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F. Achleitner, S. Hittmeir, Ch. Schmeiser:
"On Nonlinear Conservation Laws Regularized By A Riesz-Feller Operator";
in: "Hyperbolic Problems: Theory, Numerics, Applications", herausgegeben von: American Institute of Mathematical Sciences; American Institute of Mathematical Sciences, Springfield, MO 65801-2604, USA, 2014, ISBN: 1-60133-017-0, S. 241 - 248.

Scalar one-dimensional conservation laws with nonlocal di ffusion
term are considered. The wellposedness result of the initial-value problem with essentially bounded initial data for scalar one-dimensional conservation laws with fractional Laplacian is extended to a family of Riesz-Feller operators. The main interest of this work is the investigation of smooth traveling wave solutions. In case of a genuinely nonlinear smooth flux function we prove the existence of such traveling waves, which are monotone and satisfy the standard entropy condition. Moreover, the dynamic nonlinear stability of the traveling waves under small perturbations is proven, similarly to the case of the standard di ffusive regularization, by constructing a Lyapunov functional. Apart from summarizing our results in the article Achleitner et al. (2011), we provide the wellposedness of the initial-value problem for a larger class of Riesz-Feller operators.

Nonlocal evolution equation, fractional derivative, traveling wave.

http://publik.tuwien.ac.at/files/PubDat_228935.pdf