We consider a fractal scalar conservation law, that is to say, a conservation law modified by a fractional power of the Laplace operator, and we propose a numerical method to approximate its solutions. We make a theoretical study of the method, proving in the case of an initial data belonging to L ∞ ∩ BV that the approximate solutions converge in L ∞ weak-* and in L P strong for p < ∞, and we give numerical results showing the efficiency of the scheme and illustrating qualitative properties of the solution to the fractal conservation law.

/pdf/a-numerical-method-for-fractal-conservation-laws-1q3noq8b17.pdf

A numerical method for fractal conservation laws

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Nonlinear Diffusion Equations

Simultaneous denoising and enhancement of signals by a fractal conservation law

We study the existence of travelling-waves and local well-posedness in a
subspace of $C_b^1(\R)$ for a nonlinear evolution equation recently
proposed by Andrew C. Fowler to describe the dynamics of dunes.
The travelling-waves we obtained however, were more bore-like than
solitary-wave-like.

http://www.dspace.uce.edu.ec/bitstream/25000/14673/1/EXISTENCE%20OF%20TRAVELLING-WAVE%20SOLUTIONS%20AND%20LOCAL.pdf

Existence of travelling-wave solutions and local well-posedness of the fowler equation

http://www.hyp2012.eu/jdownloads/Slides/achleitner-slides.pdf

On nonlinear conservation laws with a nonlocal diffusion term

We investigate a non-local, non-linear conservation law, first introduced by A.C. Fowler to describe morphodynamics of dunes, see [6, 7]. A remarkable feature is the violation of the maximum principle, which allows for erosion phenomenon. We prove well posedness for initial data in $L^2$ and give an explicit counterexample for the maximum principle. We also provide numerical simulations corroborating our theoretical results.

/pdf/a-non-monotone-nonlocal-conservation-law-for-dune-4l5xtk3z6o.pdf

A non-monotone nonlocal conservation law for dune morphodynamics

We introduce a kinetic formulation for scalar conservation laws with nonlocal and nonlinear diffusion terms. We deal with merely L1 initial data, general self-adjoint pure jump Levy operators, and ...

https://hal.archives-ouvertes.fr/hal-02320423/document

Nonlocal dissipation measure and L1 kinetic theory for fractional conservation laws

This paper has three contribution: (i) nonstandard and optimal contraction results for $L^1$ viscosity solutions of the Hamilton-Jacobi-Bellman equation
\begin{equation*}
\partial_t \varphi=\sup_\xi \{b(\xi) \cdot D \varphi+{\rm tr}(a(\xi) D^2\varphi)\},
\end{equation*}
(ii) nonstandard and optimal contraction results for $L^\infty$ entropy solutions of the anisotropic degenerate parabolic equation
\begin{equation*}
\partial_t u+{\rm div} F(u)={\rm div} (A(u) D u),
\end{equation*}
and (iii) rigorous identification of a new duality relation between these notions of generalized solutions. More precisely, we obtain a quasicontraction principle for the first equation in the weakest possible $L^1$ type Banach setting where stability holds in general. We rigorously identify this topology and show that our contraction estimate is optimal for model HJB equations. For the second equation, we obtain a weighted $L^1$ contraction principle for possibly nonintegrable $L^\infty$ solutions. Here we identify the optimal weight in general. It is interestingly a solution of the first equation, and it is the precise formulation of this result that leads to the above mentioned duality.

Nathaël Alibaud

Papers

A non-monotone nonlocal conservation law for dune morphodynamics

Nonlocal dissipation measure and L1 kinetic theory for fractional conservation laws

Optimal and dual stability results for $L^1$ viscosity and $L^\infty$ entropy solutions