F. Achleitner, S. Hittmeir, Ch. Schmeiser:
"On nonlinear conservation laws with a nonlocal diffusion term";
Journal of Differential Equations, 250 (2011), 4; S. 2177 - 2196.

Kurzfassung englisch:
Scalar one-dimensional conservation laws with a nonlocal diffusion
term corresponding to a Riesz-Feller differential operator are
considered. Solvability results for the Cauchy problem in L∞ are
adapted from the case of a fractional derivative with homogeneous
symbol. The main interest of this work is the investigation of
smooth shock profiles. In the case of a genuinely nonlinear smooth
flux function we prove the existence of such travelling waves,
which are monotone and satisfy the standard entropy condition.
Moreover, the dynamic nonlinear stability of the travelling waves
under small perturbations is proven, similarly to the case of
the standard diffusive regularisation, by constructing a Lyapunov

Nonlocal evolution equation, Fractional derivative, Travelling wave

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