Fractional derivatives and smoothing in nonlinear conservation laws
The authors investigate solutions of the Riemann problem \[ \frac{\partial}{\partial t}\int_0^t k(t-s)(u(s,x)- u_0(x))dx+ (\sigma(u))_x (t,x)=0, \qquad t>0, \quad x\in\mathbb{R}, \tag \(*\) \] where \(k\) is locally integrable, nonnegative and nonincreasing on the positive reals, but unbounded near zero. The climatic theorem states that if \(\sigma\) is continuous and satisfies \(\sigma(v)- \sigma(w)\geq c(v-w)\) for some constant \(c>0\) and all \(v,w\) with \(v\geq w\), then the solution of \((*)\) with \(u_0(x)=1\) if \(x<0\) and \(=0\) if \(x\geq 0\) is continuous on \((0,\infty)\times \mathbb{R}\setminus \{(0,0)\}\). Several other properties of these solutions are established such as the absolute continuity of the mapping \(x\mapsto \sigma(u(t,x))\). A limiting case occurs when \(k\) is replaced by the Dirac operator. This yields the familiar law \(u_t+ \sigma(u)_x=0\) which, as the authors point out, has discontinuous solutions (even for smooth initial conditions). A corollary to the main result is that this conservation law has continuous solutions if \(u_t\) is replaced by \(\frac{\partial^\alpha u}{\partial t^\alpha}\) where \(0<\alpha<1\). The proofs are clearly written, although lengthy and detailed; especially the proof of the overall continuity of the solution.
scientific article; zbMATH DE number 811755