Abstract
Nonlocal nonlinear evolution equations of the form, \( u_t+\mathcal{L}u+\nabla \cdot f(u)=0, \) where \(-\mathcal{L}\) is the generator of a Lévy semigroup on \(L^1({\mathbf {R}}^n)\), are encountered in continuum mechanics as model equations with anomalous diffusion. In the case when \(\mathcal L\) is the Laplacian and the nonlinear term, f(u) is quadratic, the equation boils down to the classical Burgers equation that has been studied in Volume 2, and Chapter 19 of the present volume where it has been analyzed in the context of passive tracer transport in Burgers turbulence.
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Notes
- 1.
See, also, Fritz and Hairer [9], based chiefly on the 2014 Fields-Medal-winning work of Martin Hairer.
- 2.
Work in progress. A description of a great variety of physical phenomena that require Lévy flight tools for their analysis can be found in the Proceedings volume of the International Workshop “Lévy Flights and Related Topics in Physics” (see Shlesinger, Zaslavsky and Frisch [34]) held in Nice, France, in 1994. More recently, another conference on the same topic was held at the Wroclaw University of Technology, Wroclaw, Poland, in 2016.
- 3.
Based on Biler, Karch, and Woyczyński [4].
- 4.
Based on Biler, Karch, and Woyczyński [5], where complete details of the proofs can be found.
- 5.
Recall that the Sobolev space \(W^{2,2}({{\mathbf {R}}})\) is the space of twice differentiable functions with square integrable derivatives of order one and two.
- 6.
- 7.
Based on results in Górska and Woyczyński [7].
- 8.
See Górska and Penson [41].
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Saichev, A.I., Woyczyński, W.A. (2018). Nonlinear and Multiscale Anomalous Fractional Dynamics in Continuous Media. In: Distributions in the Physical and Engineering Sciences, Volume 3. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-92586-8_23
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DOI: https://doi.org/10.1007/978-3-319-92586-8_23
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