Nonlocal dissipation measure and L¹ kinetic theory for fractional conservation laws

Abstract

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 Lévy operators, and locally Lipschitz nonlinearities of porous medium kind possibly strongly degenerate. The cornerstone of the formulation and the uniqueness proof is an adequate explicit representation of the dissipation measure associated to the diffusion. This measure is a $L_{\text{loc}}^1$ function in our nonlocal framework. Our approach is inspired from the second order theory unlike the cutting technique previously introduced for bounded entropy solutions. The latter technique no longer seems to fit the kinetic setting. This is moreover the first time that the more standard and sharper tools of the second order theory are faithfully adapted to fractional conservation laws.

KEYWORDS AND PHRASES:

• Kinetic formulation
• dissipation measure
• fractional conservation law
• nonlinearity of porous medium kind
• nonlocal and nonlinear diffusion
• pure jump Lévy operator
• well-posedness
• L¹ data

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 35R11
• 35L65
• secondary 35K59
• 35D99

Notes

1 Take e.g. a regular version of $\xi \mapsto \sum _{k\ge 1}{(|\xi |-r_k)}^{+}$ where $\sum _{k\ge 1}{\int }_{\left|u_0\right|\ge r_k}|u_0|<\infty$ for some fixed $0=r_1<r_2<\dots$

Additional information

Funding

The publication has been prepared with the support of the “RUDN University Program 5-100.” This research was supported by the “French ANR project CoToCoLa, no. ANR-11-JS01-006-01.” The authors are grateful to the anonymous referee for pointing out a certain inaccuracy to us, the correction of which has improved the thoroughness and clarity of the paper. The second author thanks J.-Ph. Anker for a helpful discussion.