)-decay rates, with \(2\leq p\leq +\infty \), for zero mass perturbations of degenerate scalar viscous shock waves using energy methods are obtained. The Matsumura-Nishihara weight function is used. Basic estimates with interpolation inequalities of Nash-type are shown to get sharp decay rates for the integral of the perturbation. In contrast with some other works on this problem the method offered by the author is elementary and applies to shocks of all orders of degeneracy.
scientific article; zbMATH DE number 5925601
scientific article