Non-uniqueness of weak solutions for the fractal Burgers equation

Abstract

The notion of Kruzhkov entropy solution was extended by the first author in 2007 to conservation laws with a fractional laplacian diffusion term; this notion led to well-posedness for the Cauchy problem in the $L^{\infty }$-framework. In the present paper, we further motivate the introduction of entropy solutions, showing that in the case of fractional diffusion of order strictly less than one, uniqueness of a weak solution may fail.

23 pages

References in corpus (4)

• Well-posedness of the Cauchy problem for the fractional power dissipative equations
• Global well-posedness of the critical Burgers equation in critical Besov spaces
• On convergence of solutions of fractal Burgers equation toward rarefaction waves
• The discontinuous Galerkin method for fractal conservation laws

Cited by in corpus (6)

• Entropy Solution Theory for Fractional Degenerate Convection-Diffusion Equations
• Continuous dependence estimates for nonlinear fractional convection-diffusion equations
• Optimal continuous dependence estimates for fractional degenerate parabolic equations
• Asymptotic behaviour of solutions to fractional diffusion-convection equations
• Large-time behaviour for anisotropic stable nonlocal diffusion problems with convection
• Two classes of nonlocal Evolution Equations related by a shared Traveling Wave Problem