## Fractional derivatives and smoothing in nonlinear conservation laws.(English)Zbl 0885.45005

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.

### MSC:

 45K05 Integro-partial differential equations 35L65 Hyperbolic conservation laws